AP Physics C

Free Response Problems “SHM”

1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) =

0.8Cos(πt). Find the following:

a. The time for one complete oscillation.

b. The spring constant.

c. The maximum speed of the mass.

d. The maximum force on the mass.

e. The position, speed, acceleration of the mass at time t = 1s.

2. An object undergoes SHM with a period 2 s and amplitude 0.1 m. At time t = 0 the object is

instantaneously at rest at x = 0.1 m.

a. Write the equation for the position of the object as a function of time.

b. Find the maximum speed and acceleration of the object.

c. Write the equations giving the velocity and acceleration of the object as a

function of time.

d. Sketch the following graphs: x(t), v(t), a(t).

3. A small block of mass 0.2 kg is attached to a horizontal spring with a spring constant k = 200 N/m

and placed on a horizontal frictionless table. The other end of the spring is fixed to a wall. When

the block is 0.01 m from its equilibrium, it is observed to have a speed of 0.4 m/s.

a. Find the total energy of the object.

b. Find the maximum displacement of the object from its equilibrium position.

c. Find the maximum speed attain by the object during its oscillation.

d. Find the location of the object where its kinetic energy equals potential energy.

4. A 1.2 kg block is dropped from a height of 0.5 m above an uncompressed spring. The spring has

a spring constant k = 160 N/m and negligible mass. The block strikes the top end of the spring

and sticks to it.

a. Find the speed of the block when it strikes the top end of the spring.

b. Find the period of oscillations of the block.

c. Find the compression of the spring when the speed of the block reaches its

maximum value.

d. Find the maximum compression of the spring.

e. Find the amplitude of oscillations of the block.

5. A 6 kg block is fastened to a vertical spring with a spring constant of 900 N/m. A 4 kg block is at

rest on the top of a 6 kg block, and they are both placed on top of the spring.

a. Determine the compression of the spring when two blocks are at rest.

The blocks are slightly pushed down and released. They begin to oscillate.

b. Determine the frequency of oscillations.

c. Determine the magnitude of the maximum acceleration that the blocks can

attain and still remain in contact at all times.

d. Determine the maximum compression of the spring beyond the compression

found in part (a) without causing the blocks to exceed the acceleration in part

(c).

e. Determine the maximum speed of the blocks if the spring is compressed to the

distance found in part (d)

6. A small block of mass m1 rests on but is not attached to a larger block of mass m2. Block m2 is

placed on a horizontal frictionless surface. The maximum friction force between the blocks is f. A

spring with a spring constant k is attached to the large block m2 and to the wall.

a. Determine the maximum horizontal acceleration of block m2 where m1 doesn’t

slip on the surface of m2.

b. Determine the maximum amplitude for simple harmonic motion of two blocks if

they are to move together.

c. The two blocks are pulled to the right to the maximum amplitude found in part

(b) and released. Describe the friction force between the blocks during first half

of the period of oscillations.

d. The two blocks are pulled to the right a distance greater than the maximum

amplitude and then released.

i. Determine the acceleration of m1 at the instant when

the blocks are released.

ii. Determine the acceleration of m2 at the instant when

the blocks are released.

7. A group of students conducted an experiment to determine the spring constant of an elastic

spring. In the experiment they were using the mass-spring oscillating system shown above. A

disk of mass m is attached to a horizontal spring. The disk has a small hole that is used to place it

on a frictionless horizontal rod. When the disk is displaced from its equilibrium point and

released it undergoes SHM. The students were able to vary the mass by exchanging the disks

with different masses and in each trial they measured the time for ten complete oscillations.

The data is shown below.

M (kg) T10 (s) T (s) T2 (s2)

0.1 4.4

0.2 6.3

0.3 7.7

0.4 8.9

0.5 9.9

a. For each trial, calculate the period and the square of the period. Use a

reasonable number of significant figures. Enter these results in the table above.

b. On the axes below, plot the square of the period versus mass using an

appropriate x and y scale. Draw the best –fit line for this data.

c. A disk with an unknown mass is set to SHM. The time for ten oscillations is 7 s.

From the graph, find the mass of the disk. Write your answer with a reasonable

number of significant figures.

d. From the graph, find the spring constant.

e. Is it possible to use this device to measure mass aboard satellite orbiting Earth?

8. An experiment to measure the acceleration due to gravity g was conducted in a physics lab.

Students were using a simple pendulum, meter stick, and stopwatch. The pendulum consisted of

a ball of mass m at the end of a string of length L. During the experiment, the students were

changing the length of the string and recording the time for ten complete oscillations. The data

is shown below.

L (m) t10 (s) T (s) T2 (s2)

0.2 8.9

0.4 12.7

0.6 15.5

0.8 17.9

1.0 20.0

a. Calculate the period and the square of the period for each trial and enter these

results in the table above. Use a reasonable number of significant figures.

b. On the graph below, plot the period squared versus length using an appropriate

x and y scale. Draw the best –fit line for this data.

c. Assuming that the pendulum undergoes SHM according to your best-fit line,

determine the value of the acceleration due to gravity. Explain your answer.

d. If the experiment was performed in the accelerating upward elevator, how

would it change the answer to part (c)? Explain.

9. An elastic spring with is placed on a horizontal platform. A pan of mass M is attached to the top

end of the spring, compressing it a distance d. A piece of clay is dropped from a height h onto

the pan. The piece of clay strikes the pan and sticks to it.

a. What is the speed of the clay just before it hits the pan?

b. What is the speed of the pan just after the clay strikes it?

c. What is the period of oscillation?

d. How much is the spring stretched at the moment when the speed of the pan is a

maximum?

e. If a smaller in diameter spring is placed inside the first one so two springs can

support the pan, how would it change the period of oscillations?

