physics final ***

2023

EEK (5 questions)

2023 QUESTION 1

A cart on a horizontal surface is attached to a spring. The other end of the spring is attached to a wall. The cart is initially held at rest, as shown in Figure 1. When the cart is released, the system consisting of the cart and spring oscillates between the positions x = + L and x = −L. Figure 2 shows the kinetic energy of the cart-spring system as a function of the system’s potential energy. Frictional forces are negligible.

2023 1A. On the graph of kinetic energy K versus potential energy U shown in Figure 2, the values for the x-intercept and y-intercept are the same. Briefly explain why this is true, using physics principles.

the maximum kinetic energy and maximum potential energy are the same due to energy conservation

2023
When the cart is at +L and momentarily at rest, a block is dropped onto the cart, as shown in Figure 3. The block sticks to the cart, and the block-cart-spring system continues to oscillate between −L and +L. The masses of the cart and the block are m0 and 3m0, respectively.

1B. The frequency of oscillation before the block is dropped onto the cart is f1 . The frequency of oscillation after the block is dropped onto the cart is f2. Calculate the numerical value of the ratio f2/f1.

f1 = (1/2π) (√k/m)

f2 = (1/2π) (√k/4m) =(1/4π) (√k/m)

f2/f1 = ((1/4π) (√k/m)) / ((1/2π) (√k/m)) = 1/2

2023 1ci

The dashed line in Figure 4 shows the kinetic energy K versus potential energy U of the block-cart-spring system after the block is dropped onto the cart. This graph is identical to the graph shown in Figure 2 for the cart-spring system before the block is dropped onto the cart. Explain why these two are the same.

No work is done on the system

The maximum spring potential energy is the same (spring stretched same distance)

The force exerted on the system by block is perpendicular to the direction of motion

2023 1cii

After the block is dropped onto the cart, consider a system that consists only of the cart and the spring. On Figure 4, sketch a solid line that shows the kinetic energy of the system that consists of the cart and the spring but not the block after the block is dropped onto the cart

2023 QUESTION 2

Students conduct an experiment to determine the acceleration a of a cart. The cart is released from rest at the top of the ramp at time t = 0 and moves down the ramp. The x-axis is defined to be parallel to the ramp with its origin at the top, as shown in the figure. The students collect the data shown in the following table.

2023 2ai
Indicate which quantities could be graphed to yield a straight line whose slope could be used to determine the acceleration a of the cart. You may use the remaining columns in the table, as needed, to record any quantities (including units) that are not already in the table.

position vs time squared

2023 2aii

On the following grid, plot the appropriate quantities to create a graph that can be used to determine the acceleration a of the cart as it rolls down the ramp. Clearly scale and label all axes (including units), as appropriate. Draw a straight line that best represents the data.

2023 2aiii

Using the line you drew in part (a)(ii), calculate an experimental value for the acceleration a of the cart as it rolls down the ramp

find slope of graph (0.75)

2023 2bi

The students are asked to determine an experimental value for the acceleration due to gravity gexp using their data.

What additional quantities do the students need to measure in order to calculate gexp from a

Accept one of the following:

-The angle θ with the horizontal

- The height h and length L of the ramp

Scoring Note: Stating only the height needs to be measured can earn this point if an energy approach is used.

2023 2bii

Write an expression for the value of gexp in terms of a.

Accept one of the following:

• The angle θ with the horizontal

• The height h and length L of the ramp

Scoring Note: Stating only the height needs to be measured can earn this point if an energy approach is used.

2023 2ci.

The students calculate the value of gexp to be significantly lower than the accepted value of 9.8 m/s².

What is a physical reason, other than friction or air resistance, that could lead to a significant difference in the experimentally determined value of gexp

-A physical factor in the materials used (e.g., the wheels have nonnegligible rotational inertia, the ramp was bumpy, the wheels were wobbly or not perfectly round, the base of the ramp was not level, the floor was not level.)

-A physical factor in the environment (e.g., the room was being accelerated, elevator, the experiment was performed at high elevation or on a different planet.)

-A physical error in measurement collection (e.g., time, position, or angle was measured incorrectly.)

2023 2cii

Briefly explain how the physical reason you identified in part (c)(i) would lead to the decrease in the experimentally determined value of gexp

-Correctly indicating the functional dependence between the physical factor in the materials used and exp g (e.g., if the rotational inertia of the rotating wheels is nonnegligible, the cart will have a smaller acceleration and exp g will be smaller.)

