Lectures and Textbook (Physics II)
When we're pushing a bicycle wheel, what three things matter in determining whether it spins or not, and how powerfully it does so? What concept encompasses all of these components?
1. The magnitude of the force 2. The position of the application of the force 3. The angle at which the force is applied Torque!
What two conditions must an object in mechanical equilibrium satisfy?
1. The net external force must be zero (Sum F = 0) 2. The net external torque must be zero (Sum t = 0) (In other words, the object must be in translational and rotational equilibrium)
What is translational equilibrium?
1/2 of mechanical equilibrium. The sum of all forces acting on an object must be zero, so the object has no translational acceleration, a = 0
What is rotational equilibrium?
1/2 of mechanical equilibrium. The sum of all torques on the object must be zero, so the object has no angular acceleration, a = 0.
Kepler's second law
A line drawn from the Sun to any planet will sweep out equal areas in equal times
When does a particle moving on a curved path have centripetal acceleration? When does it have tangential acceleration?
A particle moving on a curved path always has centripetal acceleration, since the direction of the velocity changes. The magnitude of the velocity could also be changing - in this case, the particle also has tangential acceleration. Note: this is not always
Equations for work - what are you assuming with all of these?
All of these assume that force F is constant W = Fd (this only gives magnitude of work done on an object when the force is constant and parallel to the displacement, which must be along a line) W = FxΔx (work done on an object by a constant force F during a linear displacement along the x-axis, where F is the x-component of the force F) W = (Fcosθ)d (work done on an object by a constant force F during a linear displacement not parallel to any axis, where d is the magnitude of the displacement and θ is the angle between the vector F and the displacement vector Δx)
Kepler's first law
All planets move in elliptical orbits with the sun at one focus
How do we calculate center of gravity?
Calculate center of gravity in the x, y, and z directions xcg = mixi/mi ycg = miyi/mi zcg = mizi/mi
What would happen if the net force causing centripetal acceleration were to vanish?
If the force were to vanish, the object would immediately leave its circular path and move along a straight line tangent to the circle at the point where the force vanished
Units for work?
Newton-meter or Joules
PE of a spring
PEs = 1/2kx^2
Two particles collide in a head on collision, m1 < m2. Which one experiences the greater change in magnitude of velocity? What happens re: momentum?
The magnitude of the change of velocity of the lighter particle is greater than that of the heavier particle The momentum of each object changes during the collision, but the total momentum of the system is constant
Equation relating gravitational work and the gravitational potential energy
Wg = -(PEf - PEi) = -(mghf - mghi) The work done by gravity is one and the same as the negative of the change in gravitational potential energy
Why is k negative in Hook's law?
When x is positive (stretched spring), the spring force is directed to the left
Why are spring forces conservative?
Work done by an applied force in stretching or compressing a spring can be recovered by removing the applied force, so like gravity, the spring force is conservative, as long as losses through internal frictionn can be neglected
How are potential energy and work done by conservative forces related?
Work done by conservative forces can be recast as potential energy, which depends only on the beginning and end points of a curve, not the path taken. The change in potential energy of a body associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the initial and the final positions. (Potential energy is another way of looking at the work done by conservative forces!)
Tangential acceleration
at = ar
Types of collisions
elastic, inelastic, perfectly inelastic
Conversion factors from degrees to radians
1 rad = 360 degrees/2pi = 57.3 degrees θ[rad] = pi/180 degrees * θ[degrees]
5 steps in solving equilibrium problems?
1. Diagram the system 2. Draw a force diagram 3. Apply sum of torques = 0 4. Apply sum of force in x = 0, sum of force in y = 0 5. Solve the systems of equations
What will you be doing if you have a glancing collision problem?
1. breaking momentum into components, and remembering that MOMENTUM IS CONSERVED 2. probably solving a system of equations OR just recall that you can find magnitude of force and direction with the square root thing and the tan-1, so just find x and y components of final and proceed
G
6.67 * 10^-11 Nm^2/kg^2
What will you be doing if you are evaluating a perfectly elastic collision?
9/10, you will be solving a system of equations
Why doesn't a cheerleader jumping upwards from rest violate conservation of momentum?
