2010 AP Oscillations: Last Chapter of Mechanics!
Oscillations
Sine of the times!
1. An object is in equilibrium when the net force and the net torque on it is zero.
Which of the following statements is/are correct for an object?
a. Any object in equilibrium is at rest
b. An object in equilibrium need not be at rest
c. An object at rest must be in equilibrium.
2. An object can oscillate around
a. Any equilibrium point
b. Any stable equilibrium point
c. Certain stable equilibrium points
d. Any point, provided the forces exerted on it obey Hooke’s law
e. Any point
3. What was Hooke’s Law?
a. If a spring is cut in half, what happens to its spring constant?
b. What if there is two springs with the same k in parallel?
c. What if there are two springs with the same k in series?
x=-A(-xm)
x=0
x=A(xm)
d. What is the acceleration of a block on a spring? Where in the above
diagram does the block have positive acceleration? Negative acceleration?
Positive velocity? Negative velocity?
e. Through what total distance does a block on a spring travel in one period?
A/2
A
2A
4A
2010 AP Oscillations: Last Chapter of Mechanics!
f. Where is the stable equilibrium point for this block? Where does it reach
maximum velocity?
g. How is frequency related to angular frequency?
4. Fill in max, zero or constant
Velocity
Kinetic
Energy
Potential
energy
Force
0
xmax
Acceleration ETotal
x=0
x=xm
-xmax
5. Draw a sketch of kinetic energy and potential energy as a function of
displacement.
6. When the displacement in SHM is one half the amplitude xm, what fraction of the
total energy is
a. Kinetic energy
b. Potential energy
2010 AP Oscillations: Last Chapter of Mechanics!
c. At what displacement, in terms of the amplitude is the energy of the
system half kinetic and half potential energy?
7. To stretch a certain nonlinear spring by an amount x requires a force F given by
F=40x-6x2, where F is in newtons and x is in meters. What is the change in
potential energy when the spring is stretched 2 meters from its equilibrium
position?
a. 16J
b. 28 J
c. 56 J
d. 64 J
e. 80 J
x=-A
x=0
x=A
A block on a horizontal frictionless plane is attached to a spring. The block oscillates
along the x axis with simple harmonic motion of amplitude A.
8. Which of the following statements about the block is correct?
a. At x=0, its velocity is zero.
b. At x=0, its acceleration is at a maximum
c. At x=A, its displacement is at a maximum
d. At x=A, its velocity is at a maximum
e. At x=A, its acceleration is zero.
9. Which of the following statements about energy is correct?
a. The potential energy of the spring is at a minimum at x=0
b. The potential energy of the spring is at a minimum at x=A
c. The kinetic energy of the block is at a minimum at x=0
d. The kinetic energy of the block is at a maximum at x=A
e. The kinetic energy of the block is always equal to the potential energy of the
spring.
2010 AP Oscillations: Last Chapter of Mechanics!
10. Which of the following is true for a system consisting of a mass oscillating on the
end of an ideal spring?
a. The kinetic and potential energies are equal at all times
b. The kinetic and potential energies are both constant
c. The maximum potential energy is achieved when the mass passes through
its equilibrium position
d. The maximum kinetic energy and maximum potential energy are equal,
but occur at different times
e. The maximum kinetic energy occurs at maximum kinetic energy occurs at
maximum displacement of the mass from its equilibrium position.
11. A 1.0 kg mass is attached to the end of a vertical ideal spring with a force constant
of 400 N/m. The mass is set in simple harmonic motion with an amplitude of 10
cm. The speed of the 1.0 kg mass at the equilibrium position is
a. 2 m/s
b. 4 m/s
c. 20 m/s
d. 40 m/s
e. 200 m/s
12. A 4.0 kg block is suspended from a spring with a spring constant of 500 N/m. A
