Physics I Final Exam
Example: A carousel is initially at rest. At t = 0 it is given a constant angular acceleration .06 rad/s^2 which increases its angular velocity for 8.0 s. At t = 8, determine (a) the angular velocity of the carousel, and (b) the linear velocity of a child located 2.5 m from the center
(a) w = 0 + .06(8) = .48rad/s (b) v = 2.5(.48) = 1.2 m/s
1 revolution
2π radians
elastic collisions
A collision in which no kinetic energy is lost - conservation of momentum AND kinetic energy - initial KE = final KE - .5mava^2 + .5mbvb^2 = .5mava'^2 = .5mbvb'^2
friction
A force that opposes motion between two surfaces that are in contact
weight
A measure of the force of gravity on an object Fg = mg
vector
A quantity that has magnitude and direction
acceleration vs. velocity
Acceleration defines how quickly velocity changes vs. velocity defines us how quickly position changes
Newton's Second Law
Force = mass x acceleration - only valid in inertial reference frames - only applies to ONE object, not equal and opposite objects
distance
How far an object travels
An ice skater performs a pirouette by pulling her outstretched arms close to her body. What happens to her rotational kinetic energy about the axis of rotation?
It will increase - the moment of inertia will decrease because the radius is getting smaller - to conserve momentum, the velocity gets bigger - because angular velocity is squared, increasing velocity will have a greater impact than decreasing inertia
kinetic energy equation
KE = 1/2mv^2
mechanical energy
Kinetic or potential energy associated with the motion or position of an object E = U + K
causal laws
Laws describing causal relationships. Such laws specify the conditions that are necessary and sufficient to produce a certain event. Knowledge of causal laws allows both the prediction and control of events. - the universe works in a causal manner
finding the tension between two boxes
Look at the mass of the box with only tension acting as a force and use that mass to find the tension between boxes - the tension between boxes is the same for box A and box B
mechanical energy equations
ME = KE+PE 0 = ΔKE + ΔPE 0 = (KE2 - KE1) + (PE2 - PE1) ΔKE = ΔPE KE1 + PE1 = KE2 + PE2
gravitational mass
Mass as used in the law of universal gravitation; the quantity that measures an object's response to gravitational force - equal to inertial mass
Example: What will be the speed of a solid sphere of mass M and radius R when it reaches the bottom of an incline if it starts from rest at a vertical height H and rolls without slipping?
MgH = .5Mv^2 + .5Iw^2 M's cancel gH = .5v^2 + .5(2/5R^2)w^2 gH = .5v^2 + .5(2/5R^2)(v^2/R^2) R's cancel gH = .5v^2 + .5(2/5v) v = root(10/7gH)
A force F is applied to a dumbbell for a time interval Δt, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater center-of-mass speed?
Neither case - momentum is equal to the change in the speed of center of mass - it is also equal to force x time - because the force and the time are the same, there is no change in center of mass speed
unit of force
Newton
velocity has both x and y components
Vox is CONSTANT Voy changes throughout the projectile, but is equal to 0 at the highest point
To find the maximum height of an object if it is being launched from the ground in a projectile motion...
Voy = Vsin(theta) t = Voy / a (this is the time it takes to reach the MAX height) y = Voy(t) - .5at^2
How to find Voy
Vsin(theta)
horizontal motion equations
Vx = Vox x = xo + Vox(t)
work equation when traveling a distance (d) to a height (h)
W = Fh + height of the ramp determines the work, not the path!
will the center of mass change when a person stands up?
Yes because the distribution of weight is different in each case
strain
change in length / original length - the percentage that something has deformed - unitless
negative torque
clockwise rotation - look at the angle from the lever arm/axis to the force
positive torque
counterclockwise rotation - look at the angle from the lever arm/axis to the force
angular displacment
defined by angles, it is the initial angular position minus the final angular position
rolling without slipping
depends on the static friction between the rolling object and the ground - the friction is static because the rolling object's point of contact with the ground is at rest each moment
relative velocity
describes the velocity of an object with respect to a frame of reference - we need vector addition when velocities are not all relevant to the same point of reference
moment of inertia equations
does not only depend on the mass, but also how it is distributed - when the mass is concentrated farther away from the access, the rotational rotation is greater
If you are using two pulleys, what happens to the tension force?
doubles 2T - mg = ma
Sig fig rule 3
for numbers with decimal points, zeros to the right of non-zero integers AND to the right of the decimal point are significant + ex: 9.00 = 3 + ex: 760.000 = 6
stress
force/area - pressure applied to an object - SI is N/m^2
conservative forces
forces that are path independent and do not dissipate energy - gravity - elastic potential energy - electric
equation of gravity at earth's surface
g = G(mE/rE^2)
what force acts on every object, even those in static equilibrium?
gravity. There must be other forces acting on it to make the net force 0
speed
how far an object travels in a time interval, regardless of direction
angular position
how far the object has rotated
instantaneous acceleration
how fast a velocity is changing at a specific instant
unit of Newton
kg*m/s^2
standard unit of mass
kilogram (kg)
what happens to the kinetic energy if mass is doubled?
kinetic energy is doubled - able to do double the work - the distance of work would also quadruple
what happens to the kinetic energy is speed is doubled?
kinetic energy is quadrupled - able to do 4x as much work - the distance of work would also quadruple
Example: A particular bird's eye can just distinguish objects that subtend an angle no smaller than about .0003 rad. How small an object can the bird just distinguish when flying at a height of 100 m
l = rӨ = 100(.0003) = .03m
Sig Fig Addition/Subtraction Rule
least number of decimal places + ex: 10.0 + .01 = 10.0
interactive: Consider two objects with different masses. Is it possible for the two objects to simultaneously have the same momentum and the same kinetic energy?
only if velocity is zero
tan(theta)
opposite/adjacent - use this once you have found both the x and y components of the vector
sin(theta)
opposite/hypotenuse
is the torque and angular acceleration proportional or disproportional?
proportional - as you increase one, you increase the other
interactive: A rubber ball and a lump of putty have equal mass. They are thrown with equal speed against a wall. The ball bounces back with nearly the same speed with which it hit. The putty sticks to the wall. Which object experiences the greater momentum change?
putty: pi = mv pf = 0 Δp = -mv ball: pi = mv pf = m(-v') Δp = -mv' - mv = more negative the ball experiences a greater change in momentum
what happens to the centripetal acceleration if you double the velocity?
quadruple acceleration
scalar quantities
quantities that describe magnitude but do not include direction + ex: mass, time, temperature
How do you find the degrees from radians?
radian(360°/2π)
measuring a satellites speed
recorded in period - T = 2nr/v
is work a scalar or vector?