• Correctly indicating the functional dependence between the physical factor in the environment and exp g (e.g., if the experiment was performed at a high elevation, the acceleration will be smaller and exp g will be smaller.)

• Correctly indicating the functional dependence between the physical error in the measurement collection and exp g (e.g., if the angle of the ramp is smaller than the measured value, the cart will have a smaller acceleration and exp g will be smaller.)

2023 2d
The students want to confirm that the acceleration is the same whether the cart rolls up or down the ramp. The students start the cart at the bottom and give the cart a quick push so that it rolls up the ramp and momentarily comes to rest. The x-axis is still defined to be parallel to the ramp with the origin at the top.

(d) On the following graphs, sketch the position x and velocity v as functions of time t that correspond to the scenario shown while the cart moves up the ramp. (graph is x vs t, and v vs t w neg x axis visible))

2023 QUESTION 3
A small block of mass m0 is attached to the end of a spring of spring constant k0 that is attached to a rod on a horizontal table. The rod is attached to a motor so that the rod can rotate at various speeds about its axis. When the rod is not rotating, the block is at rest and the spring is at its unstretched length L, as shown. All frictional forces are negligible.

2023 3ai

At time t = t 1, the rod is spinning such that the block moves in a circular path with a constant tangential speed v1 and the spring is stretched a distance d1 from the spring’s unstretched length, as shown in Figure . 1 At time t = t2, the rod is spinning such that the block moves in a circular path with a constant tangential speed v2 and the spring is stretched a distance d2 from the spring’s unstretched length, where d > d12 , as shown in Figure 2.

On the following dots, which represent the block at the locations shown in Figure 1 and Figure 2, draw the force that is exerted on the block by the spring at times t = t 1 and t = t2. The spring force must be represented by a distinct arrow starting on, and pointing away from, the dot. Note: Draw the relative lengths of the vectors to reflect the relative magnitudes of the forces exerted by the spring at both times.

both arrows rightward, with the one at t2 being longer.

2023 3aii

Referencing d1 and d2, describe your reasoning for drawing the arrows the length that you did in part (a)(i).

spring stretched farther = more force

2023 3aiii

i. Is the tangential speed v1 of the block at time t = t 1 greater than, less than, or equal to the tangential speed v2 of the block at time t = t2 ?

_____ v1 > v2 _____ v1 < v2 _____ v1 = v2

Justify your answer without using equations.

v1 < v2

the spring force is the net force

the net force is related to the speed

f=kx

2023 3bi

Consider a scenario where the block travels in a circular path where the spring is stretched a distance d from its unstretched length L

Determine an expression for the magnitude of the net force Fnet exerted on the block. Express your answer in terms of m0, k0, L, d, and fundamental constants, as appropriate

F_net= k_od

2023 3bii.

Derive an equation for the tangential speed v of the block. Express your answer in terms of m0, k0, L, d, and fundamental constants, as appropriate.

F = ma

kx = ma

kx = m(v²/r)

k_od = (mv²)/(L+d)

v= √((k_od)(L+d))/m)

2023 3c

Does your equation for the tangential speed v of the block from part (b)(ii) agree with your reasoning from part (a)

Yes. d directly proportional to v. more stretched distance d = more tangential velocity v.

2023 QUESTION 4
A block of unknown mass is attached to a long, lightweight string that is wrapped several turns around a pulleymounted on a horizontal axis through its center, as shown. The pulley is a uniform solid disk of mass M and radius R. The rotational inertia of the pulley is described by the equation 1 2 I = MR 2 . The pulley can rotate about its center with negligible friction. The string does not slip on the pulley as the block falls. When the block is released from rest and as the block travels toward the ground, the magnitude of the tension exerted on the block by the string is FT.

2023 4a

Determine an expression for the magnitude of the angular acceleration aD of the disk as the block travels downward. Express your answer in terms of M , R, FT, and physical constants as appropriate.

Torque = (Inertia)(angular acceleration)

rF=½ MR²a

a = T_net/I

_{a = (rFsinø)/(MR²/2)}

_{a=(2Fsinø)/(MR)}

_{a=(2Fsin(90))/(MR)}

_{a= (2F)/(MR)}

2023 4b

Consider scenarios 1 and 2 at the end of time interval Δt. In a clear, coherent paragraph-length response that may also contain equations and drawings, explain why the change in angular momentum of both pulleys is the same but the change in rotational kinetic energy is greater for the disk.

torque is same for both pulleys

impulse / change in momentum are the same bc torque and t are same (T△t)

Rotational inertia (I) is different for disk and hoop

I different but momentum (Iw) same = different w / a / ø

greater displacement ø = more work

greater w or a = greater kinetic energy

2023 QUESTION 5
A rod with a sphere attached to the end is connected to a horizontal mounted axle and carefully balanced so that it rests in a position vertically upward from the axle. The center of mass of the rod-sphere system is indicated with a ƒ, as shown in Figure 1. The sphere is lightly tapped, and the rod-sphere system rotates clockwise with negligible friction about the axle due to the gravitational force.

