Physics Test 2 - Chapters 6-8
Potential energy has increased when...?
If work is done but no kinetic energy is gained Ex. - if a force is applied to lift a crate, the gravitational potential energy has increased
A 0.5-kg ball has a velocity of 30 m/s (a) what is the kinetic energy of the ball? (b) how much work would be required to stop the ball?
(a) KE = (1/2)mv^2 KE = (1/2)(0.5)(30^2) KE = 225 J (b) -225 J
Newton's Second Law
- (Total change in velocity/ time) - Fnet = m * a = m *(delta v / delta t)
Two football players traveling at right angles to one another collide and stick together. What will be their direction of motion after the collision?
- Add the individual momentum vectors to get the total momentum of the system before the collision. - The final momentum of the two players stuck together is equal to the total initial momentum
If one pole-vaulter can run faster than another, will the faster runner have an advantage in the pole vault?
- Speed provides more kinetic energy for him to convert into potential energy - The height he is able to reach will depend on the vaulter's strength and ability to do work on his own body - A vaulter's skill at converting all of the kinetic energy into potential energy will help determine the height that he reaches
How is a lever an example of a simple machine?
- a small force applied to one end delivers a large force to the rock - a small force acting through a large distance moves the rock a small distance
How is a pulley an example of a simple machine?
- a small tension applied to one end delivers twice as much tension to lift the box - a small tension acting through a large distance moves the box a small distance
The resistance to change in rotational motion depends on
- the mass of the object - the square of the distance of the mass from the axis of rotation
What is the work done by the vertical component of force?
0 - the vertical component of force has a numerical value for force but does not follow the direction of motion (0 meter distance)
Linear Motion
An object moving from one point to another in a straight line Displacement = d V = D/T A = (delta V)/T
Conservation of Angular Momentum
Angular momentum is conserved if the net external torque acting on a system is zero angular momentum before = angular momentum after If T(net)= 0 L = constant = KE = (1/2)(Rotational Inertia: I)(Rotational Velocity: w)^2
Average Force
Average Force = (1/2)kx k = spring constant x = amount of compression or extension relative to the equilibrium position
Kinetic Energy
Definition: Energy associated with an object's motion - work done = change in kinetic energy Formula: KE = (1/2)mv^2
Momentum
Definition: Momentum is an object's mass multiplied by it's change in velocity Momentum (P) = m * (delta V)
Elastic Potential Energy
Definition: The energy gained when work is done to stretch a spring - an increase in elastic potential energy = work done = average force * distance Spring constant = k (# that describes the stiffness of a spring)
Power
Definition: The rate of doing work Formula: P: w*t Units: 1 watt (W) = 1 J/s
Impulse
Definition: average force acting on an object multiplied by the time interval over which the forces act Impulse = F * (delta T)
Conservative Forces
Definition: forces for which the energy can be completely recovered; the total energies of a system remain constant (Ex.- gravity and elastic forces)
Restoring Force
Definition: the force that exerts a push or a pull back towards equilibrium
Elastic Force
Definition: the force that results from stretching or compressing an object
Work
Definition: work is equal to the force applied times the distance moved - only forces parallel to the motion do work Formula: W=F*d Units: 1 Joule (J) = 1 Nm
A 100-kg fullback moving at 5 m/s downfield collides with a 75-kg defensive back moving 4 m/s upfield. The defensive back hangs on to the fullback, and the two players move together after the collision. What is the initial momentum of each player?
Fullback = m * (delta v) P = 100(5) P = 500 kg*m/s Defensive back = m * (delta v) P = 75(-4) P = -300 kg*m/s
Gravitational Potential Energy
Gravitational Potential Energy = mgh m - mass g - gravity h - height
Rotational Displacement
How far the unit rotates - measured in degrees or radians
Rotational Velocity
How fast a unit is turning - measured in revolutions per minute/ degrees per second
Two 0.2-kg masses are located at either end of a 1-m long, very light and rigid rod. What is the rotational inertia of this system about an axis through the center of the rod?
I = mr^2 I = (0.2)(0.5^2) I = 0.05 (for one side) I = 0.05*2 I = 0.10 Kg*m^2
Conservation of Momentum (Newton's Third Law)
If the net external force acting on an object is zero, the total momentum of a system is conserved
Potential Energy
Implies storing energy to use later for alternate purposes
Difference between impulse and momentum
Impulse = left side of manipulated equation Momentum (change in) = right side of manipulated equation Manipulated Equation = multiply both sides of Newton's Second Law by the time interval in which both forces act Fnet * (delta t) = m * (delta v)
When two masses are brought in closer to the student's body while spinning in a chair, his rotational velocity_______________ to compensate for the___________ in rotational inertia.
Increases; decreases
Rotational Motion
Involves an object rotating around an axis Displacement = theta V = (theta) / t = w Acceleration = (delta w) / t
Is the kinetic energy after 5, 20,000 kg railroad cars collide (v= 3 m/s) equal to the original kinetic energy of car 5 (m = 20,000, v = 15 m/s?