scalar → there is no direction - work done by the force in which the force vector is less than 90° from the displacement vector is positive - work done by the force in which the force is greater than 90° from the displacement is negative - if the force vectors are perpendicular, then there is no work being done
Is energy scalar or vector?
scalar
standard unit of time
second (s)
what must be present for an object to roll instead of slip?
static friction
potential energy
storred energy
fracture
stress exceeds the limit and object breaks - forces exceed maximum limits (stress) and the material breaks
shear stress
stress that occurs when forces act in parallel but opposite directions, pushing parts of a solid in opposite directions - shear modulus
mechanics
study of motion
to find the time it takes to reach the highest point if a ball is thrown straight up...
t = - vo / a
To find the total time it takes for an object to reach the ground if t is being launched from the ground in a projectile motion...
t = 2(Voy / a)
what direction is linear velocity in a rotating object?
tangential
energy
the ability to do work - it is always conserved - can be transferred from one system to another - work is when energy is transferred
torque
the angular acceleration of an object moving along an axis - τ = rFsinΘ - SI units is Nm
interactive: A compact car and a large truck collide head on and stick together. Which undergoes the larger acceleration during the collision?
the car because the forces of both are the same - the acceleration must increase to compensate for the smaller mass
why does the speed of a satellite not change even though there is centripetal force?
the centripetal force and velocity are perpendicular
when two objects collide, what can we say about the change in momentum?
the change in momentum of one object is equal and opposite to the change in momentum of the other object. (This is true along any axis with no external forces or with insignificant impulse
impulse
the change in momentum over a really small time interval during a collision - Δp = FΔt - F is the average force over the collision
kinematics
the description of HOW things move
base quantity
the essential physical aspects of a measuring system that can be used alone or in combination to describe what is being measured + ex: mass, time, length
example: two kids start at the top of a kill and slide down. Kid 1 takes a steeper path and kid 2 takes a shallower path. which rider makes it to the bottom first?
the first kid because they transform their potential energy into kinetic energy before kid 2
order of magnitude
the order of magnitude of a quantity is the number rounded to the nearest power of 10 - used to make estimations when precise calculations take too much time/resources or do not have enough data
Velocity
the speed of an object in a particular direction - (xf - xi) / t - vector - defined in terms of displacement - same as speed if the motion is one directional - positive = right - negative = left
average speed
total distance divided by total time
how is the angular position measured?
using the angle from the axis to the point and distance along the circumference from the point to the axis
tensile/compressive stress
when an object has two equal forces either stretching them in opposite directions or compressing it - young's modulus
when does w = v/r?
when an object is rolling, not slipping
when do object feel weightlessness?
when their acceleration is equal to gravity - a = -9.8m/s^2
torque equation
τ = Fr τ = r(mrα) = mr^2α τ = Iα
torque in static equilibrium problems on flat planes
τ = Fx
Tail to Tip Method
Sum of vectors via combining two vectors from tail to tip.
work-energy theorem
The work done on an object by another object in motion equals the change in kinetic energy of the object W = ΔK W = .5m2v2^2 - .5m1v1^2
Steve (S) and his brother Mark (M) are riding on a merry-go-round. Mark is on the edge and Steve is near the center. What is true about their speed and angular velocity?
They have the same angular velocity but not the same speed
what is the final vector equal to?
V^2 = Vx^2 + Vy^2
How to find the Vox
Vcos(theta)
If there are two boxes connected on a string, how does the acceleration of box A compare to box B?
The acceleration of box A = the acceleration of box B
when is mechanical energy conserved?
In the absence of non-conservative forces such as friction, drag, air resistance, etc.
angular velocity
The angular displacement of an object divided by the time needed to make the displacement. - represented by w - w = 2n/T - v = w x T
Power done on rotating object
P = W/d = τӨ/t = τw
Do you exert less force if you push or pull an object?
Pull because you are decreasing the normal force instead of increasing it
Noninertial Reference Frames
Reference frames in which inertia (Newton's first law) does not hold + ex: when a car brakes, you will eventually stop even though there is no force being directly applied to you
satellites
Satellites are technically falling (accelerating) toward Earth, but their speed keeps it moving in a circle - if they are moving too fast, they will release from earths orbit - if they are moving too slow, they will fall back to earth
dynamics
The description of what causes things to move --> why objects move as they do
Tension force
The force that is transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends - the forces pulling on the cord must add up to 0
force
the interaction between two objects either as a push or a pull - an action capable of accelerating an object - vector: has direction and magnitude + ex: gravity, friction (static or kinetic), normal, tension
dynamics
the study of why things move
example: The projectile, of mass m, is fired into a large block (of wood or other material) of mass M, which is suspended like a pendulum. As a result of the collision, the pendulum and projectile together swing up to a maximum height h. Determine the relationship between the initial horizontal speed of the projectile, v, and the maximum height h.
using the conservation of momentum, we find: - mv + M(0) = (m + M)v' using the conservation of mechanical energy after the objects collide, we find: .5(m + M)v'^2 + 0 = 0 + (m + M)gh v' = root(2gh) plugging this into the first equation, we find: mv = (m + M)(root(2gh))
Example: A 150-g ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.600 m The ball makes 2.00 revolutions in a second. What is its centripetal acceleration?
v = 2nr / T T = 1/f T = 1/2 = .5 v = 2n(.6) / .5 = 7.54 a = (7.54)^2/.6 = 94.7 m/s^2
velocity of an object in space
v = root(G x mE/r)
velocity equation in elastic collisions
va - vb = vb' - va' va - vb = -(va' - vb')
Example: A simple clutch consists of two cylindrical plates that can be pressed together to connect two sections of an axle, as needed, in a piece of machinery. The two plates have masses Ma = 6 and Mb = 9 and with equal radii .6m They are initially separated. Plate Ma is accelerated from rest to an angular velocity of 7.2rad/s in time 2s. Calculate (a) the angular momentum of and (b) the torque required to accelerate Ma from rest to w1
(a) .5(6)(.6)^2(7.2) = 7.8 (b) τ = L / t = 7.8/2 = 3.9
Example: A centrifuge rotor is accelerated for 30 s from rest to 20,000 rpm (revolutions per minute). (a) What is its average angular acceleration? (b) Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?
(a) 20,000 rpm → 2100 rad/sec 2100 = 0 + a(30) = 70 (b) 2100^2 = 0^2 + 2(70)(Ө) = 31500 rad → 5000 revs
example: two small "weights," of mass 5.0 kg and 7.0 kg, are mounted 4.0 m apart on a light rod (whose mass can be ignored). Calculate the moment of inertia of the system (a) when rotated about an axis halfway between the weights, and (b) when rotated about an axis 0.50 m to the left of the 5.0-kg mass
(a) I = 5(2)^2 + 7(2)^2 = 48 (b) I = 5(.5)^2 + 7(4.5)^2 = 143
Example: For the child on the rotating carousel, determine that child's (a) tangential (linear) acceleration, (b) centripetal acceleration, (c) total acceleration, and (d) angle of acceleration.