50 g bullet is fired into the block from directly below with a speed of 150 m/s and
becomes embedded in the block. Find the amplitude of the resulting simple
harmonic motion.
13. Comparing simple harmonic motion with uniform circular motion
a. Draw a clock hand at 3 o’clock.
b. Draw a line from the end of the clock hand to the x axis. Describe the
motion of the point on the x-axis as the clock hand travels
counterclockwise.
c. Relate the amplitude (the length of the clock hand) of the motion to the
position of the point on the x axis.
2010 AP Oscillations: Last Chapter of Mechanics!
d. Draw a graph of the position of the point on the x axis as a function of
time.
xmax
x
-x
- max
t
½T
T
e. Write the equation for the two motions given in class.
f. What if the blocks don’t travel at the same angular velocity?
g. Substitute angular displacement to relate to angular velocity/frequency?
Substitute in for angular velocity/frequency so that it is terms of period.
h. Write the equations for the two motions given in class.
i. What if the motion starts with an angular displacement?
j. Write the equations for the two motions given in class.
2010 AP Oscillations: Last Chapter of Mechanics!
x
Refer to the graph below of the displacement x versus t for a particle in simple harmonic
motion.
t
K
K
K
K
14. Which of the following graphs shows the kinetic energy K of the particle as a
function t for one cycle of motion?
a.
b.
c.
d.
t
t
t
t
K
e.
t
15. Which of the following graphs shows the kinetic energy K of the particle as a
function of its displacement x?
c.
k
e.
x
k
k
x
d.
x
k
b.
k
a.
x
x
2010 AP Oscillations: Last Chapter of Mechanics!
A particle moves in a circle in such a way that the x and y coordinates of its motion are
given in meters as functions of time t in seconds by:
x= 5 cos (3t)
y= 5 sin (3t)
16. What is the period of revolution of the particle?
a. 1/3 s
b. 3 s
2
s
c.
3
3
s
d.
2
e. 6 s
17. Which of the following is true of the speed of the particle?
a. It is always equal to 5 m/s
b. It is always equal to 15 m/s
c. It oscillates between 0 and 5 m/s
d. It oscillates between 0 and 15 m/s
e. It oscillates between 5 and 15 m/s
18. Find velocity as a function of time.
a. Graph velocity as a function of time.
v vmax
-vmax
t
½T
T
x
2010 AP Oscillations: Last Chapter of Mechanics!
5
4
3
2
1
0
-1 0
-2
-3
-4
-5
0.25 0.5 0.75
1
1.25 1.5 1.75
2
t
19. A particle moves in simple harmonic motion represented by the graph above.
Which of the following represents the velocity of the particle as a function of
time?
a. v(t) = 4cost
b. v(t)= cost
c. v(t)= -cost
d. v(t)= sint
e. v(t)= -4sint
20. Below is the v vs t graph for a particle m, undergoing SHM. Sketch the
corresponding graph of K vs t of the particle
velocity versus time for SHM
Velocity (m/s)
1.5
1
0.5
0
-0.5 0
2
4
6
8
-1
-1.5
time (s)
K
t
½T
T
21. How is angular frequency related to the spring constant and mass?
2010 AP Oscillations: Last Chapter of Mechanics!
22. Find the period of simple harmonic motion.
23. When a mass m is hung on a certain ideal spring, the spring stretches a distance d.
If the mass is then set oscillating on the spring, the period of oscillation is
proportional to
d
d
m
m2 g
g
b.
c.
d.
e.
g
mg
g
d
d
24. Two objects of equal mass hang from independent springs of unequal spring
constant and oscillate up and down. The spring of greater spring constant must
have the
a. smaller amplitude of oscillation
b. larger amplitude of oscillation
c. shorter period of oscillation
d. longer period of oscillation
e. lower frequency of oscillation
a.
k
g=10m/s2
1.0 kg
A 0.1 kilogram block is attached to an initially unstretched spring of force constant
k=40 newtons per meter as shown above. The block is released from rest at time t=0.
25. What is the amplitude of the resulting simple harmonic motion of the block?
a. 1/40 m
b. 1/20 m
c. ¼ m
d. ½ m
e. 1 m
26. At what time after release will the block first return to its position?

