Simple Harmonic Motion Test

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Simple Harmonic Motion Test - 5

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Question 1
1 / -0

A small sphere carrying a charge ‘q’ is hanging in between two parallel plates by a string of length L. Time period of pendulum is T₀. When parallel plates are charged, the electric field between the plates is E and time period changes to T. The ratio T/T₀ is equal to

A

B

C

D
None of these

Solution

Compare this system with a simple pendulum oscillating only under gravity. Calculate effective acceleration in a vertically downward direction.

Step 1: Draw a diagram as follows.

Step2: Calculate effective acceleration g'.

Net downward force mg' = mg+QE

Step3 : Calculate timeperiod by using the effective acceleration.
Question 2
1 / -0

A simple pendulum has a time period T₁ when on the earth’s surface and T₂ when taken to a height R above the earth’s surface, where R is the radius of the earth. The value of T₂/T₁ is:

A
1

B
√2

C
4

D
2

Solution

If acceleration due to gravity is g at the surface of Earth, then at height R its value becomes
Question 3
1 / -0

A particle executes linear simple harmonic motion with an amplitude of of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then, its time period in seconds is

A
√5/π

B
√5/2π

C
4π/√5

D
2π/√3

Solution

magnitude of velocity of particle when it is at displacement x from mean position

Also, magnitude of acceleration of particle in SHM

= ω²x

Given, when x=2cm

|v|=|a|

⇒Angular velocity ω = √5/2

So, Time period of motion
Question 4
1 / -0

When two displacements represented by y₁=asin(ωt) and y₂=bcos(ωt) are superimposed,the motion is -

A
not a simple harmonic

B
simple harmonic with amplitude a/b

C
simple harmonic with amplitude √a² + b²

D
simple harmonic with amplitude (a+b)/2

Solution

Given,

y₁= a sin ωt

y₂= b cos ωt = b sin (ωt + π/2)

The resultant displacement is given by;

Hence, the motion of superimposed wave is simple harmonic with amplitude √a²+b² .
Question 5
1 / -0

The displacement of a particle along the r-axis is given by x = asin²ωt,t. The motion of the particle corresponds to

A
simple harmonic motion of frequency ω/π

B
simple harmonic motion of frequency 3ω/2π

C
non simple harmonic motion

D
simple harmonic motion of frequency ω/2π

Solution

For the given displacement x a sin²ω/t,

a ∝-x is not satisfied

Hence, the motion of the particle is non simple harmonic motion.

Note : The given motion is a periodic motion with a time period
Question 6
1 / -0

The period of oscillation of a mass M suspended from a spring of negligible mass is T. If along with it another mass M is also suspended, the period of oscillation will now be

A
T

B
T/√2

C
2T

D
√2T

Solution
Question 7
1 / -0

A body performs simple harmonic motion about x=0 with an amplitude a and a time period T. The speed of the body at x = a/2 will be:

A

B

C

D

Solution
Question 8
1 / -0

Which one of the following equations of motion represents simple harmonic motion?

where k, k₀, k₁ and αα are all positive.

A
Acceleration = -k₀x+k₁x²

B
Acceleration = -k (x+a)

C
Acceleration = k (x+a)

D
Acceleration =kx

Solution

Acceleration ∝−(displacement).

Here, y=x+a

∴ acceleration =-k(x+a)
Question 9
1 / -0

Two simple harmonic motions of angular frequency 100 rad s ^-1 and 1000 rad s⁻¹ have the same displacement amplitude. The ratio of their maximum acceleration will be

A
1:10

B
1:10²

C
1:10³

D
1:10⁴

Solution

Acceleration of simple harmonic motion is
Question 10
1 / -0

A point performs simple harmonic oscillation of period T and the equation of motion is given by x= a sin (ωt +π/6). After the elapse of what fraction of the time period the velocity of the point will be equal to half to its maximum velocity?

A
T/8

B
T/6

C
T/3

D
T/12

Solution

Velocity is the time derivation of displacement. Writing the given equation of a point performing SHM

Differentiating Eq. (i),w.r.t. time, we obtain

It is given that v = aω/2, so that

Thus, at T/12 velocity of the point will be equal to half of its maximum velocity.