Simple Harmonic Motion Test

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Simple Harmonic Motion Test - 22

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Question 1
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A hollow metal sphere is filled with water through a small hole in it. It is hung by a long thread and is made to oscillate. Water slowly flows out of the hole at the bottom. Select the correct variation of its time period.

A
The period will go on increasing till the sphere is emptied.

B
The period will go on decreasing till the sphere is emptied.

C
The period will not be affected at all.

D
The period will increase first, then decrease to the initial value after the sphere is emptied.

Solution

Initially, the center of mass is at the center of the sphere. As the water flows out of the sphere, the C.O.M. descends down increasing the effective length. Thus, the time period increases but when the water drains out completely, the C.O.M. again shifts to the center of the sphere and the time period decreases to its initial value.
Question 2
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A uniform rod of mass m and length l is suspended about its end. The time period of small angular oscillations is:

A

B

C

D

Solution

As we know, for a rod:
Question 3
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A uniform disc of mass M and radius R is suspended in a vertical plane from a point on its periphery. Its time period of oscillation is:

A

B

C

D

Solution

As we know, for a physical pendulum:

I is the moment of inertia from the point of suspension.

Using parallel axis theorem:
Question 4
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A block of mass m hangs from three springs having the same spring constant k. If the mass slightly displaced downwards, the time period oscillations will be:

A

B

C

D

Solution

Here, the upper two springs are in parallel and the lower one is in series with their combination.
Question 5
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The SHM of a particle is given by the equation x = 2 sinωt + 4 cosωt. Its amplitude of oscillations is:

A
4 units

B
2 units

C
6 units

D
2√5 units

Solution
Question 6
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The displacement of a particle is represented by the equation y = 3 cos (π/4 − ωt). The motion of the particle:

A
simple harmonic with period 2π/ω

B
simple harmonic with period π/ω

C
periodic but not simple harmonic

D
non-periodic

Solution

It represents simple harmonic motion with time period.
Question 7
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The displacement of a particle is represented by the equation y = sin³(ωt). The motion is

A
non-periodic

B
periodic but not simple harmonic

C
simple harmonic with period 2π/ω

D
simple harmonic with period π/ω

Solution

Given the equation of displacement of the particle, y=sin³ωt

We know sin3θ = 3sinθ − 4sin³θ

Hence, the motion is not SHM.

As the expression is involving sine function, hence it will be periodic.

Also sin³ωt=(sinωt)³

Hence, y=sin³ωt represents a periodic motion with period 2π/ω.
Question 8
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The relation between acceleration and displacement of four particles are given below.

Which one of the particles is exempting simple harmonic motion?

A
a₁ = + 2x

B
a₁ = + 2x²

C
a₁ = - 2x²

D
a₁ = - 2x

Solution

Hint: Use the relation between the acceleration and displacement in SHM.

For the motion to be SHM, the acceleration of the particle must be proporrtional to negative displacement.

i.e.,

a∝-y

We should clear that acceleration should be linear with the displacement.
Question 9
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A particle is acted simultaneously by mutually perpendicular simple harmonic motion x=acosωt and y = asinωt. The trajectory of motion of the particle will be:

A
an ellipse

B
a parabola

C
a circle

D
a straight line

Solution

It is an equation of circle. Thus, trajectory of motion will be circle.
Question 10
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The displacement of a particle varies with time according to the relation y = asinωt + bcosωt.

A
The motion is oscillatory but not SHM.

B
The motion is SHM with amplitude a + b.

C
The motion is SHM with amplitude a² +b²

D
The motion is SHM with amplitude

Solution