A student takes a video of the rod rotating from the vertically upward position to the vertically downward position. Figure 2 shows five frames (still shots) that the student selected from the video. Note: these frames are not equally spaced apart in time.

2023 5ai

In which frame is the angular acceleration of the rod-sphere system the greatest? Justify your answer.

Frame C

-This is the instant when the lever arm is greatest.

-This is when the angle between radius vector and weight force vector is most perpendicular.

2023 5aii

In which frame is the rotational kinetic energy of the rod-sphere system the greatest? Briefly justify your answer

Frame E

• Work or energy (e.g., this is when the maximum work has been done on the system by gravity.)

• Angular momentum (e.g., the torque due to gravity is clockwise the entire time, causing the rod to gain angular momentum.)

• Kinematics (e.g., the rod speeds up the entire time.)

2023 5bi

Derive an expression for the change in kinetic energy of the rod-sphere-Earth system from the moment shown in Frame A to the moment shown in Frame E. Express your answer in terms of M , L, and fundamental constants, as appropriate.

Ei = Ef / Ugi+Ki = Ugf +Kf

y=3L/2

Kf = Ug = 3MgL/2

2023 5bii

Briefly explain why the rod and sphere gain kinetic energy, even if Earth is not included in the system.

earth not in system, then gravitational force is external = does work = change in kinetic energy

2022

SOIUP)(DGIJONKM WHHBIOUWGJO:KLMS

2022 QUESTION 1

Two blocks are connected by a string that passes over a pulley, as shown above. Block 1 is on a horizontal surface and is attached to a spring that is at its unstretched length. Frictional forces are negligible in the pulley’s axle and between the block and the surface. Block 2 is released from rest and moves downward before momentarily coming to rest.

k0 is the spring constant of the spring. M1 is the mass of block 1. M2 is the mass of block 2. Δy is the distance block 2 moves before momentarily coming to rest

2022 1ai

Block 2 starts from rest and speeds up, then it slows down and momentarily comes to rest at a position below its initial position. In terms of only the forces directly exerted on block 2, explain why block 2 initially speeds up and explain why it slows down to a momentary stop.

The direction of the acceleration and the direction of the net force are the same. The block speeds up when the gravitational force is greater than the tension and then slows down because the tension becomes larger than the gravitational force.

2022 1aii

Derive an expression for the distance Δy that block 2 travels before momentarily coming to rest. Express your answer in terms of k0, M1, M2, and physical constants, as appropriate.

y = 2mg / k

2022 1b

Indicate whether the total mechanical energy of the blocks-spring-Earth system changes as block 2 moves downward. ___ Changes ___ Does not change Briefly explain your reasoning.

Does not change

No external forces, so mech energy stays the same

2022 1c

Consider the system that includes the spring, Earth, both blocks, and the string, but not the surface. Let the initial state be when the blocks are at rest just before they start moving, and let the final state be when the blocks first come momentarily to rest. Diagram A at left below is a bar chart that represents the energies in the scenario where there is negligible friction between block 1 and the surface. The shaded-in bars in the energy bar charts represent the potential energy of the spring and the gravitational potential energy of the blocks-Earth system, Us and Ug, respectively, in the initial and final states. Positive energy values are above the zero-point line (“0”) and negative energy values are below the zero-point line.

Complete diagram B (at right above) for the scenario in which friction is nonnegligible. The energies for the initial state are already provided. Shade in the energies in the final state using the same scale as in diagram A.

Shaded regions should start at the solid line representing the zero-point line. Represent any energy that is equal to zero with a distinct line on the zero-point line

2022 QUESTION 2

Two identical moons, Moon A and Moon B, orbit a planet. The mass m0 of each moon is significant, but less than the mass mp of the planet. At some point in their orbits, the planet and the two moons are aligned as shown in the figure.