(a) at = αr = .06(2.5) = .15m/s2 (b) ar = rw^2 = 2.5(.48)^2 = .58 (c) a = root(.58^2 + .15^2) (d) tan(at/ar) = tan(.15/.58) = 15°
Example: A bicycle slows down uniformly from v = 8.4m/s to rest over a distance of 115 m. Each wheel and tire has an overall diameter of 68.0 cm. Determine (a) the angular velocity of the wheels at the initial instant (b) the total number of revolutions each wheel rotates before coming to rest; (c) the angular acceleration of the wheel; and (d) the time it took to come to a stop.
(a) v = wr = 8.4 = w(.34) = 24.7 (b) rev = d/2πr = 115/2π(.34) = 53.8 rev (c) 0 = 24.7^2 + 2(α)(53.8) = -.902 (d) 0 = 24.7 + -.902(t) = 27.4 sec
Suppose a football was punted, and left the punter's foot at a height of 1.00 m above the ground. How far did the football travel before hitting the ground?
+ set -1 as y -1m = 0 + voy(t) + .5at^2 + Use the quadratic formula to find that t = 2.53sec + Plug in the t into the range formula x = vox(2.53)
scaling a vector
- The multiplication of a vector by a positive scalar c changes the vector by a magnitude of c but not the direction - The multiplication of a vector by a negative scalar c changes the vector by a magnitude of c but the direction is opposite
Example: why is the painter able to carry a painting on a sled?
- The painter will move forward if his force upon the ground is greater than his force upon the sled - the sled will move if the force upon the assistant is greater than its force upon the ground
relationship between speed and moment of inertia
- as the moment of inertia increases, the velocity decreases - as the moment of inertia decreases, the velocity increases
types of forces in nature
- gravitational - electromagnetic - strong nuclear - weak nuclear
work energy principle
- if net work is positive, then the object's kinetic energy increases by an amount W - if the net work is negative, then the object's kinetic energy decreases by an amount W + ex: a net force exerted on an object opposite to the object's direction, the speed and therefore kinetic energy will decrease + ex: the force a nail exerts on a hammer is negative F, so KE is negative
what is required to do work?
- the object must move - the force must be applied to the object - the direction of the force vector must have a component parallel to the direction of motion
What is true about vector components for vectors in two dimensions?
- they are parallel to the x and y axes - they are right angles to each other - they add together like vectors to give the vector itself
interactive: A golf ball is fired at a bowling ball, initially at rest. After the collision, the golf ball bounces back elastically. Which of the following is true? A) The golf ball had a greater momentum change than the bowling ball and more momentum at the end. B) The golf ball had a greater momentum change than the bowling ball and less momentum at the end. C) The golf ball had less momentum change than the bowling ball and less momentum at the end. D) The golf ball had the same momentum change as the bowling ball and more momentum at the end. E) The golf ball had the same momentum change as the bowling ball and less momentum at the end.
- they experience the same momentum change because momentum is conserved - the golf ball has less momentum at the end because now its momentum is -p
when is no work done?
- when a force is perpendicular to the motion - when there is no movement - normal force never does work
interactive: A car accelerates from 0 to 30 mph in 1.5 s. How long does it take for it to accelerate from 0 to 60 mph assuming the power of the engine to be independent of velocity and neglecting friction?
- when the velocity doubles, the kinetic energy increases by 4x - when kinetic energy increases by 4x, the work increases by 4x - if the power is the same in both cases, the time needs to increase by 4 to compensate - 1.5 x 4 = 6sec
If the relative velocity of object A to object B is 10 m/s, what is the relative velocity of object B to object A?
-10 m/s ALWAYS the opposite sign
what is the acceleration for free falling objects, neglecting air resistance?
-9.8 m/s^2
Rotational kenetic energy
.5Iw^2
equation relating momentum and kinetic energy
.5mv^2 = p^2 / 2m = (mv)^2 / 2m p = root(2mK)
to find how long it takes the ball to be thrown straight up and then back down...
0 = 0 + vo(t) .5at^2
what is the net energy of the universe?
0 because energy is conserved - a closed and isolated system + final energy - initial energy = 0
when a ball is thrown straight up and down, what is the velocity at the very top?
0 m/s
How to solve a relative velocity equation
1) find all the components of the two known vectors 2) determine what relative velocity you are solving for (are you adding or subtracting the known quantities?) 3) add or subtract the x components 4) add or subtract the y components 5) V^2 = Vx^2 + Vy^2 theta = tan-1(Vy / Vx)
steps for solving kinematic equations
1) read and reread the problem 2) decide what object is being studied and the time interval Initial t is usually 0 3) draw a diagram 4) write down the known quantities Starting from rest means v0 = 0 , t0 = 0 , and x0 = 0 5) principles of physics → do they apply? 6) choose which equation which best involves the known quantities 7) carry out the calculation 8) is the answer reasonable? 9) check your units
Example: Several objects roll without slipping down an incline of vertical height H, all starting from rest at the same moment. The objects are a thin hoop (or a plain wedding band), a spherical marble, a solid cylinder (a D-cell battery), and an empty soup can. In addition, a greased box slides down without friction. In what order do they reach the bottom of the incline?
1) the block because the kinetic energy is being converted only into translational motion 2) sphere 3) solid cylinder 4) empty can 5) hoop - the speed does not depend on the mass nor the radius - only the moment of inertia (how mass is distributed) matters
what angle is the force of friction?
180, so it makes the entire work negative
sidereal period
27.32 days for the full moon to return from sun's frame
synodic period
29.53 days for the full moon to return from Earth's frame
Example: A bike wheel rotates 4.50 revolutions. How many radians has it rotated?
4.50(2π) = 9π
What is the angle at which a projectile must be launched to obtain maximum range?
45 degrees
interactive: A woman stands on the edge of a cliff. She throws a stone vertically downward with an initial speed of 10 m/s. The instant before the stone hits the ground below, it has 450 J of kinetic energy. If she were to throw the stone horizontally outward from the cliff with the same initial speed of 10 m/s, how much kinetic energy would it have just before it hits the ground?
450 J because the path does not matter
interactive: A spring loaded toy dart gun shoots a dart straight up in the air and the dart reaches a maximum height of 24 m. The same dart is shot straight up a second time but this time the spring is compressed only half as far. How far up does the dart go this time neglecting friction?
