Simple Harmonic Motion Test

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Simple Harmonic Motion Test - 9

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Question 1
4 / -1

A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

A

B

C

D

Solution

Let the piston be displaced through distance x towards left, then volume decreases, pressure increases. If ΔP is increase in pressure and ΔV is decrease in volume, then considering the process to take place gradually (i.e. isothermal)

This excess pressure is responsible for providing the restoring force (F) to the piston of mass M
Question 2
4 / -1

One end of a spring of force constant k is fixed to a vertical wall and the other end is connected to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance x₀ from the body. The spring is now compressed by 3x₀ and released. The time after which the block will strike the other wall is

A

B

C

D

Solution

Amplitude of the motion, A = 2 × 0

We know that the time needed to cover from compressed

position to mean position = T/4

For the displacement from the mean position to

X_O time taken t will be

Y = Asinwt

∴ X_O = 2X_Osinwt
Question 3
4 / -1

Two particles move parallel to x-axis about the origin with same amplitude and frequency. At a certain instant they are found at a distance a/3 from the origin on opposite sides but their velocities are found to be in same direction.Then the phase difference between two particles will be

A

B
180⁰

C
45⁰

D
None of the above

Solution

Let phase difference between them is ϕ .
Question 4
4 / -1

A particle performs SHM with a period T and amplitude a.The mean velocity of the particle over the time interval during which it travels a distance a/2 from the extreme position is

A
a/T

B
2a/T

C
3a/T

D
a/2T

Solution

Since the particle is starting from extreme position. We can take its equation as x = a cos ωt

at t = t₁ particle is at extreme position, i.e. x = a, we have

at t = t₂ particle is at a distance a/2 from extreme position, i.e. x = a/2, we have a/2 = a cos ωt₂
Question 5
4 / -1

A body is executing SHM, At a displacement x, its potential energy is E₁ and at a displacement y, its potential energy is E₂. Find its potential energy E at displacement ( x + y)

A

B

C
none of these

D
all of these

Solution
Question 6
4 / -1

Two identical springs are attached to a small block P. The other ends of the springs are fixed at A and B. When P is in equilibrium the extension of top spring is 20 cm and extension of bottom spring is l0 cm. The period of small vertical oscillations of P about its equilibrium position is (use g = 9.8 m/s₂)

A
2π/7 sec

B
π/7 sec

C
2π/5 sec

D
none of these

Solution
Question 7
4 / -1

If two simple pendulums first of bob mass M₁ and length l₁, second of bob mass M₂ and length l₂. Given M₁=M₂ and l₁=2l₂. If the vibrational energies of both are same, then which of the following is correct ?

A
Amplitude of B is smaller than A

B
Amplitude of B is greater than B

C
Amplitude will be same

D
None of the above

Solution
Question 8
4 / -1

A particle is executing a motion in which its displacement as a function of time is given by x = 3sin ⁡(5πt + π/3) + cos ⁡(5πt + π/3) where x is in m and t is in s. Then the motion is

A
simple harmonic with time period 0.2 s

B
simple harmonic with time period 0.4 s

C
simple harmonic with amplitude 3 m

D
not a simple harmonic but a periodic motion

Solution
Question 9
4 / -1

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions. Find the phase difference between the individual motions.

A
3π/2

B
2π/3

C
2π

D
π

Solution
Question 10
4 / -1

one end of a spring of force constant ′k′ is fixed to a vertical wall and the other to a block of mass ′in′ resting on a smooth horizontal surface. There is another wall at a distance ′x₀′ from the block. The spring is then compressed by 2x₀ and released. The time taken by the block to strike the other wall is

A

B

C

D

Solution

According to the question,