2022 2a

The following dots represent the two moons when they are at the locations shown in the previous figure. On each dot, draw and label the forces (not components) exerted on Moon A and on Moon B. Each force must be represented by a distinct arrow starting on, and pointing away from, the appropriate dot.

2022 2bi

Consider the net gravitational force exerted on each moon due to the planet and the other moon. i. Justify why the magnitude of the net force exerted on Moon A could be larger than the magnitude of the net force exerted on Moon B

For indicating the force vectors on Moon A point in the same direction and therefore, the magnitudes add, while the force vectors on Moon B point in opposite directions and the magnitudes are to be subtracted

2022 2bii

Justify why the magnitude of the net force exerted on Moon B could be larger than the magnitude of the net force exerted on Moon A.

2022 2c

Derive expressions for both of the following quantities. Express your answers in terms of m0, mp, RA, RB, and physical constants, as appropriate.

-The net force FA exerted on Moon A

-The net force FB exerted on Moon B

**memorize

2022 2di

Could the expressions in part (c) support your reasoning in part (b)(i) ? ___ Yes ___ No Explain your reasoning

Yes. For the net force on Moon A, both force terms have the same sign, so they add, while for the net force on Moon B, the two terms have opposite signs, so they have a canceling effect

2022 2dii

Could the expressions in part (c) support your reasoning in part (b)(ii) ? ___ Yes ___ No Explain your reasoning

The gravitational force has an inverse relationship with distance. If Moon A is very far away, net FA from part (c) will be small. If Moon B is close to the planet while Moon A is far away, the force toward the planet would be big while the force toward the moon would be small such that netFB could be larger than FA .

2022 QUESTION 3

A wheel is mounted on a horizontal axle. A light string is attached to the wheel’s rim and wrapped around it several times, and a small block is attached to the free end of the string, as shown in the figure. When the block is released from rest and begins to fall, the wheel begins to rotate with negligible friction. Two students are discussing how different forms of energy change as the block falls. One student says that the kinetic energy of the block increases as it falls. The second student says that this is because gravitational potential energy is converted to kinetic energy. The students decide to test whether the decrease in gravitational potential energy is equal to the increase in the block’s kinetic energy from when the block starts moving to immediately before it reaches the floor.

2022 3a

Design an experimental procedure that the students could use to compare the increase in the block’s translational kinetic energy with the decrease in the gravitational potential energy of the block-Earth system as the block falls. In the table, list the quantities that would be measured in your experiment. Define a symbol to represent each quantity and list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table. In the space to the right of the table, describe the overall procedure. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table. If needed, you may include a simple diagram of the setup with your procedure

2022 3b

Explain how the students could determine the kinetic energy of the block immediately before it reaches the floor using the quantities you indicated in the table in part (a).

2022 3c

The graph above represents both the change in the gravitational potential energy of the block-wheel-Earth system and the translational kinetic energy gained by the block as functions of the block’s falling distance d. On the graph, draw a line or curve to represent the rotational kinetic energy of the wheel as a function of the block’s falling distance d.

2022 3di

The students also measure the angular velocity w of the wheel as the block falls and determine the rotational kinetic energy KR of the wheel. The students then make a graph of KR as a function of w², as shown.

i. On the above graph, draw a straight line that best represents the data

2022 3dii

Using the line you drew for part (d)(i), calculate an experimental value for the rotational inertia of the wheel.

2022 QUESTION 4

A student has a piece of clay and a rubber sphere, both of the same mass. Both objects are thrown horizontally at the same speed at identical blocks that are at rest at the edge of identical tables, as shown, where friction between the blocks and the table is negligible. After the collisions, both blocks fall to the floor.

In Case A, the clay sticks to Block A after the collision. In Case B, the rubber sphere bounces off of Block B after the collision.

2022 4a

In the figure at left above, the arrow represents the momentum immediately after the collision for the clay-block system in Case A. In the figure at right above, draw an arrow starting on the dot to represent the momentum of the sphere-block system immediately after the collision in Case B. If the momentum is zero, write “zero” next to the dot. The momentum, if it is not zero, must be represented by an arrow starting on, and pointing away from, the dot. The length of the vector, if not zero, should reflect the magnitude of the momentum relative to Case A.

2022 4b

After the clay and Block A collide, Block A lands a horizontal distance dA from the edge of the table. Does Block B land on the floor at a horizontal distance from the edge of the table that is greater than, less than, or equal to dA ? In a clear, coherent, paragraph-length response that may also contain equations and/or drawings, explain your reasoning. Neglect any frictional effects due to the table or air resistance.