6 because PE = .5kx^2 - decreasing x by 2 decreases the height by 4x
static equilibrium
A condition where there are no net external forces acting upon a particle or rigid body and the body remains at rest or continues at a constant velocity. - no acceleration or angular acceleration Fx = 0 Fy = 0 τ = 0
youngs modulus
A measure of the stiffness of an elastic material and defined by stress/strain - the change in length is in the same direction as the original length
parallelogram method
A method used to find the resultant of two vectors in which you place the vectors at the same initial point, complete a parallelogram, and draw the diagonal.
work
A result of a force moving an object a certain distance - transfer of energy into or out of a system - unit is Joule
closed system
A system in which no matter or energy is allowed to enter or leave
isolated system
A system that exchanges neither matter nor energy with its surroundings. - no external forces acting on it
Bulk Modulus
A term that describes a substance's resistance to compression under pressure - there are forces acting in all directions - occurs when there is a volume change - stress / (change in volume / original volume)
does 0 acceleration mean 0 velocity?
A zero acceleration does not mean that the velocity is zero, such as when you are going at a constant speed on a highway
does 0 velocity mean 0 acceleration?
A zero velocity does not mean that the acceleration is zero, such as when you start your car, you have to start at v = 0
Newton's First Law
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force - an object will only undergo acceleration if acted upon by an outside force - when acceleration = 0
Kepler's Second Law
As a planet moves around its orbit, it sweeps out equal areas in equal times
Interactive: In which system is there a decrease in potential energy? A) A boy stretches a spring. B) A child slides down a sliding board. C) A crate rests at the bottom of an inclined plane. D) A car ascends up a steep hill. E) More than one of the above
B - the child goes from a high position to a low position
if one object is dropped straight down and the other is projected horizontally, which will reach the ground first?
Both will reach the ground at the same time
interactive: In which of the following situations will there be an increase in kinetic energy? A) A projectile approaches its maximum height B) A box is pulled across a floor at a constant speed. C) A child is pushing a merry-go-round causing it to rotate faster. D) A satellite travels in a circular orbit around a planet at a fixed altitude. E) A stone at the end of a string is whirled in a horizontal circle at a constant speed.
C - in order to make the merry-go-round go faster, the velocity must increase - an increase in velocity means there is an increase in the kinetic energy
example: A rocket is shot into the air. At the moment the rocket reaches its highest point, a horizontal distance d from its starting point, a prearranged explosion separates it into two parts of equal mass. Part I is stopped in midair by the explosion, and it falls vertically to Earth. Where does part II land? Assume g = constant.
CM = 2d if the first part falls at the very top of the object's trajectory 2d = m1(d) + m2(xf) / M m2 = 3m 2d = m1(d) + 3m1(xf) / m1 + 3m1 2d = m(d) + 3m1(xf) = 4m1 2d = d + 3xf / 4 2d = .25d + .75xf 1.75d = .75xf xf = 2.33d
nonuniform circular motion
Circular motion that involves a change in angular speed during a time interval - force is exerted at an angle - F is split into Ftan and Fr - a is split into atan and ar
right hand rule
Common method used to determine the direction of the magnetic force vector. Thumb points in the direction of object's velocity, palm points in the direction of the acting force.
displacement
Distance and direction of an object's change in position from the starting point - how far an object is from the starting point - vector because it has magnitude and direction
Newton's Law of Universal Gravitation
Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining two particles
interactive: You lift a 10 N physics book up in the air a distance of 1 meter at a constant velocity of 0.5 m/s. The work done by you is what?
F = 10 d = 1m Θ = 0 because you exert a the normal force, which is in the same direction as displacement W = 10(1)cos(0) = 10 J
interactive: You lift a 10 N physics book up in the air a distance of 1 meter at a constant velocity of 0.5 m/s. The work done by gravity is what?
F = 10 d = 1m Θ = 180 because gravity is opposite the displacement W = 10(1)cos(180) = -10 J
interactive: A 50 kg woman runs up a flight of stairs in 5 s. Her net upward displacement is 5 m. What power did the woman exert while she was running?
F = 50(9.8) = 490 d = 5 t = 5 p = (490 x 5) / 5sec = 500 W = .5kW
what equation should be used to solve for static equilibrium?
F = ma (newtons second)
example: For a top player, a tennis ball may leave the racket on the serve with a speed of 55m/s. If the ball has a mass of 0.060 kg and is in contact with the racket for about 4 ms estimate the average force on the ball.
F = mΔv / Δt v1 = 0 v2 = 55 F = .06(55 - 0) / .004 s = 800N
interactive: Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?
F = p / Δt it would require the same amount of time because the force and the momentum are the same
Force equation (momentum)
F = Δp / Δt F = mΔv / Δt
how many newtons are required to beat static friction?
Ff = Fn(us) - anything more will move the object - the difference + kinetic friction is used to find the acceleration
Example: A 1000-kg car rounds a curve on a flat road of a radius of 50 m at a speed of 15m/s. Will the car follow the curve, or will it skid?
Fn = mg = 9.8(1000) = 9800 m(ar) = 1000((15)^2/50) = 4500N Ff = Fn(uf) =9800(.25) = 2450N The car will skid because the ground cannot exert sufficient force to keep it moving on the curve
when there is a banked curve, what is the force in the y direction?
Fncos(Ө) - mg = 0
banked curve equation
Fntan(Ө) = v^2/rg
Newton's Third Law
For every action there is an equal and opposite reaction - every object is elastic to some degree, which is what allows inanimate objects to exert a force back onto an object + ex: an ice skater pushes against a wall. The force she exerts does not make her go backward because she is pushing against the wall. It is the force of the wall on the skater that makes her move backward + ex: when a rocket launches, it is the force the rocket exerts on the gas that allows the gas to push the rocket upward with equal force
nonconservative forces
Forces that its work depends on the path. - friction - push/pull - tension - air resistance - motor or rocket propulsion
Example: a box weighs 10kg and you apply 40N of force. The us = .4 and the uk = .3. What is the acceleration of the box?
Fs = Fn(us) Fn = mg = 10(9.8) = 98 Fs = 98(.4) = 39 N With a force of 40N, the box will move. Therefore... Fk = Fn(uk) = 98(.3) = 29 40 - 29 = 11N a = 11N / 10kg = 1.1 m/s^2
Velocity, acceleration, and rotational of axis
If rotation is counterclockwise.. - Velocity increasing = acceleration is upward - Velocity decreasing = acceleration downward If rotation is clockwise... - Velocity increasing = acceleration downward - Velocity decreasing = acceleration upward
plastic limit
If the object is stretched beyond the elastic limit, it enters the plastic region → remains permanently deformed
What happens to the range if we double the initial velocity?
