2010 AP Oscillations: Last Chapter of Mechanics!

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?

1. Any object in equilibrium is at rest
2. An object in equilibrium need not be at rest
3. An object at rest must be in equilibrium.

2.  An object can oscillate around

1. Any equilibrium point
2. Any stable equilibrium point
3. Certain stable equilibrium points
4. Any point, provided the forces exerted on it obey Hooke’s law
5. Any point

3.  What was Hooke’s Law?

1. If a spring is cut in half, what happens to its spring constant?

1. What if there is two springs with the same k in parallel?

1. What if there are two springs with the same k in series?

1. 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?

1. Through what total distance does a block on a spring travel in one period?

A/2 A 2A 4A

1. Where is the stable equilibrium point for this block? Where does it reach maximum velocity?

1. How is frequency related to angular frequency?

4.  Fill in max, zero or constant

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

1. Kinetic energy

1. Potential energy

1. 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?

1. 16J
2. 28 J
3. 56 J
4. 64 J
5. 80 J

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.

10.  Which of the following is true for a system consisting of a mass oscillating on the end of an ideal spring?

1. The kinetic and potential energies are equal at all times
2. The kinetic and potential energies are both constant
3. The maximum potential energy is achieved when the mass passes through its equilibrium position
4. The maximum kinetic energy and maximum potential energy are equal, but occur at different times
5. 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

1. 2 m/s
2. 4 m/s
3. 20 m/s
4. 40 m/s
5. 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

1. Draw a clock hand at 3 o’clock.

1. 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.

1. Relate the amplitude (the length of the clock hand) of the motion to the position of the point on the x axis.
2. Draw a graph of the position of the point on the x axis as a function of time.

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1. Write the equation for the two motions given in class.

1. What if the blocks don’t travel at the same angular velocity?

1. Substitute angular displacement to relate to angular velocity/frequency? Substitute in for angular velocity/frequency so that it is terms of period.

1. Write the equations for the two motions given in class.

1. What if the motion starts with an angular displacement?

1. Write the equations for the two motions given in class.

Refer to the graph below of the displacement x versus t for a particle in simple harmonic motion.

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.

e.

15.  Which of the following graphs shows the kinetic energy K of the particle as a function of its displacement x?

a. b. c. d.

e.

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

c. 

d. 

e.  6p 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.

1. Graph velocity as a function of time.

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) = 4cospt

b.  v(t)= pcospt

c.  v(t)= -p2cospt

d.  v(t)= -4sinpt

e.  v(t)= -4psinpt

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

21.  How is angular frequency related to the spring constant and mass?

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

a. b. c. d. e.

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 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?

1. 1/40 m
2. 1/20 m
3. ¼ m
4. ½ m
5. 1 m

26.  At what time after release will the block first return to its position?

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[(4p rad/s)t+ p/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?

29.  The equation of motion of a simple harmonic oscillator is , where x is displacement and t is time. The period of oscillation is

1. 6p b. c. d. e.

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 wt and y=A sin wt

where A=1.5 meters and w = 2.0 radians per second. What is the magnitude of the particle’s acceleration?

1. zero
2. 1.3 m/s2
3. 3.0 m/s2
4. 4.5 m/s2
5. 6.0 m/s2

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.

32.  What is the period for a simple pendulum?

33.  Alice performs a simple pendulum lab. The purpose of the lab is to:

1. Verify the relationship between the period T and the length l of the simple pendulum T=2π(1/g)1/2
2. Determine the value of the acceleration due to gravity g

The data collected by the student is given below:

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

1. ¼ T
2. ½ T
3. T
4. 2T
5. 4T

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.

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 q that the pendulum makes with the vertical as a function of time t?

a.

b.

c.

d.

e.

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