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Simple Harmonic Motion - Multiple Choice Questions

Q1. A mass M hangs in equilibrium on a spring. M is made to oscillate about the equilibrium position by pulling it down 10 cm and releasing it. The time for M to travel back to the equilibrium position for the first time is 0.50 s. Which line, A to D, is correct for these oscillations?

 
Amplitude /cm
Period /s
A
10
1.0
B
10
2.0
C
20
2.0
D
20
1.0

 

It is pulled down 10 cm therefore the maximum displacement from the equilibrium (amplitude) is 10 cm.

To travel from the displacement to the equilibrium position is a quarter of the whole cycle (A to midpoint to-A to midpoint to A (again)) - therefore period is 2.0 s

Choice B

Q2. A particle oscillates with undamped simple harmonic motion. Which one of the following statements about the acceleration of the oscillating particle is true?

A

It is least when the speed is greatest.

Speed is greatest when x = zero (therough the centre of the swing).

If x = zero acceleration = zero, so this is true.
B

It is always in the opposite direction to its velocity.

It is always in the opposite direction to displacement, not velocity - not true.
C

It is proportional to the frequency.

It is proportional to frequency squared, so not true.
D

It decreases as the potential energy increases.

That would mean it decreased as x went from zero to A. It is at a maximum when x = A and zero when x = zero, so this is not true.

 

Q3. Which one of the following statements is true when an object performs simple harmonic motion about a central point O?

A

The acceleration is always away from O.

Acceleration is sometimes away from and somtimes towards O, but if it is towards then displacement is away from - and vice versa!
B

The acceleration and velocity are always in opposite directions.

Look at the graph - this is not the case.
C

The acceleration and the displacement from O are always in the same direction.

No - they are in opposite directions.
D

The graph of acceleration against displacement is a straight line.

True - acceleration is directly proportional to the displacement (with a negative constant of proportionality)

 

Q4. A particle of mass m executes simple harmonic motion in a straight line with amplitude A and frequency f. Which one of the following expressions represents the total energy of the particle?

A
22 mfA2
B
2π2 mf2A2
C
42 m2f2A
D
42 mf2A2

 

Total energy is composed of kinetic energy and potential energy.

When PE is zero all of the energy is KE and vice versa.

Therefore the maximum velocity can be used to work out the energy

E = 1/2 mv2

= 1/2 m (2fA)2

= 2m2f2A2

 

Q5. A body moves with simple harmonic motion of amplitude A and frequency b/2 .

What is the magnitude of the acceleration when the body is at maximum displacement?

A
zero
B
42Ab2
C
Ab2
D
42Ab-2

 

At maximum displacement x = A.

acceleration = -(2f)2x

= -42b2A/42

= -b2A

as we are asked for the magnitude only the minus sign can be ignored.

 

Q6. A simple pendulum and a mass-spring system are taken to the Moon, where the gravitational field strength is less than on Earth.

Which line, A to D, correctly describes the change, if any, in the period when compared with its value on Earth?

 
period of pendulum
period of mass-spring system
A
decrease
decrease
B
increase
increase
C
no change
decrease
D
increase
no change

Pendulum

For the pendulum g is inversely proportional to T2.

Therefore if g decreased the period would increase.

Spring

The spring system's period does not depend on g.

It depends on the mass (which is the same wherever the body is) and the spring constant.

The spring constant depends on the internal arrangment of the spring's particles.

That is unaltered by position in a graviational field.

 

Q7. A simple pendulum and a mass-spring system both have the same time period T at the surface of the Earth. If taken to another planet where the acceleration due to gravity was half that on Earth, which line, A-D, in the table gives correctly the new periods?

 
simple pendulum
mass-spring
A
T
B
T
C
D

Pendulum

For the pendulum g is inversely proportional to T2.

Therefore if g halved the period would increase (by a factor of root 2).

Spring

The spring system's period does not depend on g.

It depends on the mass (which is the same wherever the body is) and the spring constant.

The spring constant depends on the internal arrangment of the spring's particles.

That is unaltered by position in a graviational field.

 

Q8. Which one of the following statements is not true for a body vibrating in simple harmonic motion when damping is present?

A
 The damping force is always in the opposite direction to the velocity.
B
 The damping force is always in the opposite direction to the acceleration.
C
 The presence of damping gradually reduces the maximum potential energy of the system.
D
 The presence of damping gradually reduces the maximum kinetic energy of the system.

 

A damping force is a form of friction - it opposes movement and will therefore be in the opposite direction to velocity - so A is true.

Acceleration and velocity are not always opposite (see the graphs below) , so B is NOT true.

A damping force will transfer kinetic energy into heat energy thus gradually reducing the total energy of the SHM system - therefore C and D are true.

 

Q9. The frequency of a body moving with simple harmonic motion is doubled. If the amplitude remains the same, which one of the following is also doubled?

A
 the time period
B
 the total energy
C
 the maximum velocity
D
 the maximum acceleration

 

Time period would halve as T = 1/f

Energy is related to v2 (KE = 1/2 mv2) - therefore enegy would in crease by a factor of root 2.

Max velocity would double as v and f are proportional.

Acceleration relates to square of frequency so that is incorrect. It would increase by a factor of four.

 

Q10. The time period of a pendulum on Earth is 1.0 s. What would be the period of a pendulum of the same length on a planet with half the density but twice the radius of Earth?

A
0.5 s
B
1.0 s
C
1.4 s
D
2.0 s

 

Mass Effect

Assuming the planet is a sphere - volume would be 8x that of earth (cubed relationship) if radius was twice as great.

Density = mass/volume, therefore mass depends on the product of density and volume.

Volume increases mass by a factor of eight, but density decreases it by a factor of 2.

Therefore the overall effect is that mass of the planet is four times that of Earth.

increasing g by a factor of 4

Radius effect

g has a inverse square relationship with radius, so doubling the radius will

decrease g by a factor of 4

Overall effect

Planet will havethe same g as Earth

∴ T will be the same 1.0 s

Choice B