It quadrupled
Is there ever a point during projectile motion that the velocity is zero?
No, because there will always be a value for the x component of velocity
when solving force problems along an inclined slope, is it is always best to define the x and y axes parallel to the horizontal and vertical directions?
No, it is usually better to define the x and y axes by the incline so that the normal force and Ff are on coordinate planes. - Fg will be split into vectors
will an object ever be able to escape the pull of earth?
No, the pull will only get smaller
gravitational potential energy
Potential energy that depends on the height of an object + W = mgy2 - mgy1 - change in PE is equal and opposite to the work of gravity
angular momentum (L)
Product of rotational inertia and rotational velocity
inertial reference frame
Reference frame that has a zero or constant velocity (both speed and direction) - in most cases, earth is the reference frame
A solid sphere (S) with IS=(2/5)MR2, a thin hoop (H) with IH =MR2, and a solid disk (D) with ID=(1/2)MR2, all with the same radius, are allowed to roll down an inclined plane without slipping. In which order will they arrive at the bottom? (The fist one down listed first).
S, D, H because S has the fewest moment of inertia and H has the highest
The record playing on the turntable is rotating clockwise as seen from above. After turning it off, the turntable is slowing down, but hasn't stopped yet. The direction of the acceleration at point P is pointed where?
South east
Example: A 0.150-kg ball on the end of a 1.10-m-long cord (negligible mass) is swung in a vertical circle. Determine the minimum speed the ball must have at the top of its arc so that the ball continues moving in a circle
T + mg = m(v^2/r) T = 0 at minimum speed because we only want gravity acting on it mg = m(v^2/r) g = v^2/r 9.8 x 1.1 = 10.78 root(10.78) = 3.28 m/s^2
radius of an object in space
T = the amount of time it takes the object to revolve around the planet
Angular velocity
The angular displacement of an object divided by the time needed to make the displacement. - specified in rads/sec - positive when rotating counterclockwise and negative when rotating clockwise - the same for each point in a rotating object
Example: why does a skater's speed get faster when they tuck their arms in?
The angular momentum of the skater is conserved, and so when the moment of inertia decreases, the speed increases
A hollow cylinder of mass M and radius R rolls down an inclined plane. A block of mass M slides down an frictionless inclined plane at the same angle as the other one. If both objects are released at the same time, what will happen?
The block will reach the bottom first because all of its energy is being converted into translational kinetic energy, whereas the cylinder must convert some of its energy into rotational as well. They both will reach the bottom with the same kinetic energy
derived quantities
The combinations of fundamental quantities to form + ex: velocity, acceleration, force, momentum, work, and power.
Coriolis Force
The force that, owing to the rotation (spin) of the earth, deflects objects to the right (clockwise) in the Northern Hemisphere and to the left (counter-clockwise) in Southern Hemisphere.
perturbations
The gravitational forces of other planets that can cause small variations in a planet's motion
Example: Next, plate Mb initially at rest but free to rotate without friction, is placed in firm contact with the freely rotating plate and the two plates then both rotate at a constant angular velocity which is considerably less than. Why does this happen, and what is w2?
The moment of inertia increases, and so the angular velocity must decrease to compensate and maintain the same angular momentum
A ball on a string is rotating in a circle. The string is shortened by pulling it through the axis of rotation. What happens to the angular velocity and the tangential velocity of the ball?
The moment of inertia is directly related to the radius. As the radius decreases, so too does the moment of inertia. Because angular momentum is conserved, the angular velocity would increase The tangential is directly related to the angular velocity. As the angular velocity increases, so does the tangential velocity
why does the moon rise an hour later each day?
The moon revolves around the earth every 24hr, 50min → the moon rises nearly an hour later each day
Frequency
The number of complete rotations that occur in a certain amount of time - f = 1/T - SI units are 1/s
what does it mean when the net force = 0?
The object is moving at a constant speed or is at rest
what does it mean when the direction of velocity and acceleration are different?
The object is slowing down
force of gravity
The only downward force acting on a free falling object is gravity and it causes objects to accelerate at a force of 9.8 m/s2
Kepler's First Law
The orbit of each planet around the Sun is an ellipse with the Sun at one focus - they appear circular because the eccentricity of the planet is very small
lever arm
The perpendicular distance from the axis of rotation to a line drawn along the direction of the force - the axis is always an extension of the lever arm - a force applied exactly perpendicular to the lever arm is the most effective
acceleration
The rate at which velocity changes - vector
what are cars limited by when going up a hill?
The rate in which it can do work
angular acceleration
The rate of change of angular velocity - - the same for each point in a rotating object
Shear Modulus
The ratio of shear stress to shear strain - the change in length is perpendicular to the original length
Kepler's Third Law
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
Universal constant
The symbol G in the formula for Newton's law of gravitation is - 6.67 x 10^-11
inertia
The tendency of an object to resist a change in motion
what happens when a spinning wheel is turned upside down?
The wheel will turn in the opposite direction and the person will turn in the original direction of the wheel because of the conservation of angular momentum - Lwheel = Iw - -Lwheel+ Lperson = Lwheel - Lperson = 2Lwheel
How to set up a relative velocity equation...
Vac = Vab + Vbc - if you are given Vac, then you must subtract either Vab components or Vbc components from Vac components to solve
tangential velocity
Velocity that is parallel (tangent) to a curved path - perpendicular to centripetal acceleration - always in the direction of motion that is tangent to the circle - 2nr/T
vertical motion equations
Vy = Voy - at y = yo + Voy(t) - .5at^2 Vy^2 = Voy^2 - 2a(y - yo)
Work done on rotating object
W = Fd = FrӨ = τӨ
interactive: Two objects are sitting at rest on a "frictionless" air hockey table. Object A has twice the mass of object B. Both objects start at rest and are pushed with the same force for the same distance. Which statement is true?
W = Fd = K kinetic energy would not change because the force and the distance remain constant p = F / t it would take more time to push more mass the same distance with the same force, so they would have different momentums
work equation
W = force x distance W = FdcosӨ - Ө is the angle between the direction of the force and displacement of the object
If there are two accelerations acting on an object, how do we know which direction the object is moving?