A cheerleader jumping upwards from rest might appear to violate conservation of momentum, because initially her momentum is zero and suddenly she's leaving the ground with velocity v. The flaw in this reasoning lies in the fact that the cheerleader isn't an isolated system. In jumping, she exerts a downward force on Earth, changing its momentum. This change in Earth's momentum isn't noticeable, however, because of Earth's giant mass compared to the cheerleader's. When we define the system to be the cheerleader and the Earth, momentum is conserved
Conservative vs nonconservative forces
A force is conservative if the work it does moving an object between two points is the same no matter what path is taken
What is a radian? What are the units?
A radian is a unit of angular measure. It is defined as the arc length s along a circle dived by the radius r θ = s/r Radians are dimensionless, or you can use degrees
What's the easiest way to balance a rigid object in a uniform gravitational field?
A rigid object in a uniform gravitational field can be balanced by a single force equal in magnitude to the weight of the object, as long as the force is directed upward through the object's center of gravity
System
A system is a collection of objects interacting via forces or other processes that are internal to the system
Where is a velocity vector in uniform circular motion?
A vector tangential to the circle, pointing in the direction of instantaneous motion
Recoil
Action and reaction, together with the accompanying exchange of momentum between two objects, is responsible for recoil. Everyone knows that throwing a baseball while standing straight up without bracing one's feet against Earth, is a good way to fall over backwards. This reaction, an example of recoil, also happens when you fire a gun or shoot an arrow. Conservation of momentum provides a straightforward way to calculate such effects.
What do you do when two or more torques act on an object at rest, what do you do?
Add the torques!
An object can have a centripetal acceleration only if...
An object can have centripetal acceleration only if some external force acts on it
Angular displacement
An object's angular displacement, Δθ, is the difference in its final and initial angles
Average instantaneous angular speed
Average angular speed, w, of a rotating rigid object is the ratio of the angular displacement to the time interval wav = (θf - θi)/(tf-ti) = Δθ/Δt
Why does torque increase if a woman leans back on a seesaw?
Because her center of gravity moves backward, so r increases
Why do even very tiring tasks not constitute work, per the physics definition?
Because if no force is exerted / no displacement takes place, there is no physics 'work.' Ex: truck driver, student pressing against a wall for hours
Why do we care about torque? Why isn't our current understanding of force good enough? Can you give an example?
Because point of application of force matters! If a tennis ball is struck with a strong horizontal force acting through its center of mass, it may travel a long distance before hitting the ground, far out of bounds. Instead, the same force applied in an upward, glancing stroke will impart topspin to the ball, which can cause it to hand in the opponent's court
Why do we talk about static friction even though a car is moving in a circle?
Because the force of friction is point toward the center of the circle, and the car isn't moving that way
Why is jumping up and down not a productive way to lose weight?
Because the muscles are at most 25% efficient at producing KE from chemical energy (muscles always produce a lot of internal energy and KE as well as work - that's why you sweat when you work out), they use up to 4 times the 350J of chemical energy in one jump. The chemical energy comes from the food we eat, with one food calorie equal to 4200 J. So total energy supplied by the body as internal energy and KE in one jump is only about 1/3 of a food calorie!
Why are doorknobs located at the outer edge to doors? What would happen if you moved it to the middle of the door?
Because when a force F is applied perpendicular to the outer edge of the door, we multiply F * r to get torque, and we want maximum torque, so maximum length for r. Basically - we want the door to open as fast as possible. If the same perpendicular force was applied at a point nearer to the hinge, there would be a smaller angular acceleration.
Elastic collisions
Both momentum and KE are conserved Macroscopic collisions fall between nearly elastic and perfectly inelastic collisionns
What if the applied force F isn't perpendicular to the position vector r?
Break F into components - the component of the force perpendicular to r will cause the object to rotate. F*sin(θ) Think of a door - maximum torque is when force is applied perpendicular to the door. At this point, F*sinθ = F*1. Force applied parallel to the door (pulling to the right or something) = F*sin(180) = F*0
Centripetal force (+ provide examples)
Centripetal force is a classification that includes forces acting toward a central point. It is not a force in itself Ex: Tension in a string Gravity Force of friction
What does a net force causing centripetal acceleration do to the velocity vector? In what direction does it act?
Changes the direction of the velocity vector Toward the center of the circle
How do we choose axes for torque when calculating equilibrium?