Block b travels a greater distance than A

The momentum of the clay-block and sphere-block systems before the collision is the same for both cases and because momentum does not change in the collision; it is the same after the collision also. The sphere in Case B bounces off the block, so it has less (or negative) momentum after the collision than the clay in Case A. In order for the systems in both cases to have the same momentum after the collision, Block B must have greater momentum, and therefore greater speed, than Block A. The blocks take the same amount of time to fall, so the horizontal distance traveled by Block B (launch speed x time to fall) is greater than dA .

2022 QUESTION 5

A spring of unknown spring constant k0 is attached to a ceiling. A lightweight hanger is attached to the lower end of the spring, and a motion detector is placed on the floor facing upward directly under the hanger, as shown in the figure above. The bottom of the hanger is 1.00 m above the motion detector. A 0.50 kg object is placed on the hanger and allowed to come to rest at the equilibrium position. The spring is then stretched downward a distance d0 from equilibrium and released at time t = 0. The motion detector records the height of the bottom of the hanger as a function of time. The output from the motion detector is shown in the graph on the following page.

2022 5a
Using the information given and information taken from the graph, caluclate the spring constant.

around 12.6 N/m

2022 5bi

At time 0.75 s, the object-spring-Earth system has a total kinetic energy K0 and a total potential energy U0. At 1.13 s, the object-spring-Earth system again has a total kinetic energy K0 and a total potential energy U0.

i. Explain how a feature of the graph indicates that the total kinetic energy of the system is the same at these two times.

The magnitude of the slope of the graph is the same at both times, this means the speed and, therefore, the kinetic energy is the same at both times. OR The object is the same distance from equilibrium at both times, so the kinetic energy must be the same.

2022 5bii

ii. Briefly explain why the total potential energy of the system is the same at these two times

The total energy of the system is constant, so if K is the same at both times, U must be also. OR The total energy of the system is constant, and equal energy is transferred from gravitational potential to spring potential

2022 5ci

) The experiment is repeated with a spring of spring constant 4k0 and that has the same length as the original spring. The 0.50 kg object is hung from the new spring and allowed to come to rest at a new equilibrium position. i. Determine the new equilibrium position above the motion detector.

90 cm / 0.90 m

2022 5cii
ii. The object is again pulled down the same distance d0 from the equilibrium position and released. On the following graph, draw a curve representing the motion of the object after it is released. Label the vertical axis with an appropriate numerical scale. A grid for scratch (practice) work is also provided

2021

ACK!! (5 questions)

2021 QUESTION 1
A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parked cars, as shown above. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp, jumps over six cars, and lands on a second ramp.

H0 is the vertical distance between the top of the first ramp and the launch point. q0 is the angle of the ramp at the launch point from the horizontal. X0 is the horizontal distance traveled while the cyclist and bicycle are in the air. m0 is the combined mass of the stunt cyclist and bicycle.

2021 1a.

Derive an expression for the distance X0 in terms of H0, q0, m0, and physical constants, as appropriate

Ei = Ef (conservation of energy)
mgh = ½ mv²

v = √(2gh)

vertical:

x=1/2 at²+ vsinøt

0=1/2 (g)t+ vsinø

t= -2vsinø/-g

t= (2√(2gh)sinø)/g

horizontal:

x=vt

x=(cosø√2gh)(2sinø(√2gh))/g

x=4sinøcosøgh/g

x= =4hsinøcosø

Using the range equation to get X= 2Hsin(2θ) is sufficient to earn the second and third points.

2021 1b.

If the vertical distance between the top of the first ramp and the launch point were 2H0 instead of H0, with no other changes to the first ramp, what is the maximum number of cars that the stunt cyclist could jump over? Justify your answer, using the expression you derived in part (a).

12 cars

x directly proportional to h using equation

initial 6 × 2 = 12

2021 1c.

On the axes below, sketch a graph of the vertical component of the stunt cyclist’s velocity as a function of time from immediately after the cyclist leaves the ramp to immediately before the cyclist lands on the second ramp. On the vertical axis, clearly indicate the initial and final vertical velocity components in terms of H0, q0, m0, and physical constants, as appropriate. Take the positive direction to be upward.

2021 QUESTION 2
A group of students is investigating how the thickness of a plastic rod affects the maximum force Fmax with which the rod can be pulled without breaking. Two students are discussing models to represent how Fmax depends on rod thickness. Student A claims that Fmax is directly proportional to the radius of the rod. Student B claims that Fmax is directly proportional to the cross-sectional area of the rod—the area of the base of the cylinder, shaded gray in the figure above.