We don't know. Just because one acceleration might be of greater magnitude than another does not indicate which direction the object was originally moving
work equation for nonconservative forces
Wnc = (KE1 - KE2) + (PE2 - PE1) Wnc = Fdcosθ = -Ffrd - the initial mechanical energy is reduced by the amount Ffrd KE1 + PE1 = KE2 + PE2 + Ffrd
net work equation
Work done by the all external forces Wn = Wg + Wfn + Wfr + Wf
subtraction of vectors
a - b = a + (-b)
sum of total acceleration
a = root(atan^2 + ar^2)
acceleration
a change in the state of motion
law
a descriptive statement or equation that reliably predicts events under certain conditions - takes the form of a relationship (equation) or statement - experimentally valid over a large range of phenomena - do not say how nature SHOULD behave but rather how it DOES behave - cannot be tested in all possible cases
free body diagram
a diagram showing all the forces acting on a single object - if there is more than one object in the problem, then more than one free body diagram is required - forces are represented by vectors
what is required to change momentum?
a force - Δp = F / Δt
centripetal force
a force that acts on a body moving in a circular path and is directed toward the center around which the body is moving. - NOT a new force --> sum of all forces - F = m(ar) - F = m(v^2/r)
static friction
a friction force that acts on objects that are not moving - opposite of the applied force - not always proportional to the normal force - generally greater than kinetic friction
model
a kind of analogy or mental image of the phenomena in terms of something else we are already familiar with - provides a structural similarity to the phenomena being studied
inertial mass
a measure of an object's resistance to any type of force - resistance to acceleration
contact force
a push or pull on one object by another that is touching it
stable equilibrium
a slight change from equilibrium is restored back the equilibrium position + ex: a ball swinging from a pendulum
unstable equilibrium
a slight change from equilibrium moves farther from the equilibrium position + ex: a pencil that is balanced on its tip is moved
dimensional analysis
a technique of problem-solving that uses the units that are part of measurement to help solve the problem - are used to work out relationships and see if they are correct
principle
a truth, a rule, or a law that is used in general statements
Radian
a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius. - l = rӨ
example: A 60-kg jogger runs up a long flight of stairs in 4.0s. The vertical height of the stairs is 4.5 m. (a) Estimate the jogger's power output in watts and horsepower. (b) How much energy did this require?
a) only force is the force of gravity = mgy = 2646. Divided by time (4s) = 661.5 W b) to convert to energy, multiple by the time = 2646 J
kinematic equations
acceleration is CONSTANT a = (v - v0) / t v = vo + at x = xo + vt x = xo + vo(t) + .5at^2 v^2 = vo^2 + 2a(x - xo)
centripetal acceleration
acceleration toward the center of a curved or circular path - causes an object to turn - v^2/r - NOT a vector
Accuracy vs. Precision
accuracy: how close experimental value is to accepted value + ex: if a scale is manufactured with a 2% error, then the accuracy of its measurement would be 2% of the true value precision: how closely measured values agree with each other - refers to the repeatability of the measurement using a given instrument
vector addition
adding or combining quantities that have magnitude and direction - the sum of the resultant is the sum of the individual vectors
Sig fig rule 1
all non-zero integers are significant + ex: 256 = 3 + ex: 6362.1 = 5
rotational motion
all points in the object move in circles and the center of these circles lie on the axis of rotation - a straight line drawn from the axis to any point in the object sweeps out the same angle in the same interval
center of mass equation
always use the origin as the reference for the x and y distance of the points we are looking at
uncertainty
an estimate of how much a measured or calculated value differs from a true value - numerical value is assumed to be one or a few units in the last digit specified + ex: 8.8 has an uncertainty of .1 because both are in the tens place
Theory
an explanation using an integrated set of principles that organizes observations and predicts behaviors or events - accepted or rejected based on the results of the observation and experimentation - a new theory is only accepted when it explains a greater range of phenomena than does the older one - they can quantitatively predict phenomena
why does friction exist?
because surfaces are not perfectly smooth and microscopic bumps will impede motion
Suppose two vectors each have length 3.0 units. What is the range of possible lengths for the vector representing the sum of the two?
between 0 and 6 because the minimum value the vectors can add to are 0 (opposite directions) and the maximum value the vectors can add to are 6 (same direction)
momentum and kinetic energy
both are conserved in a completely elastic collision, but not in an inelastic collision
what happens during a collision?
both objects are deformed, often considerably because of the large forces involved - the force each exerts on the other usually jumps from zero at the moment of contact to a very large force, then back to zero after contact
A force F is applied to a dumbbell for a time interval Δt, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy?
case b because it also has rotational kinetic energy, whereas case a only has translational kinetic energy
inelastic collisions
collisions in which kinetic energy is not conserved - some of the initial kinetic energy is converted into other types of energy - completely inelastic collisions are when two object stick together → maximum amount of kinetic energy is transformed
radial acceleration
component of acceleration of a body in angular motion directed toward the center of curvature - represents change in direction
will a farther planet move faster or slower than a closer planet?
faster
interactive: Suppose you wanted to ride your mountain bike up a steep hill. Two paths lead from the base to the top, one twice as long as the other. Compared to the average force you would exert if you took the short path, the average force you exert along the longer path is what?
half as small - work is the same, and so if the distance is twice as long, then you need to half the force to make them equal
what happens to the centripetal acceleration if you double the radius?
halves acceleration
tangential acceleration (atan)
has a magnitude equal to the rate of change of the magnitude of acceleration - causes an object to speed up or slow down - if the object is increasing speed, atan and velocity are parallel - if the object is decreasing speed, atan and velocity are perpendicular - atan = ∆v/∆t
vector quantities
have magnitude and direction + ex: displacement, velocity, force, acceleration, momentum
example: on a roller coaster, the potential energy will only reach maxim energy at the original height. What happens if the second hill is the same height as the first?
it will come to rest at an equal height
example: on a roller coaster, the potential energy will only reach maxim energy at the original height. What happens if the second hill is taller than the first?
it will reach maximum height equal to the first hill, and then slide back down
if you double the mass, what happens to the kinetic energy if force and time are constant?
it would decrease because velocity had to be halved to compensate for the increase in mass
will a farther plant take longer or shorter to orbit?
longer
will a farther plant travel longer or shorter in an orbit?
longer
measure of inertia
mass - objects with a large mass require a larger force to achieve the same change in motion - objects with a small mass require a small force to achieve the same change in motion
collisions in the x axis
mavax + mbvbx = mavax'cosӨa + mbvbx'cosӨb
collisions in the y axis
mavax + mbvbx = mavax'sinӨa + mbvbx'sinӨb
Standard unit of length
meter (m)
Example: 65-kg woman descends in an elevator that briefly accelerates at 0.20g downward. During this acceleration, what is her weight and what does the scale read?
mg - Fn = ma mg - Fn = m(.2g) Fn = mg - m(.2g) Fn = .8mg The scale would show .8 of her total weight = 52kg
when is momentum conserved?
momentum of the universe is always conserved - momentum is only conserved in a system when there are no net external forces on the system
interactive: A ball drops some distance and gains 30 J of kinetic energy. Do not ignore air resistance. How much gravitational potential energy did the ball lose?