Choose a pivot point and axes that render at least one torque equal to zero (think of the ladder)
What does the value of torque depend on?
Chosen axis of rotation. Torques can be computed around any axis, regardless of whether there is some actual, physical rotation axis present. Once this point is chosen, however, it must be used consistently throughout a given problem
If the disk rotates clockwise, what direction is the angular velocity?
Clockwise
What do we do with momentum when the trajectory of an object is not parallel to an axis?
Components, baby! px = mvx py = mvy
How do we consider ME when only conservative forces are present? How is this different than when nonconservative forces are present?
Conservative forces, so we know that ME is conserved: (KE + PEg + PEs)i = (KE + PEg + PEs)f Nonconservative forces, so we know there will be a chance in ME, since Wnc = ΔE. Wnc = (KEf - KEi) + (PEgf - PEgi) + (PEsf - PEsi)
If the disk rotates counterclockwise, what direction is the angular velocity?
Counterclockwise
What do tangential speed and acceleration depend upon?
Distance from a given point to the axis of rotation
Connect inertia and centripetal force
Due to inertia, objects want to continue moving in straight lines (per instantaneous velocity vectors). But the tension force (or whatever force! friction, etc) prevents motion along a straight line by exerting a radial force on the object that makes it follow the circular path
Why do we use angular variables to describe a rotating disc?
Each point on the disc undergoes the same angular displacement in any given time interval
How do we evaluate non-isolated systems?
Energy can cross the system boundary in a variety of ways. The total energy of the system changes. The change in the total amount of energy in the system is equal to the total amount of energy that crosses the boundary of the system.
Every point on a rotating object has the same.... ? Does not have the same... ?
Every object on a rotating object has the same angular motion Every point on the rotating object does not have the same linear motion
How can you determine centripetal force?
Fnet = mac Fnet stands for any net force that keeps an object following a circular path. This Fnet is also called the centripetal force
What must be true of an object moving through space in mechanical equilibrium?
For an object to be in mechanical equilibrium, it must move through space at a constant speed and rotate at a constant angular speed.
What does force cause? What does torque cause?
Force causes accelerations. Torque causes angular accelerations
Frictional work
Frictional work is really important in life because doing most other kinds of work is impossible without it! Ex: An eskimo depends on surface friction to pull his sled. Otherwise, the rope would slip in his hands and exert no force on the sled, while his feet slid out from underneath him and he fell on his face. "Work done by friction" we will interpret this phrase as applying to mechanical energy alone.
Equation for spring force (Fs)
Fs = -kx
Which changes, g or G? Why?
G does not change. g varies with altitude
Describe the motion of a slinky down stairs in terms of energy
Gravitational PE is converted to other forms of energy as it moves
Is gravity a conservative or non-conservative force? Why?
Gravity is a conservative force, because the path taken doesn't matter. For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises h meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, "Guess what — our potential energy compared to before is mgh greater." "Mine, too," you say, looking out over the view. You pull out this equation: This equation basically states that the actual path you take when going vertically from hi to hf doesn't matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn't matter, gravity is a conservative force.
How does angular acceleration relate to the speed of the object?
If angular velocity and acceleration are in the same direction, the object is rotating faster. If angular velocity and angular acceleration are in opposite directions, the object is slowing down.
What's up with momentum if the net force on an object is zero?
If net force on an object is zero, the object's momentum doesn't change. In other words, the linear momentum of an object is conserved when Fnet = 0.
If rotation is speeding up, what's happening with angular acceleration/angular velocity? What if it's slowing down?
If rotation is speeding up, angular acceleration is in the same direction as angular velocity. If rotation is slowing down, angular acceleration is in the opposite direction to angular velocity.
Newton's law of universal gravitation (equation + voiceover)
If two particles with masses m1 and m2 are separated by a distance r, then a gravitational force acts along a line joining them, with magnitude given by: F = G * m1m2/r^2 If extended objects, r is a distance between the two centers
Why is relative velocity relevant to discussions of collisions? Give the equation too
If we are trying to solve an elastic collision problem, and we have ONE DIMENSIONAL EQUATIONS and ELASTIC COLLISIONS, we can use the relative velocity equation instead of the conservation of KE (get rid of quadratic!) v1i - v2i = -(v1f - v2f) Relative velocity before equals negative of relative velocity after This is literally your best friend with elastic collisions. Comes from combining linear momentum and KE equations. Replace KE eq. with this equation and solve system of 2 linear equations
Define 3 analogies between linear and rotational motion. What do these relationships imply?
