AP Physics

Harmonic Motion

Multiple Choice

Test E

1G/1Y

€

kA = mg

A =mgk

A =0.1Kg( ) 10 m

sec2( )40 N

m( )A = 1

40 m

0.1Kg

g = 10 msec 2

€

k = 40 Nm

What is the amplitude of the resulting simple harmonic motion of the block?

A 0.10-Kg block is attached to a spring, initially unstretched, of force constant as shown below. The block is released from rest at t = 0 sec.

k = 40 Nm

2

2G/2Y€

T = 2π mk

= 2π 0.1Kg40 N

m

T = 2π 1Kg400 N

m

T =2π20

=π

10sec

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

3

3G/23Y

D It moves to the right of x3 and does not return.

4

A conservative force has the potential energy function U(x) shown by the graph below. A particle moving in one dimension under the influence of this force has Kinetic Energy 3.0 J when at position X1. Which of the following is a correct statement about the motion of the particle?

4G/5Y

€

x = 3sin 10πt − 0.5( )

ω =10π = 2πf

f =10π2π

= 5Hz

The motion of a particle is described by the equation:

€

x = 3sin 10πt − 0.5( )where x is in meters, t is in seconds, and the phase angle (φ) is in radians. What is the frequency of the particle?

5

5G/24Y

€

k =Fx

=250N0.5m

= 500 Nm

W = Favg ⋅ x = 125N ⋅0.5m = 62.5J

W = ΔU = 12 kx 2 = 1

2 500 Nm( ) 0.5m( )2 = 62.5J

An archer pulls her bow string back 0.5 meters by exerting a force that increases uniformly form zero to 250 N. What is the equivalent spring constant of the bow, and how much work is done in pulling back on the bow string?

6

6G/25Y (0,a)

€

O€

y

€

v

€

x

€

xo

€

L = r × pL = mvdsinθL = mva

7

A particle of mass m moves with a constant speed v along the dashed line y = a. When the x-coordinate of the particle is xo, the magnitude of the angular momentum of the particle with respect to the origin (O) of the system is

A rod of negligible mass is pivoted at a point that is off-center, so that l1 is different from length l2.

The figures above show two cases in which masses are suspended from the ends of the rod. In each case the unknown mass m is balanced by a known mass, M1 or M2, so that the rod remains horizontal. What is the value of m in terms of the known masses?

8

7G/15Y

7G/15Y

€

τCCW =τCW

ml1 = M1l 2

M2l1 = ml 2

l1 =M1l 2

m

€

M2M1l 2

m= ml 2

M1M2 = m2

M1M2 = m

9

8G/22Y€

T = 2π mk

2T = 2π 2mk

C

10

A mass m hanging on a spring with a spring force constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation will

9&10G/3&4Y

€

m

max displacement

€

Uminimum

11

A block on a horizontal frictionless plane is attached to a spring , as shown above. The block oscillates with simple harmonic motion of amplitude A.9/3. Which of the following

statements about the block is correct?

10/4. Which of the following statements about energy is correct? A

C

11G/6Y

€

T = 2π mk

T 2 = 4π 2 mk

k =4π2m

T2

k =4π2 5Kg( )

π2

k = 20 Nm

€

ETotal = 12 kA2

2 ETotal( )

k= A2

2 ETotal( )k

= A

2 40J( )20 N

m

= A

2m = A

12

A 5-Kg mass is attached to a spring and executes simple harmonic motion with a period of π seconds. If the total energy of the system is 40 Joules, the amplitude of oscillation in meters is?

12G/17Y

€

F = maFm

= a

F3m

=a3

13

A body of mass M accelerates under the action of a Force F. The same Force F applied to a second object of mass 3M produces an acceleration. What is the acceleration of the second object compared to the acceleration of the first object?

13G/18Y

€

ω 2 =ωo2 + 2αθ

ω = 2αθω = 2α2πω = 2 απ

14

A turntable that is initially at rest is set in motion with a constant angular acceleration α. What is the angular velocity of the turntable after it has made one complete revolution?

14G/16Y

€

KB = UTop

12 m vo

2

2

= mgH

14

vo2

2g

= H

€

vo

€

vo

2

€

vo

€

KB = UTop

12 mvo

2 = mgH2

vo2

2g

= H2

4H = H2

€

30°

15

A spring loaded gun can fire a projectile to a height H if it is fired at an angle of 30° with respect to the horizontal. If the same gun is pointed straight up, what maximum height can now be reached by the projectile?