KE = (1/2)mv^2 KE = (1/2)(20,000)(15^2) KE = 2250 kJ (initial) KE = (1/2)(100,000)(3) KE = 450 kJ (final)
If the student flips the axis of the wheel, reversing the direction of its angular-momentum vector, what is the rotational velocity of the student and stool about their axis after the wheel is flipped? L = 60 kg*m^2/s I of wheel= 2 kg* m^2 I of wheel/student/platform = 6 kg*m^2
L = I*w --> w = L/I w = 2(60)/6 w= 120/6 w = 20 rad/s
A student sits on a stool holding a bicycle wheel with a rotational velocity of 5 rev/s about a vertical axis. The rotational inertia of the wheel is 2 kg·m2 about its center and the rotational inertia of the student and wheel and platform about the rotational axis of the platform is 6 kg·m2. What is the initial angular momentum of the system?
L= I*w L = (2)(5)(2pi) L = 62.8 which is approximately 60 kg*m^2/s (2 pi is included because it is equal to one full rotation)
Simple Machine
Multiplies the effect of an applied force (Examples: pulley, lever)
A ball is being twirled in a circle at the end of a string. The string provides the centripetal force needed to keep the ball moving in the circle at constant speed. Does the force exerted by the string on the ball do work on the ball?
No - the tension in the string acts perpendicular to the instantaneous direction of the motion
A man pushes very hard for several seconds on a rock, but the rock does not budge. Has the man preformed any work on the rock?
No work has been done on the rock because the rock did not move (distance = 0)
Does the normal force from the floor pushing upward on the block do any work?
No, the normal force is perpendicular to the block so it does no work- the vertical component
Partially Inelastic Collision
Objects collide but do not stick together - some energy is lost
Elastic Collision
Objects colliding and bouncing off of each other with no decrease in the magnitude of its velocity
Perfectly Inelastic Collision
Objects stick together after a collision - Energy is not conserved/ greatest portion of energy lost
Simple Harmonic Motion
Occurs when energy of a system repeatedly changes from potential energy to kinetic energy and back again
Four railroad cars, all with the same mass of 20,000 kg, sit on a track. A fifth car of identical mass approaches them with a velocity of 15 m/s. This car collides and couples with the other four cars. What is the initial momentum of the system?
P = m * v P = 20,000(15) P = 300,000 kg*m/s
Elastic Potential Energy in a Spring
PE (U) = (1/2)kx^2
The distance from the fulcrum to the point of application of the force must be measured in a direction _______________ to the line of action of the force
Perpendicular
Rotational Acceleration
Rate of change of rotational velocity - measured in revolutions per second per second / rotations per second per second Rotational Acceleration = Torque/ Rotational Inertia
A 3 N weight is placed against a 5 N weight on a beam. If the 5 N weight is placed 20 centimeters to the right of the fulcrum what is the torque produced by the 5 N weight? How far away must the 3 N weight from the fulcrum to balance the system?
T = F(l) T = - (5)(0.2) T = -1 Nm (the equation becomes negative because the force applied is to the right of the fulcrum) T = F(l) --> l = T/F l = +1/3 l = 0.33 m = 33 cm
Rotational Inertia / Moment of Inertia
Resistance of an object to change in its rotational motion -The rotational counterpart to mass Formula: I=mr^2 Rotational Inertia = mass * the square of the distance from the axis
Angular Momentum (Rotational Momentum)
Rotational Inertia * Rotational Velocity L = I*W
Two forces (80 N counter clock-wise and 50 N clockwise) are applied to a merry-go-round with a radius of 1.2 m. What is the torque about the axle of the merry-go-round due to the 80 N force? What is the torque about the axle due to the 50 N force?
T = F(l) T = +80(1.2) T = 96 Nm T = -50(1.2) T = -60 Nm
A 50N force is applied to the end of a wrench that is 24 cm long. The force applied is in the direction perpendicular to to the handle. What is the torque applied to the nut by the wrench? What would the torque be if the force was applied halfway up the handle?
T = F(l) T = 50 (0.24) T = 12 N*m T= 50 (0.12) T = 6 Nm
An 80-N plank is placed on a dock with 3 m between the left end of the plank and the pivot point on dock and 1 m between the right end of the dock and the pivot point. A 150-N boy standing on the plank walks out slowly from the edge of the dock. What is the torque exerted by the weight of the plank about the pivot point at the edge of the dock? How far from the edge of the dock can the 150 N boy walk until the plank is just on the verge of tipping?
T = F(l) T = 80(1) T = +80 Nm T = F(l) --> l= T/F l = 80/150 l = 0.53 m
At the low point in its swing, a pendulum bob with a mass of 0.5 kg has a velocity of 7 m/s. The kinetic energy at the low point is 12.3 J. If you ignore air resistance, how high will the bob swing above the low point before reversing direction?
TE = PE + KE 12.3 = mgh 12.3 = 0.5(9.8)h 12.3 = (4.9)h h = 2.5 m
A box is moved from the floor up to a table top but gains no speed in the process. Is there work done on the box? If so what has happened to the energy added to the system?