Implication across the board is that acceleration (linear or angular) is constant v = vi + at Δx = vit + 1/2at^2 v^2 = vi^2 + 2aΔx w = wi + at Δθ = wit + 1/2at^2 w^2 = wi^2 + 2aΔθ
In what direction does impulse act?
Impulse is a vector quantity with the same direction as the constant force acting on the object
What does Newton's third law tell us about impulses in a collision?
Impulses, like forces, are equal and opposite in a collision
What is conserved in a perfectly inelastic collision? What else can we know about these things?
In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same
How do we apply conservation of momentum to collisions? (voiceover)
In an isolated system, with no net external forces present, the momentum of each object will change. We can therefore use Newton's third law during the collision (for every action/force in nature there is an equal and opposite reaction) Notice! That this also means that impulses are equal and opposite
Conservation of mechanical energy (voiceover + equations)
In any isolated system of objects interacting only through conservative forces, the total mechanical energy E = KE + PE of the system remains the same at all times E = KE + PE 1/2mvi^2 + mghi = 1/2mvf^2 + mghf
Center of gravity At this point, a force may be applied to cause...
In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration.
Why is mass of car important in a collision?
Injury is caused by a change in velocity, and the more massive vehicle undergoes a smaller velocity change in a typical accident
Why is it hard to walk to the center of a moving merry go round? What forces are acting on you when you do this? What would happen if you let go?
It's hard to hold onto a merry go round and walk towards the center! Feels like there's a center-fleeing force pulling you back. In reality, the rider is exerting a centripetal force on his body with his hand and arm muscles. In addition, a smaller centripetal force is exerted by the static friction between his feet and the platform. If the rider's grip slipped, he wouldn't be flung radially away; rather, he would go off on a straight line, tangent to the point in space where he let go of the railing. The rider lands at a point that is farther away from the center, but not by "fleeing the center" along a radial line. Instead, he travels perpendicular to a radial line, traversing an angular displacement while increasing his radial displacement.
How is linear momentum related to kinetic energy?
KE = p^2/2m
What happens to KE in inelastic collisions?
KE is not conserved! Some of the KE is converted into other types of energy such as heat, sound, work due to permanently deform an object
How do we deal with glancing collisions, aka two dimensional collisions?
Linear momentum in x and y directions must be conserved. So, break momentum into components! m1v1ix + m2v2ix = m1v1fx + m2v2fx m1v1iy + m2v2iy = m1v1fy + m2v2fy
When is linear momentum conserved? When is KE conserved?
Linear momentum is conserved in any collisions as LONG AS it's an isolated system KE is only conserved in elastic collisions
When can we use conservation of momentum?
Momentum is conserved for isolated systems (not individual objects), which include all of the objects interacting with each other. Assumes only internal forces are acting during the collision, and can be generalized to any number of objects.
When are center of gravity and center of mass equivalent?
Most of the time, except when g changes (think of a skyscraper)
Units of torque
N*m
If torque is zero, does that mean force is zero?
NOPE. Recall that there are two important factors that you need to evaluate when considering equilibrium!
How do you determine net work?
Net work done on an object as it undergoes some displacement is the sum of the amount of work done by each force
Relate net torque, net force, linear motion, and rotational motion
Object will move with constant velocity unless acted upon by a net force Rate of rotation of an object doesn't change unless the object is acted upon by a net torque
Identify the centripetal force in a stone being twirled around on a string at an angle. How about on a flat table?
On flat table: centripetal force = tension force, T Twirled around at an angle: centripetal force = horizontal component of tension force, T
Equations for potential energy
PE = mgh (gravitational potential energy associated with an object located near the surface of the Earth is the object's weight mg times its vertical position h above the earth) IMPORTANT: Choose a reference point where y = 0, but once this is chosen, it must remain fixed for a given problem
Is potential energy a property of a system or an object?
Potential energy is a property of a system, because it's due to the relative positions of interacting objects in the system.