2021 2a.
The students have a collection of many rods of the same material. The rods are all the same length but come in a range of six different thicknesses. Design an experimental procedure to determine which student’s model, if either, correctly represents how Fmax depends on rod thickness.

In the table below, list the quantities that would be measured in your experiment. Define a symbol to represent each quantity, and also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.
Describe the overall procedure to be used, referring to the table. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup.

For measuring the radius or diameter of rods with different radii using an appropriate tool

For measuring force using an appropriate tool

For a plausible/practical way to directly or indirectly determine Fmax for a given rod

For attempting to reduce experimental uncertainty in an experiment that involves breaking the rods

use force probe and pull on rod with measured radius until it breaks

2021 2b.

For a rod of radius r0, it is determined that Fmax is F0, as indicated by the dot on the grid below. On the grid, draw and label graphs corresponding to the two students’ models of the dependence of Fmax on rod radius. Clearly label each graph “A” or “B,” corresponding to the appropriate model.

2021 2c

The table below shows results of measurements taken by another group of students for rods of different thicknesses.

On the grid below, plot the data points from the table. Clearly scale and label all axes, including units. Draw either a straight line or a curve that best represents the data.

2021 2d. Which student’s model is more closely represented by the evidence shown in the graph you drew in part (c) ?

____ Student A’s model: Fmax is directly proportional to the radius of the rod.

____ Student B’s model: Fmax is directly proportional to the cross-sectional area of the rod. Explain your reasoning.

Student B’s model: Fmax is directly proportional to the cross-sectional area of the rod. Explain your reasoning.

F max proportional to R² and inc as R² inc.

2021 QUESTION 3

A student of mass MS, standing on a smooth surface, uses a stick to push a disk of mass MD. The student exerts a constant horizontal force of magnitude FH over the time interval from time t =0 to t = tf while pushing the disk. Assume there is negligible friction between the disk and the surface.

2021 3ai.)

i.A student of mass MS, standing on a smooth surface, uses a stick to push a disk of mass MD. The student exerts a constant horizontal force of magnitude FH over the time interval from time t -

0 to t - tf while pushing the disk. Assume there is negligible friction between the disk and the surfac Assuming the disk begins at rest, determine an expression for the final speed vD of the disk relative to the surface. Express your answer in terms of FH, tf , MS, MD, and physical constants, as appropriate.

F=ma

F= (m)(v/t)

v_D= Ft/M

2021 3aii.
Assume there is negligible friction between the student’s shoes and the surface. After time tf , the student slides with speed vS. Derive an equation for the ratio vD / vS. Express your answer in terms of MS, MD, and physical constants, as appropriate

momentum of system same before and after collision

pi = pf

m_sv_s= m_dv_d

vd / vs = ms/md

2021 3b.

Assume that the student’s mass is greater than that of the disk (MS > MD). On the grid below, sketch graphs of the speeds of both the student and the disk as functions of time t between t = 0 and t = 2tf. Assume that neither the disk nor the student collides with anything after t = tf . On the vertical axis, label vD and vS. Label the graphs “S” and “D” for the student and the disk, respectively

0-tf: 2 straight lines with positive slope (slope D > slope S) that begin at origin

tf-2tf: straight horizontal lines continuining from before

all values of D > S

2021 3ci.

The disk is now moving at a constant speed v1 on the surface toward a block of mass MB, which is at rest on the surface, as shown above. The disk and block collide head-on and stick together, and the center of mass of the disk-block system moves with speed vcm. i. Suppose the mass of the disk is much greater than the mass of the block. Estimate the velocity of the center of mass of the disk-block system. Explain how you arrived at your prediction without deriving it mathematically.

For stating or mathematically representing that if the disk is much more massive, then the block will have little effect on the motion of disk 1

For stating or mathematically representing that when M D >> M B , cm v = 1 v

2021 3cii.
Suppose the mass of the disk is much less than the mass of the block. Estimate the velocity of the center of mass of the disk-block system. Explain how you arrived at your prediction without deriving it mathematically.

When M D << M B , v cm = 0

block much more massive, then will barely move when disk collides

2021 3ciii.

Now suppose that neither object’s mass is much greater than the other but that they are not necessarily equal. Derive an equation for vcm. Express your answer in terms of v1, MD, MB, and physical constants, as appropriate.

MDv1 = (MD+MB)(vcm)

vcm = (MDv1)/ (MD+MB)

2021 3civ.