s
a.
40

s
b.
20

s
c.
10

s
d.
5

s
e.
4
2010 AP Oscillations: Last Chapter of Mechanics!
27. Write the equation for the acceleration of an object in simple harmonic motion.
28. The position of a particle is given by
x(t)=(5.0 cm)sin[(4 rad/s)t+ /3 rad] where t is in seconds
Is this simple harmonic motion?
What is the frequency?



What is the period?
What is the amplitude?
At t=2.0 seconds, what is the displacement?
At t=2.0 seconds, what is the velocity?
At t=2.0seconds, what is the acceleration?
At what time does it first reach x=0?
What is the maximum acceleration?

2010 AP Oscillations: Last Chapter of Mechanics!
29. The equation of motion of a simple harmonic oscillator is
displacement and t is time. The period of oscillation is
9
3
2
a. 6
b.
c.
d.
2
2
3
d 2x
 9 x , where x is
dt 2
e.
2
9
30. Two blocks (m=1.0 kg and M=10kg) and a spring (k=200 N/m) are arranged on a
horizontal, frictionless surface. The coefficient of static friction between the two
blocks is 0.40. What amplitude of simple harmonic motion of the spring-blocks
system puts the smaller block on the verge of slipping over the larger block?
31. A particle moves in the xy-plane with coordinates given by
x=A cos t and y=A sin t
where A=1.5 meters and  = 2.0 radians per second. What is the magnitude of the
particle’s acceleration?
a. zero
b. 1.3 m/s2
c. 3.0 m/s2
d. 4.5 m/s2
e. 6.0 m/s2
2010 AP Oscillations: Last Chapter of Mechanics!
A 2-kilogram block is dropped from a height of 0.45 meter
above an uncompressed spring, as shown above. The spring has
an elastic constant of 200 newtons per meter and negligible
mass. The block strikes the end of the spring and sticks to it.
a. Determine the speed of the block at the instant it hits the end of
the spring.
b. Determine the period of the simple harmonic motion that ensues.
c. Determine the distance that the spring is compressed at the instant the speed of the
block is maximum.
d.Determine the maximum compression of the spring.
e. Determine the amplitude of the simple harmonic motion.
2010 AP Oscillations: Last Chapter of Mechanics!
32. What is the period for a simple pendulum?
33. Alice performs a simple pendulum lab. The purpose of the lab is to:
a. Verify the relationship between the period T and the length l of the simple
pendulum T=2π(1/g)1/2
b. Determine the value of the acceleration due to gravity g
The data collected by the student is given below:
Trial Number
Length of simple pendulum Time for 10 oscillations
(m)
1
0.408
13.2
2
0.508
13.8
3
0.608
16.0
4
0.708
16.5
5
0.808
18.0
6
0.908
19.0
7
1.008
20.5
State at least two different ways the data above can be analyzed graphically for the
purpose of the lab. Show how the value of g can be determined directly from the slope of
the straight line graph.
34. A simple pendulum of length l, whose bob has mass m, oscillates with a period T.
If the bob is replaced by one of mass 4m, the period of oscillation is
a. ¼ T
b. ½ T
c. T
d. 2T
e. 4T
2010 AP Oscillations: Last Chapter of Mechanics!
You are given a long, thin, rectangular bar of known mass M and length
l with a pivot attached to one end. The bar has a nonuniform mass
density, and the center of mass is located a known distance x from the
end with the pivot. You are to determine the rotational inertia Ib of the bar
about the pivot by suspending the bar from the pivot, as shown above,
and allowing it to swing. Express all algebraic answers in terms of Ib , the
given quantities, and fundamental constants.
(a)i. By applying the appropriate equation of motion to the bar, write the
differential equation for the angle θ the bar makes with the vertical.
ii. By applying the small-angle approximation to your differential equation, calculate
the period of the bar’s motion.
(b) Describe the experimental procedure you would use to make the additional
measurements needed to determine Ib . Include how you would use your measurements to
obtain Ib and how you would minimize experimental error.
(c) Now suppose that you were not given the location of the center of mass of the bar.
Describe an experimental procedure that you could use to determine it, including the
equipment that you would need.
2010 AP Oscillations: Last Chapter of Mechanics!
Questions 35 and 36: A simple pendulum has a period of 2 s for small amplitude
oscillations.
35. The length of the pendulum is most nearly
a. 1/6 m
b. ¼ m
c. ½ m
d. 1 m
e. 2 m
36. Which of the following equations could represent the angle that the pendulum
makes with the vertical as a function of time t?