Power (definition, equations)
Power is the time rate of energy transfer. Using work as the energy transfer method, we can write power as: Pavg = W/Δt P = Fv (both the force and velocity v must be parallel, but can change with time)
Units of angular speed
Radians per second or sec^-1
How do you solve problems involving elastic collisions?
Since we know that both momentum and KE are conserved, we have 2 equations - can do system of equations! m1v1i + m2v2i = m1v1f + m2v2f 1/2m1v1^2i + 1/2m2v2^2i = 1/2 m1v1^2f + 1/2m2v2^2f
How do you determine the center of gravity of an irregularly shaped object?
Suspend it from two different arbitrary points. The point where the intersection of the two lines (going straight from the string, through the object) occurs is the center of gravity. In fact, if an object is hung freely from any point, the center of gravity always lies straight below the point of support, so the vertical line through that point must pass through the center of gravity
Does conservation of momentum apply to systems or objects?
Systems!
T/F: energy of the universe is constantly conserved
TRUE
When is tangential acceleration = 0? When is it not?
Tangential acceleration = 0 when speed is constant. Not = 0 when speed is changing Acceleration is basically rate of change of velocity. In a rotational motion, there are two components of the net acceleration: one Normal(along the radius) and one Tangential (along the circumference). The Normal acceleration serves the purpose of causing the motion to be circular, whereas the tangential one can make the rotation faster or slower. Since a uniform motion itself defines to be a constant velocity motion (ie the rotation is going on at a constant speed), the tangential component of acceleration is zero. If the rotation is non-uniform, that means the speed of rotation is changing, thus inevitably leading to the conclusion that the tangential component of the net acceleration is non-zero.
Units of power?
The SI unit of power is the watt: 1 watt = 1 joule/second = 1kg*m^2/s^3 The unit of power int eh US system is horsepower: 1 hp = 746 W
Relate average force to momentum and explain why we'd want to do this? What law is this?
The average force can be thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the interval. Impulse momentum theorem: I = FavΔt = Δp We do this because in real life, force on an object is rarely constant. For example, when a bat hits a baseball, the force increases sharply, reaches some maximum value, and then decreases just as rapidly. In order to analyze this interaction, it's useful to define an average force Fav. The magnitude of the impulse delivered by a force during the time interval Δt is equal to the area under the force vs. time graph, or alternatively to the area of the rectangle drawn between the two time points and the value for F.
Why can we say that the center of gravity of a 10m ladder is at 5m without doing any calculations?
The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry
Centripetal acceleration (voiceover)
The centripetal component of acceleration is due to the changing direction. The centripetal acceleration always points towards the center of the circle and is therefore always perpendicular to the velocity The centripital force is the one responsible for changing the direction the object, thus causing a centripetal acceleration. The centripetal force happens when the force is perpindicular to the direction of the velocity.
What is the most important factor in an auto collision? Why? How do air bags and seat belts factor into this?
The collision time, or the time it takes for a person to come to rest. Think about the impulse-momentum theorem: I = FavΔt = Δp = mvf - mvi From this, we know that: Fav = mvf - mvi / Δt So, increasing the Δt will decrease the overall average force experienced by the individual involved in the collision. We increase this time by using air bags and seat belts?
What's keeping a car on a banked, frictionless surface from flying off?
The horizontal component of the normal force
In a force versus time curve, how would we determine impulse? What else can we say this is equal to?
The impulse imparted to the particle by the force is the area under the curve of the force versus time curve. This area is also equal to the change in momentum
Lever (moment) arm (define + equations)
The lever arm (moment arm), d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = rsinθ t = Fd = Frsinθ
How do we find the work done by gravity on an object when it moves from one position to another?
The negative of that work is the change in gravitational potential energy of the system
Work-Kinetic Energy Theorem
The net work done by all the forces acting on an object is equal to the change in the object's kinetic energy Speed will increase if work is positive. Speed will decrease if work is negative. Wnet = KEf - KEi = ΔKE
What if net torque isn't zero?
The object starts rotating at an ever-increasing rate
What if net torque is zero? Is it possible for a rotating object to experience zero net torque?
The object's rate of rotation doesn't change Yes! Because an object in motion will stay in motion unless acted on by a net force
Kepler's third law (and equation)
The square of the orbital period of a planet is directly proportional to the cube of the average distance from the Sun to the planet T^2 = 4(pi)^2/GM *r^3 Where M is the mass of the sun for the period of planets or comets, and M is the mass of Earth for the period of satellites
Mechanical energy
The sum of kinetic and potential energies
Why can we say that mechanical energy is conserved? Is it always conserved?