more than 30J because some of the energy was transferred to air resistance
translational motion
movement in which an entire molecule moves in a definite direction without rotation - they move along a straight-line path
what is the moment of inertia for a point value?
mr^2
deceleration
negative acceleration or decrease in speed - the direction of velocity and acceleration are opposite - if a car is going left and slowing down, acceleration is POSITIVE - if a car is going right and slowing down, acceleration is NEGATIVE
is mechanical energy the same as total energy?
no, it is the sum of kinetic and potential energies
if the total work done on an object is not zero, will the kinetic energy of the object change?
no, it remains constant
Does mass depend on gravity?
nope, but weight does; an object has the same mass anywhere in the entire universe
example: on a roller coaster, the potential energy will only reach maxim energy at the original height. What happens if the second hill is lower than the first?
not all of the cart's kinetic energy will be transformed to potential energy, so it will go over the other side of the hill
collisions in two dimensions
occurs when one object strikes another non-head-on - the two objects leave at different angles off of the x and y axis
if you double the mass, what happens to the momentum if force and time are constant?
p = FΔt the momentum would stay the same because we are not changing force or time - the velocity would have to be halved in order to compensate for the double mass
example: Estimate the impulse and the average force delivered by a karate blow that breaks a board. Assume the hand moves at 10m/s roughly when it hits the board.
p = mv m = 1kg v = 10m/s p = 10 kgm/s t = 2ms F = 10 / .002 = 5000N
conservation of momentum equation
pa + pb = p'a + p'b MaVa + MbVb = MaVa' + MbVb' FΔt = pfinal - pinital
interactive: A plane, flying horizontally, releases a bomb, which explodes before hitting the ground. Neglecting air resistance, the center of mass of the bomb fragments, just after the explosion will move in what direction?
parabolically
what direction is centripetal force compared to velocity?
perpendicular
Sig Fig Multiplication/Division Rule
round to least number of sig figs + ex: 6.5 x 4.00 = 26
percent uncertainty vs sig figs
sig figs are only an approximation, so it may underestimate the accuracy (uncertainty) of the calculation - add an extra digit if it gives a more realistic estimate of uncertainty - want to match the percent uncertainty of the answer to the original numbers being used
To find the angle of a projectile if it is being launched from the ground in a projectile motion...
sin(2theta) = Ra / vo^2
velocity vs time graph
slope = acceleration - average acceleration during this time interval is equal to the slope of the straight line connecting two points on the graph - instantaneous acceleration is the slope of the tangent line
Displacement vs Time Graph
slope = velocity - average velocity during this time interval is equal to the slope of the straight line connecting two points on the graph - instantaneous velocity is the slope of the tangent line
do northern or southern points on earth have faster tangential velocity?
southern
satellites at the same distance are equal in what?
speed; mass does not change the speed as long as satellites are the same distance from earth
average acceleration
the change in velocity during some measurable time interval divided by that time interval
interactive: A small car meshes with a large truck in a head-on collision. Which of the following statements concerning the magnitude of the collision force is correct?
the collision force is the same due to newton's third law - equal and opposite
projectile motion
the curved path that an object follows when thrown, launched, or otherwise projected near the surface of Earth - occurs in two direction in the absence of air resistance - a = -9/8 m/s^2 - velocity and displacement can be broken up into vector components
what happens when the net force acts sideways on an object?
the direction of the velocity will change, and that change will also cause acceleration
elastic potential energy
the energy of stretched or compressed objects - to hold a spring stretched or compressed a distance x from its natural length requires the hand to exert an external force on the spring of magnitude F + F = kx where k is the spring constant + the spring exerts a force -kx
normal force
the force perpendicular to a surface that prevents an object from falling through the surface - the minimum value of Fn is 0 because a normal force cannot pull on an object as a negative sign would indicate
kinetic friction
the force that opposes the movement of two surfaces that are in contact and are moving over each other - opposite of velocity - proportional to normal force - Fk = Fn(uk)
A golf ball is fired at a bowling ball, initially at rest. After the collision, the golf ball bounces back elastically. Compared to the bowling ball, the golf ball after the collision has how much kinetic energy?
the golf ball has a higher speed than the bowling ball after the collision - because velocity is squared, it has a greater impact on the kinetic energy than the mass - greater kinetic energy than the bowling ball
interactive: An object with mass m is raised a distance h. The system is defined as the object only. Which is true about the system?
the gravitational force does work on the system equal to -mgh - cannot include PE because it includes the earth
tangential acceleration
the instantaneous linear acceleration of an object directed along the tangent to the object's circular path - is greater for points farther away from the axis of rotation
law of conservation of energy
the law that states that energy cannot be created or destroyed but can be changed from one form to another - encompasses nonconservative forces as well
linear momentum and CM
the linear momentum of an object is the product of the object's mass and the velocity of its CM Mvcm = mava + mbva + mcvc
interactive: A compact car and a large truck collide head on and stick together. Which undergoes the larger momentum change?
the momentum change is the same for both because momentum is always conserved
what will always happen when in impulse is applied to an object?
the momentum will change
uniform circular motion
the movement of an object at a constant speed around a circle with a fixed radius - the magnitude of velocity is constant but the direction is changing, so acceleration is changing
Frequency
the number of complete wavelengths that pass a point in a given time - 1 rev = 2π - f = w / 2π
what does it mean when the direction of velocity and acceleration are the same?
the object is speeding up
what happens when a person pushes/pulls an object upwards with a force that is greater than the object's weight?
the object will accelerate
when the net force is in the opposite direction as the velocity...
the object will decrease in velocity (decelerate or accelerate in the negative direction)
when the net force is in the same direction as velocity...
the object will increase in velocity (accelerate in positive direction)
elastic limit
the object will return to its original length if the force is removed - the region from the origin to the elastic limit is called the elastic region
example: two kids start at the top of a kill and slide down. Kid 1 takes a steeper path and kid 2 takes a shallower path. what are their speeds once they reach the bottom?
the path does not matter, so both will reach the bottom of the hill with the same speed
center of mass
the point in an object that moves as if all the object's mass were concentrated at that point - the point at which the force of gravity (weight) can be considered to act - does not have to be in the physical mass of the object - CM of a perfectly symmetrical object is in the center
interactive: An object with mass m is raised a distance h. The system is defined as the object and the earth. Which is true about this situation?