M

3M

€

k = 800 Nm

€

USpring = UGravity

12 kx 2 = mgh x = h

12 kx 2 = mgx

x =2mg

k

x =2 4Kg( ) 10 m

sec2( )800 N

m( ) x =10cm

15G/19Y

16

The blocks, as shown above, are released from rest with the spring unstretched. The horizontal surface is frictionless as is the pulley. If M = 4 kg, the maximum extension of the spring before mass M starts to move up is most nearly?

m

θl

€

FRestoring = mgsinθ

FRestoring = 2Kg( ) 10 msec2( ) 0.5( )

FRestoring = 10N

€

mg

€

mgsinθ

16G/20Y

17

A certain Pendulum consists of a 2.0 Kg mass swinging at the end of a string of length l = 2.5 meters. At the lowest point in the swing, the Tension in the string is 25 Newtons. The restoring force on the mass when the pendulum makes a θ = 30° angle with the vertical is most nearly?

€

T = 2π l

g= 2π 2.5m

10 msec2

T = 2π 25m100 m

sec2

T = 2π 12( )

T = πsec

17G/21Y 18

When the same pendulum is at the highest point of its swing, its velocity is zero. The pendulum's period of motion is most nearly?

m

θl

€

mg

€

mgsinθ

l = 2.5m

θ = 30°

18G/9Y

€

I = ICM + mD2 = 12 mR2 + m R

2

2

I = 12 mR2 + 1

4 mR2 = 34 mR2

€

ω =mgD

I

ω =mg R

234 mR2

ω =2g3R

A solid disk of radius R and mass m is suspended from a pivot a distance meters above its center of mass as shown above. The angular frequency for small oscillations of the disk is most nearly:

€

R2

19

19G/13Y

€

m1

€

τ = Iατ = TNetRT2 − T1( )R = Iα

20

Two blocks are joined by a light string that passes over the pulley shown above, which has radius R and moment of inertia I about its center. T1 and T2 are the tensions in the string on either side of the pulley and α is the angular acceleration of the pulley. Which of the equations best describes the pulley’s rotational motion during the time the blocks accelerate?

20G/7Y

€

12 kA2 = 1

2 kx2 + 12 mv2

kA2 = kx2 + mv2

km

A 2 − x 2( ) = v

4003

0.01− 0.0025( ) = v

4003

7510000

= v

4 25100

= v

2 ⋅ 510

= v = 1 msec

21

A 3.0 Kg block sliding on a horizontal frictionless surface is attached to one end of a horizontal spring which has its other end fixed. The block is displaced 10 cm from its equilibrium position and released from rest. The speed of the block when it is 5.0 cm from its equilibrium position is most nearly?

€

k = 400 Nm( )

21G/8Y

€

t = 2π l

g

2t = 2π 4l

g

22

A simple pendulum consists of a 2.0 Kg bob on a string about l meters long and has a period of t seconds. The pendulum would have a period of 2t seconds if the

string were replaced by one about 4 l meters!

22G/11Y€

a = Aω 2

a = 1.5m( ) 2 radsec( )2

a = 6 msec2

A particle moves in the xy-plane with coordinates given by

€

x = Acosωt and y = Asinωt,where A = 1.5 meters and ω = 2.0 radians per second. What is the magnitude of the particle's acceleration?

23

A simple pendulum of length l is constructed from a point object of mass m suspended by a light string attached to a fixed pivot point. A small peg is placed a distance l/2 directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced 20° from the vertical and released from rest. How long does the pendulum bob take to return to its original position?

Fixed Pivot Point

Small Peg

m

23G/14Y

€

T = 12 2π l

g + 12 2π l

2g

T = π lg + π l

2g

T = π lg 1+ 2

2

24

24G/10Y 25 75 50 50 75 25 100 0

0 100

€

U J( )

€

K J( )

€

10m

€

20m

€

30m

€

40mA ball is thrown upward. At a height of 10 meters above the ground, the ball has a potential energy of 25 Joules (with the potential energy equal to zero at ground level) and is moving upward with a kinetic energy of 75 Joules. Air friction is negligible. The maximum height reached by the ball is most nearly

25

25G/12Y

€

τCCW =τCW

m1ga = m2gbam1 = bm2

26

For the wheel-and-axle system shown, which of the following expresses the condition required for the system to be in static equilibrium?