The sum of the kinetic energy and the potential energy remains constant at all times. Therefore, mechanical energy is a conserved quantity. Mechanical energy is conserved so long as we ignore air resistance, friction, etc. When we don't ignore outside forces, such as those just mentioned, mechanical energy is not conserved.
Tangential acceleration (voiceover)
The tangential acceleration of a point on a rotating object equals the distance of that point from the axis of rotation multiplied by the angular acceleration
Tangential speed (voiceover)
The tangential speed of a point on a rotating object equals the distance of that point from the axis of rotation multiplied by the angular speed)
What happens when two objects of equal mass undergo an elastic head-on collision?
The two objects will exchange velocities, almost as if they'd passed through each other. This is always the case when two objects of equal mass undergo an elastic head-on collision
In what direction is torque? How do we determine this? Is there a way to double check?
Torque is a vector, so it has direction! The direction of torque is perpendicular to the plane determined by the position vector and the force We figure out what the direction is using the right hand rule. Point your fingers in the direction of the position vector. Curl your fingers towards the force vector. The thumb will point in the direction of the torque. To double check, ask yourself in which direction torque has a tendency to rotate the object
Torque (voiceover) is...
Torque, t, is the tendency of a force to rotate an object about some axis
T/F: work always requires a system of more than just one object. Explain your choice
True! A nail can't do work on itself, but a hammer can do work on the nail by driving it into a board.
When the system is isolated and only conservative forces are present, 5 steps for determining energy:
USE CONSERVATION OF MECHANICAL ENERGY! WE LOVE CONSERVATIVE FORCES AND ISOLATED SYSTEMS. 1. Choose a level from which you will measure the PE 2. Write down the initial ME of the system 3. Sketch a final situation 4. Write down the final ME 5. Use conservation of energy to solve for whatever you want to solve for
When can we say that circular motion is uniform?
Uniform circular motion occurs when an object moves in a circular path with a constant speed
Linear momentum (equations, units)
Unit: kgm/s Equations: p = mv Fnet = ΔP/ΔT (this equation is valid when forces are constant AND not constant, provided the limit is infinitesimally small)
Impulse (definition, equations)
We know that Fnet = ΔP/Δt. From this, we know that if net force on an object is zero, the momentum doesn't change. From THIS, we know that changing an object's momentum requires the continuous application of a force over a period of time Δt. This is impulse: if a constant force F acts on an object, the impulse I delivered to the object over a time interval Δt is given by: I = FΔt Unit: kgm/s In order to change the momentum of an object, a force must be applied. The time rate of change of momentum of an object is equal to the net force acting upon it.
When is w positive? When is it negative?
We take w to be positive when θ is increasing (counterclockwise motion), and negative when θ is decreasing (clockwise motion)
When an object is lying flat, how do you determine torque?
Weight * xcg, where xcg is the value of r, the position vector (if the pivot is at one end of the object)
In the example of a stone being twirled in a circle on a string - what's an equation that could give us centripetal force?
Well you know that centripetal force is the force holding the object in a circle, so we can apply Fnet = mac Lucky us, we have an equation for ac! It's mv^2/r So Fnet = T = m*ac = m*v^2/r
What do we mean when we say that a physical quantity is conserved?
When a physical quantity is conserved, the numeric value of the quantity remains the same throughout the physical process. Although the form of the quantity may change in some way, its final value is the same as its initial value.
What remains the same throughout a rigid object?
When a rigid object rotates about a fixed axis, like a bicycle wheel, every portion of the object has the same angular speed and the same angular acceleration.
What can we know about impulse when a single constant force F acts on an object?
When a single constant force F acts on an object, we can say that: I = FΔt = Δp = mvf - mvi This is a special case of the impulse momentum theorem, showing that the impulse of the force acting on an object equals the change in momentum of that object (this equality is true even if the force is not constant, as long as the time interval is taken to be arbitrarily small)
What do we know when we are told that a single, constant force acts on an object?
When a single, constant force acts on an object, there is an impulse delivered to that object I = FΔt Unit = Ns = kgm/s
When is torque positive? When is torque negative?