the potential energy of the system is increased by mgh - the object could be raised at an angle, and so we cannot assume Fg = -mgh
linear momentum
the product of the mass and velocity of an object - vector - the direction of momentum is the direction of velocity - SI unit is kgm/s - the larger the momentum, the harder it is to stop
power
the rate at which work is done - average power = the work done divided by the time required to do it - the rate at which energy is transferred - measured in watts or horsepower - 1 W = 1.33 hp
percent uncertainty
the ratio of the uncertainty of a measurement to the measured value expressed as a percentage +/- number / value x 100 Ex: 8.8 +/-.1 = .1/8.8 x 100
efficiency
the ratio of the useful power output of an engine to the power input provided by burning gas - always less than 1 because engines are not perfectly efficient - nearly 85% of input is wasted → 15% efficient
moment of intertia
the resistance to rotation - equal to mr^2 - sum of the masses x radial position squared
interactive: Suppose you wanted to ride your mountain bike up a steep hill. Two paths lead from the base to the top, one twice as long as the other. Compared to the work you would do if you took the short path, the work you do along the longer path is what?
the same
interactive: A bullet is fired from a rifle. Neglect any external forces on the rifle. In doing so the bullet gains a certain amount of momentum and the rifle gains how much momentum?
the same amount because momentum is conserved - just opposite in direction
Constant angular acceleration
the same kinematic equations that applied to constant linear acceleration also applied to angular acceleration
interactive: Suppose you wanted to ride your mountain bike down a steep hill. Two paths lead from the top to the base, one twice as long as the other. Neglect friction and air resistance. Compared to the maximum speed you would reach if you took the short path, the maximum speed you will reach along the longer path is what?
the same; the path does not matter
interactive: A father and his much lighter daughter are balanced on a massless seesaw. If they both move forward so that they are one-half their original distance from the pivot point, what will happen to the seesaw?
the seesaw will remain balanced
neutral equilibrium
the state of an object balanced so that any small movement neither raises nor lowers its center of gravity - a slight change from equilibrium stays in a new location + ex: when a refrigerator is bumped
force and CM
the sum of all the forces acting on the system is equal to the total mass of the system times the acceleration of the center of mass Macm = maaa + mbab + mcac
what is the total linear acceleration equal to?
the tangential and radial (centripetal) acceleration
period
the time required for one cycle, - a complete motion that returns to its starting point - T = 1/f
period
the time required for one cycle, a complete motion that returns to its starting point - T = 1 / f - T = 2π / w
law of conservation of angular momentum
the total angular momentum of a rotating object remains constant if the net torque acting on it is zero
instantaneous velocity
the velocity of an object at some instant or at a specific point in the object's path - if an object moves at a constant velocity during a particular time interval, then its instantaneous velocity is the same as the average velocity
change in potential energy
the work required of an external force to move the object without acceleration between two points - equal to the negative work done by a specific force when the object is moved from one point to another
Are theories perfect?
theories are never perfect nor are they ever "proven" - no measure is perfect enough to prove something
if you drop a lighter object and a heavier object at the same time, which will hit the ground first?
they will hit at the same time because mass is not relevant to free falling objects
dissipative forces
those that reduce the mechanical energy but not the total energy of the system - nonconservative forces - work transfers energy from the system
what direction is friction when an object is undergoing uniform circular motion?
towards the center
how many objects does work involve?
two objects - one object does work ON another object by exhibiting a force ON the other object - work done ON and object is equal and opposite the work done BY an object
linear velocity
v = rw - velocity is greater for points farther away from the axis of rotation
To find the velocity after the ball returns to the thrower's hand if a ball is thrown straight up...
v = vo + at
interactive: A pitched baseball is popped straight up by the bat as shown in the figure at the right. In which direction is the impulse provided by the bat?
vf - vi that leaves us with a vector going north east
when an object is dropped, what two variables are equal to 0?
vo and xo
when does Fg = Fn?
when an object is horizontal and at rest/moving at a constant velocity
when is Fn less than Fg?
when an object is horizontal and has a force pulling/pushing upwards
when is Fn more than Fg?
when an object is horizontal and has a force pushing/pulling downwards
when will an object be stable?
when their center of mass is above its base of support - this is because the normal force can be exerted only within the area of contact - the larger the base, the more stable the object
when do you use the energy conservation equation?
when you have an object on which there are external forces KE1 + PE1 = KE2 + PE2 + Ffrd
when do you use the work-energy principle?
when you have an object on which there are no external forces KE1 + PE1 = KE2 + PE2
To find the total distance in the x direction of an object if t is being launched from the ground in a projectile motion...
x = Vox(t) - use the total time it takes for an object to reach the ground R = (vo^2sin(2theta))/g
The Vox of an object if it is being projected horizontally off a cliff...
x = xo + Vox(t) + .5at^2 a = 0 so... x = Vox(t) Vox is also the initial velocity because there is no vertical component at the start, and it is constant throughout
to find the highest point if a ball is thrown straight up...
y = (v^2 - vo^2) / 2a
The time the object is in the air if it is being projected horizontally off a cliff...
y = yo + Voy(t) + .5at^2
is it possible to do work if the net work is 0?
yes, they would just all add to 0
interactive: You are on a cart initially at rest on a track with no friction. You throw balls off the cart to the right. Is the cart put in motion?
yes, to the left
Sig fig rule 2
zeros between non-zero integers are significant + ex: 6.08 = 3 + ex: 7.007 = 4
Sig fig rule 4
zeros to the right of non-zero integers without decimal points are non significant + ex: 990 = 2 + ex: 3030 = 3
Sig fig rule 5
zeros to the right of the decimal point but the left of non-zero integers are not significant + ex: .00787 = 3 + ex: .07800 = 4
torque and angular momentum
τ = L / t
Example: A 15.0-N force is applied to a cord wrapped around a pulley of mass 4kg and radius 33cm. The pulley accelerates uniformly from rest to an angular speed of 30.0 rad/s in 3 sec. If there is a frictional torque 1.10 N at the axle, determine the moment of inertia of the pulley. The pulley rotates about its center.
τfr = - 1.1N because it is clockwise τc = 15(.33)sin(90) = 4.95 τnet = 4.95 - 1.1 = 3.85 N 30 = 0 + α(3) = 10 3.85 = I(10) = .385
Example: Consider again the pulley. But instead of a constant 15.0-N force being exerted on the cord, we now have a bucket of weight 15N (mass 1.53) hanging from the cord. We assume the cord has negligible mass and does not stretch or slip on the pulley. Calculate the angular acceleration of the pulley and the linear acceleration of the bucket. Assume the same frictional torque (1.1) acts perpendicular
τnet = 3.85N Inet = .385 + 1.53(.33)^2 = .552 3.85 = .552α = 6.98 a = αr = 6.98(.33) = 2.3
balance
occurs when the center of mass of an object is over a point of contact - stable, unstable, and neautral
elastic potential energy equation
U = 1/2 k x^2
cos(theta)
adjacent/hypotenuse
kinetic energy
motional energy
power equation
p = W / Δt p = Fv