10. A baseball bat of mass M and length L is pivoted at point P. The center of mass of the bat is

located at a distance d from the pivot. When the bat is displaced from equilibrium by a small

angle θ and released it undergoes SHM. Physics students are asked to determine the moment of

inertia of the bat with respect to the pivot point P.

a. Write the appropriate differential equation for the angle θ that can be used to

describe motion of the bat.

b. Using the analogy to a mass oscillating on a spring, determine the period of the

bat’s motion.

c. Describe an experiment that you would perform to measure the moment of

inertia of the bat. Include the list of additional materials and write a detail

procedure on how you would obtain them.

d. If the center of mass of the bat is not given in the problem, how would you set

up an additional experiment to measure the center of mass?

Free Response Answers

1. a. 2𝑠

b. 4.93𝑁

𝑚

c. 2.51𝑚

𝑠

d. 3.94 𝑁

e. 𝑥 = 0.8 𝑚, 𝑣 = 0𝑚

𝑠, 𝑎 = 7.9

𝑚

𝑠2

2. a. 𝑥 = 0.1 cos(𝜋𝑡)

b. 𝑣𝑚𝑎𝑥 = 0.31𝑚

𝑠, 𝑎𝑚𝑎𝑥 = 0.97

𝑚

𝑠2

c. 𝑣 = −0.31 sin(𝜋𝑡) , 𝑎 = −0.97 sin(𝜋𝑡)

d.

3. a. 0.026 𝐽

b. 0.016 𝑚

c. 0.51𝑚

𝑠

d. 0.011 𝑚

x

t

v

t

a

t

4. a. 3.16𝑚

𝑠

b. 0.54𝑚

𝑠

c. 0.075𝑚

𝑠

d. 0.27 𝑚

e. 0.195 𝑚

5. a. 0.11 𝑚

b. 1.15 𝐻𝑧

c. 𝑔

d. 0.11 𝑚

e. 1.04𝑚

𝑠

6. a. 𝑓

𝑚1

b. (𝑚1+𝑚2)𝑓

𝑘𝑚1

c. 𝑇ℎ𝑒 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑖𝑛𝑢𝑠𝑜𝑖𝑑𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛.

d. i. 𝑓

𝑚1= 𝑎1 ii.

𝑘𝐴′−𝑓

𝑚2= 𝑎2

7. a.

b.

c. 0.27 𝑘𝑔

d. 19.74𝑁

𝑚

e. 𝑌𝑒𝑠, 𝑎𝑠 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑒𝑝𝑒𝑛𝑑 𝑜𝑛 𝑔𝑟𝑎𝑣𝑖𝑡𝑦

M (kg) T10 (s) T (s) T2 (s2)

0.1 4.4 0.44 0.19

0.2 6.3 0.63 0.40

0.3 7.7 0.77 0.59

0.4 8.9 0.89 0.79

0.5 9.9 0.99 0.98

1.0

0.6

0.8

0

0.4

0.2

0.1 0.2 0.3 0.5 0.4

T2

m

8. a.

b.

c. 9.87𝑚

𝑠2 . 𝐹𝑜𝑟 𝑎 𝑝𝑒𝑛𝑑𝑢𝑙𝑢𝑚, 𝑇 = 2𝜋√𝑙

𝑔. 𝑇ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 𝑡𝑜 𝑔 =

4𝜋2𝑙

𝑇2 𝑎𝑛𝑑 𝑠𝑖𝑛𝑐𝑒 𝑠𝑙𝑜𝑝𝑒 =∆𝑇2

∆𝐿 , 𝑔 =

4𝜋2

𝑠𝑙𝑜𝑝𝑒 .

d. 𝑇ℎ𝑒 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑜 𝑐 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 ℎ𝑖𝑔ℎ𝑒𝑟. 𝑇ℎ𝑖𝑠 𝑖𝑠 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑔 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑎𝑓𝑓𝑒𝑐𝑡𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑢𝑝𝑤𝑎𝑟𝑑𝑠

𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛, 𝑎𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝐹𝑁 − 𝑚𝑔 = 𝑚𝑎.

9. a. √2𝑔ℎ

b. 𝑚√2𝑔ℎ

𝑚+𝑀

c. 2𝜋√(𝑀+𝑚)𝑑

𝑀𝑔

d. (𝑀+𝑚)𝑑

𝑀

e. 𝐼𝑡 𝑤𝑜𝑢𝑙𝑑 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑. 𝑇ℎ𝑖𝑠 𝑖𝑠 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑘 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝑙𝑎𝑟𝑔𝑒𝑟.

L (m) t10 (s) T (s) T2 (s2)

0.2 8.9 0.89 0.79

0.4 12.7 1.27 1.61

0.6 15.5 1.55 2.40

0.8 17.9 1.79 3.20

1.0 20.0 2.00 4

4.0

2.4

3.2

1.6

0.8

0.2 0.4 0.6 1.0 0.8

T2

L

10. a. −𝑚𝑔𝑑 sin 𝜃

𝐼=

𝑑2𝜃

𝑑𝑡2

b. 2𝜋√𝐼

𝑚𝑔𝑑

c. Rotate the bat by a small angle θ and release the bat. Time ten complete oscillations

using a stopwatch. Divide the final time by 10 to get the period (T) of the bat. Use the equation

𝑇 = 2𝜋√𝐼

𝑚𝑔𝑑 to algebraically solve for I (𝐼 =

𝑇2𝑚𝑔𝑑

4𝜋2 ).

d. Grab a fulcrum and slowly move the bat along it. When the bat balances out on the fulcrum,

the point that the fulcrum is touching is the center of mass.