When an applied force causes an object to rotate counterclockwise, the torque on the object is positive. When the force causes the object to rotate clockwise, the torque on the object is negative.
When is torque at a maximum?
When the angle between the position vector and the force vector is 90 degrees
When is the instantaneous angular speed equal to the average angular speed?
When the angular speed is constant (just like kinematics!)
Work energy theorem
Wnc + Wc = ΔKE The change in KE is equal to the sum of the work done by conservative and nonconservative forces
Equations related to the conservation of mechanical energy
Wnc + Wc = ΔKE Wc = -ΔPE Wnc = (KEf - KEi) + (PEf - PEi) = 0 KEi + PEi = KEf + PEf
Equation relating work by nonconservative forces, KE, PE When is this true?
Wnc = (KEf - KEi) + (PEf - PEi) This is true in general, even when other conservative forces besides gravity are present. The work done by these additional conservative forces will be recast as changes in PE and will appear on the right hand side along with the expression for gravitational potential energy
How to calculate work done by varying force, given a curve of displacement vs force?
Work done is equal to the area under the curve between xi and xf
When is work positive? Negative? Zero?
Work is a positive number if Fx and Δx are both positive or both negative, in which case the work increases the mechanical energy of the object. If Fx is positive and Δx is negative, or vice versa, the work is negative, and the object loses mechanical energy. Work is zero when the force exerted is perpendicular to the direction of motion (think of a man walking forward while carrying a bucket. The bucket is traveling horizontally while the upward force that his hand exerts is perpendicular to that, so no work is done. If he were to lift the bucket up a bit, then that upward force would count as work)
How do we determine whether work is positive or negative when the force is not parallel to an axis?
Work is positive or negative depending on whether cosθ is positive or negative. This, in turn, depends on the direction of F relative to the direction of Δx. When these vectors are pointing in the same direction, the angle between them is 0, and cos0 = +1, and the work is positive. Ex: when a student lifts a box, the work he does on the box is positive because the force he exerts on the box is upward, in the same direction as displacement. When vectors F and Δx are in opposite directions, the angle between them is 180, and cos180 = -1, so work is negative. Ex: the student puts the box back down, exerting an upward force to hold it up even as the displacement is in the negative direction. In general, when the part of F parallel to Δx points in the same direction as Δx, the work is positive; otherwise, it's negative
Does a car moving on a flat circular track have centripetal acceleration? Why or who not?
Yes - the force acting on it is friction between the car and the track (an object can have centripetal acceleration only if some external force acts on it)
Does the equation for work apply even when the constant force F is not parallel to the x-axis?
Yes - work is done only by the part of the force acting parallel to the object's direction of motion
Does a satellite in circular orbit around Earth have centripetal acceleration? Why or why not?
Yes- due to the gravitational force between the satellite and Earth
Equation for total acceleration of a particle moving on a curved path
a = ((at)^2 + (acp)^2)^(1/2)
Instantaneous angular acceleration
a = lim t-->0 Δw/Δt
Average angular acceleration
aav = Δw/Δt
Centripetal acceleration (equation)
acp or ar = v^2/r
Fictitious force + example
an apparent but non-existent force invented to explain the motion of objects within an accelerating (non-inertial) frame of reference Ex: it's hard to hold onto a merry go round and walk towards the center! Feels like there's a center-fleeing force pulling you back. In reality, the rider is exerting a centripetal force on his body with his hand and arm muscles. In addition, a smaller centripetal force is exerted by the static friction between his feet and the platform. If the rider's grip slipped, he wouldn't be flung radially away; rather, he would go off on a straight line, tangent to the point in space where he let go of the railing.
Equation relating g and G
g = GMe/r^2
Conservation of momentum (equation)
m1v1i + m2v2i = m1v1f + m2v2f
Units of angular acceleration
rad/s^2
Equation for displacement along an arc
s = θr
General definition of torque (define variables!)
t = rFsinθ t is torque F is the magnitude of the force r is the length of the vector from the pivot point to the point where the force acts (position vector) θ is the angle between the force and the position vector
Tangential (linear) speed
vt = wr
Instantaneous angular speed
w = lim as Δt --> 0 (Δθ/Δt)
How much potential energy is present in a relaxed spring?
x = 0, so no PEs