Consider the scenario from part (c)(i), where the mass of the disk was much greater than the mass of the block. Does your equation for vcm from part (c)(iii) agree with your reasoning from part (c)(i) ?

____ Yes ____ No

Explain your reasoning by addressing why, according to your equation, vcm becomes (or approaches) a certain value when MD is much greater than MB

Yes.

If MB very negligble and small, MD cancels with MD and leaves vcm = v1

2021 QUESTION 4

A cylinder of mass m0 is placed at the top of an incline of length L0 and height H0, as shown above, and released from rest. The cylinder rolls without slipping down the incline and then continues rolling along a horizontal surface.

2021 4a.
(a) On the grid below, sketch a graph that represents the total kinetic energy of the cylinder as a function of the distance traveled by the cylinder as it rolls down the incline and continues to roll across the horizontal surface.

2021 4b.
The cylinder is again placed at the top of the incline. A block, also of mass m0, is placed at the top of a separate rough incline of length L0 and height H0, as shown above. When the cylinder and block are released at the same instant, the cylinder begins to roll without slipping while the block begins to accelerate uniformly. The cylinder and the block reach the bottoms of their respective inclines with the same translational speed.

(b) In terms of energy, explain why the two objects reach the bottom of their respective inclines with the same translational speed. Provide your answer in a clear, coherent paragraph-length response that may also contain figures and/or equations.

-both objects start with the same potential energy

-block: accelerates and turns potential to kinetic, but some is lost to friction

-cylinder: rolls, potential is transformed into both translational and rotational KE

-amount of translational KE lost by cylinder to rotationial KE = KE lost to friction by block

2021 QUESTION 5

Two pulleys with different radii are attached to each other so that they rotate together about a horizontal axle through their common center. There is negligible friction in the axle. Object 1 hangs from a light string wrapped around the larger pulley, while object 2 hangs from another light string wrapped around the smaller pulley, as shown in the figure above.

m0 is the mass of object 1. 1.5m0 is the mass of object 2. r0 is the radius of the smaller pulley. 2r0 is the radius of the larger pulley

2021 5ai.

At time t = 0, the pulleys are released from rest and the objects begin to accelerate. i. Derive an expression for the magnitude of the net torque exerted on the objects-pulleys system about the axle after the pulleys are released. Express your answer in terms of m0, r0, and physical constants, as appropriate.

Object 1: 1.5mgr

Object 2: mg2r

(opposite directions so)

net torque = mg2r -1.5mgr

0.5mgr

2021 5aii.

Object 1 accelerates downward after the pulleys are released. Briefly explain why

object 1 gives a greater torque bc its twice as far, while 2 is 1.5 the mass which is < 2.

2021 5b.

At a later time t = t C, the string of object 1 is cut while the objects are still moving and the pulley is still rotating. Immediately after the string is cut, how do the directions of the angular velocity and angular acceleration of the pulley compare to each other?

Same direction

Opposite directions

Briefly explain your reasoning.

opposite directions

wo object 1, torque switches direction from before and thus so those acceleration

velocity remains prior direction bc needs time to switch, not immediately after

2021 5c.

On the axes below, sketch a graph of the angular velocity w of the system consisting of the two pulleys as a function of time t. Include the entire time interval shown. The pulleys are released at t = 0, and the string is cut at t = t C.

2019

EEP!!!

2019 QUESTION 1

Identical blocks 1 and 2 are placed on a horizontal surface at points A and E, respectively, as shown. The surface is frictionless except for the region between points C and D, where the surface is rough. Beginning at time At , block 1 is pushed with a constant horizontal force from point A to point B by a mechanical plunger. Upon reaching point B, block 1 loses contact with the plunger and continues moving to the right along the horizontal surface toward block 2. Block 1 collides with and sticks to block 2 at point E, after which the two-block system continues moving across the surface, eventually passing point F.

2019 1a.

On the axes below, sketch the speed of the center of mass of the two-block system as a function of time, from time At until the blocks pass point F at time Ft . The times at which block 1 reaches points A through F are indicated on the time axis.

2019 1b.

The plunger is returned to its original position, and both blocks are removed. A uniform solid sphere is placed at point A, as shown. The sphere is pushed by the plunger from point A to point B with a constant horizontal force that is directed toward the sphere’s center of mass. The sphere loses contact with the plunger at point B and continues moving across the horizontal surface toward point E. In which interval(s), if any, does the sphere’s angular momentum about its center of mass change? Check all that apply.