Yes - work is done on the box to move it the height of the table ; the work has gone into increasing the potential energy of the block-earth system
A sports car accelerates rapidly from a stop and "burns rubber." What kind of energy transformations occur in this situation?
The energy from burning the fuel is transfromed into kinetic energy for the car to move. Some of the kinetic energy of the car is transformed into heat due to the friction with the road and some is also used to burn the rubber - total energy is conserved
How does a rocket accelerate in an empty space when there is nothing to push off of?
The exhaust gases rushing out of the tail of the rocket have both mass and velocity and, therefore, momentum
Impulse - Momentum Principle
The impulse acting on an object produces a change in momentum of object that is equal to both magnitude and direction of the impulse impulse = change in momentum = (delta P)
How is the torque found if the force applied is not perpendicular to the lever arm?
The lever arm is found by drawing a line perpendicular to the fulcrum to the line of action of the force
The distance used to calculate Torque is called
The lever arm or moment arm -Distance is measured in meters
If two skaters of different masses push off of each other from rest, why will the less massive one gain a higher velocity?
The net external force acting on the system is zero so conservation of momentum applies. Before the push off the initial total momentum was zero; after the push of the final total momentum should also be zero
Newton's Second Law for Rotational Motion
The net torque acting on an object about a given axis is equal to the rotational inertia of the object about that axis times the rotational acceleration of the object. Formula: T(net)= I*a
Torque
The product of all the force and the distance from the fulcrum T= F(l) Units: N*m - tendency of weight to preduce rotation
A lever is used to lift a rock. Will the work done by the person on the lever be greater than, less than, or equal to the work done by the lever on the rock?
The work by done by the person can never be less than the work done on the rock. If there are no dissipative forces, then the forces will be equal (conservative forces)
Negative Torque
Torques that produce clockwise rotations
Positive Torque
Torques that produce counter- clockwise rotations
A horizontally directed force of 32 N is used to pull a box a distance of 2.6 m across a tabletop. How much work is done by the 32 N-force?
W = F * d W = 32(2.6) W = 83.2 J
A force of 50 N is used to drag a 4 meters across the floor. The force is directed at an angle perpendicular to the crate is 30 N, diagonal to the crate is 50 N, and horizontal to the crate is 40 N. What is the work done by the horizontal component of force?
W = F*d W = (40 N)*(4 M) W = 160 Nm or 160 J
A string is used to pull a wooden block across the floor without accelerating the block. The string makes an angle to the horizontal. Does the force applied via the string do work on the block?
Yes but only the horizontal part of the force does work - the horizontal component
Kepler's Second Law states that the radius line from the sun to the planet sweeps out equal areas in equal time, meaning that when the planet is closer to the sun in it's elliptical orbit, the planet is moving faster. How does conservation of angular momentum affect this?
When the planet moves nearer to the sun, its rotational inertia about the sun decreases. To conserve angular momentum, the rotational velocity of the planet about the sun must increase.
Work is done on a large crate to tilt the crate so that it is balanced on one edge, rather than sitting squarely in the floor as it was before. Has the potential energy of the crate increased?
Yes - The center of the crate has been lifted slightly (work). If it is released it will fall back, converting the potential energy to kinetic energy, but because it is balancing, the energy remains potential
Can a bowling ball and a tennis ball have the same momentum?
Yes - if a tennis ball with its smaller mass has a much larger velocity
When a bow and arrow are cocked, a force is applied to the string to pull it back. Is the energy of the system increased?
Yes - the work in cocking the bow and arrow has gone into elastic potential energy to the bow
If there is a frictional force opposing the motion of a block, does the frictional force do work on the block?
Yes but since the frictional force is opposing the motion of the block it does negative work on the block
How do you locate an object's center of gravity?
by finding the point where an object balances on a fulcrum for a more complex object, suspend the object from two points, drawing a line straight down from the point of suspension of each point, and finding the point of intersection between the two lines
Doing work on an object _____________ its kinetic energy
increases
Depict a graph where the horizontal position x of the mass on the spring is plotted against time as the mass moves back and forth
period (t) - time taken to complete one cycle frequency (f) - number of cycles per unit of time amplitude - the maximum distance from equilibrium
Work done in pulling a sled up a hill produces an increase in ___________ energy of the sled rider. This initial energy is converted into ___________ energy as they slide down the hill
potential; kinetic
Center of Gravity of an Object
the point about the weight of an object itself exerts no torque
A merry-go-round is accelerated at a constant rate of 0.005 rev/s^2, starting from rest. What is the rotational velocity at the end of one minute? How manyrevolutions has the merry-go-round made after one minute?
w = w(knot) + a(t) w = 0 + (0.005)(60) w = 0 + 0.30 w = 0.3 rev/s Theta = W(knot)*T + (1/2)(a)(t^2) Theta = (0)(60) + (1/2)(0.005)(60^2) Theta = 9 revs
Recoil
when a brief force between two object causes the object to move in opposite directions