a.    max sin t
2
b.    max sin  t
c.    max sin 2 t
d.    max sin 4 t
e.    max sin 8 t
37. A mass M suspended by a spring with force constant k has a period T when set
into oscillation on Earth. Its period on Mars, whose mass is about 1/9 and radius
½ that of Earth , is most nearly
a. 1/3 T
b. 2.3 T
c. T
d. 3/2 T
e. 3T
38. A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is
g, is taken to another planet, where its period is 2 s. The acceleration due to
gravity on the other planet is most nearly
a. g/4
b. g/2
c. g
d. 2g
e. 4g
39. A simple pendulum consists of a 1.0 kilogram brass bob on a string about 1.0
meter long. It has a period of 2.0 seconds. The pendulum would have a period of
1.0 second if the
a. string were replaced by one about 0.25 meter long
b. string were replaced by one about 2.0 meters long
c. bob were replaced by 0.25 kg brass sphere
d. bob were replaced by a 4.0 kg brass sphere
e. amplitude of the motion were increased.
2010 AP Oscillations: Last Chapter of Mechanics!
40. Find the period for a rotating pendulum (angular simple harmonic motion).
41. Physical Pendulum- Find the period.
42. A uniform circular disk whose radius R is 12.5 cm is suspended as a physical
pendulum from a point on its rim
a. What is its period?
b. At what radial distance r<R is there a pivot point that gives the same
period?
2010 AP Oscillations: Last Chapter of Mechanics!
43. A pendulum is formed by pivoting a long thin rod of length L and mass m about a
point on the rod that is a distance d above the center of the rod. Find the period of
this pendulum in terms of d,L,M and g, assuming small amplitude swinging.
44. A person swings on a swing. When the person sits still, the swing oscillates back
and forth at its natural frequency. If, instead, two people sit on the swing, the new
natural frequency of the swing is
a. Greater
b. The same
c. Smaller
45. When does resonance occur?
2010 AP Oscillations: Last Chapter of Mechanics!
46. A skier of mass m will be pulled up a hill by a rope, as shown above. The magnitude of
the acceleration of the skier as a function of time t can be modeled by the equations
𝜋𝑡
(0 < 𝑡 < 𝑇)
𝑎 = 𝑎𝑚𝑎𝑥 sin 𝑇
𝑎=0
(𝑡 ≥ 𝑇)
where amax and T are constants. The hill is inclined at an angle 𝜃 above the horizontal, and
friction between the skis and the snow is negligible. Express your answers in terms of given
quantities and fundamental constants.
(a) Derive an expression for the velocity of the skier as a function of time during the
acceleration. Assume the skier starts from rest.
(b) Derive an expression for the work done by the net force on the skier from rest until terminal
speed is reached.
(c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.
(d) Derive an expression for the total impulse imparted to the skier during the acceleration.
2010 AP Oscillations: Last Chapter of Mechanics!
(e) Suppose that the magnitude of the acceleration is instead modeled as
𝑎 = 𝑎𝑚𝑎𝑥 𝑒−𝜋𝑡/2𝑇
for all t > 0 , where amax and T are the same as in the original model. On the axes below, sketch
the graphs of the force exerted by the rope on the skier for the two models, from t = 0 to a time t
> T . Label the original model F1 and the new model F2 .