____ A to B ____ B to C ____ C to D ____ D to E _____ None

Explain.

C to D because there is a friction which applies net external torque about the central axis to increase / change angular momentum (NOT DECREASE) which results in change of momentum.

2019 QUESTION 2

This problem explores how the relative masses of two blocks affect the acceleration of the blocks. Block A, of mass m A, rests on a horizontal tabletop. There is negligible friction between block A and the tabletop. Block B, of mass mB , hangs from a light string that runs over a pulley and attaches to block A, as shown above. The pulley has negligible mass and spins with negligible friction about its axle. The blocks are released from rest. (a)

2019 2ai. Suppose the mass of block A is much greater than the mass of block B. Estimate the magnitude of the acceleration of the blocks after release.
Briefly explain your reasoning without deriving or using equations.

zero, negligible, small, much less than g

-Block a has large inertia, and block b is light so it barely applies a force, thus there is a lot of resistance to the force

2019 2aii. Now suppose the mass of block A is much less than the mass of block B. Estimate the magnitude of the acceleration of the blocks after release.

Briefly explain your reasoning without deriving or using equations.

9.8 or 10 m/s²

-block A has negligible mass, with neglible friction of the pulley and tabletop, so block b is essentiaally in freefall

2019 2b

Now suppose neither block’s mass is much greater than the other, but that they are not necessarily equal. The dots below represent block A and block B, as indicated by the labels. On each dot, draw and label the forces (not components) exerted on that block after release. Represent each force by a distinct arrow starting on, and pointing away from, the dot.

2019 2c

Derive an equation for the acceleration of the blocks after release in terms of mA , mB , and physical constants, as appropriate. If you need to draw anything other than what you have shown in part (b) to assist in your solution, use the space below. Do NOT add anything to the figure in part (b).

A: F_tension=m_aa

B: F= m_bg-F_tension= m_ba

F_tension = m_bg-m_ba =m_aa

m_aa + m_ba =m_bg

a (m_a+m_b) = m_bg

a = (m_bg) / (m_a+m_b)

Alternate:

F = ma

m_bg= (m_a+m_b) a

a = (m_bg) / (m_a+m_b)

2019 2d

Consider the scenario from part (a)(ii), where the mass of block A is much less than the mass of block B. Does your equation for the acceleration of the blocks from part (c) agree with your reasoning in part (a)(ii) ? ____ Yes ____ No Briefly explain your reasoning by addressing why, according to your equation, the acceleration becomes (or approaches) a certain value when m A is much less than mB .

Yes. If ma negligible, then mb cancels out on numerator and denominator and leaves g.

2019 2e

While the blocks are accelerating, the tension in the vertical portion of the string is T1. Next, the pulley of negligible mass is replaced with a second pulley whose mass is not negligible. When the blocks are accelerating in this scenario, the tension in the vertical portion of the string is T2. How do the two tensions compare to each other? ____ T2> T1 ____ T2 =T1 = ____ T2< T1 Briefly explain your reasoning

T2 > T1

acceleration of both blocks is smaller = T2 > T1

2019 QUESTION 3

A projectile launcher consists of a spring with an attached plate, as shown in Figure 1. When the spring is compressed, the plate can be held in place by a pin at any of three positions A, B, or C. For example, Figure 2 shows a steel sphere placed against the plate, which is held in place by a pin at position C. The sphere is launched upon release of the pin. A student hypothesizes that the spring constant of the spring inside the launcher has the same value for different compression distances.

2019 3ai

The student plans to test the hypothesis by launching the sphere using the launcher. i. State a basic physics principle or law the student could use in designing an experiment to test the hypothesis.

conservation of energy

2019 3aii

Using the principle or law stated in part (a)(i), determine an expression for the spring constant in terms of quantities that can be obtained from measurements made with equipment usually found in a school physics laboratory.

(1/2)kx² = mgh

k = 2mgh / x²

2019 3b

Design an experimental procedure to test the hypothesis in which the student uses the launcher to launch the sphere. Assume equipment usually found in a school physics laboratory is available. In the table below, list the quantities and associated symbols that would be measured in your experiment. Also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.

Describe the overall procedure to be used to test the hypothesis that the spring constant of the spring inside the launcher has the same value for different compression distances, referring to the table. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup.

2019 3c

Describe how the experimental data could be analyzed to confirm or disconfirm the hypothesis that the spring constant of the spring inside the launcher has the same value for different compression distances.