JEE Physics Oscillations MCQs Set 26

Answer: (b) A-3, B-4, C-1, D-2

Explanation:
1. A satellite repeats its orbit, so periodic \( (A-3) \).
2. A small-angle pendulum is SHM \( (B-4) \).
3. A string vibrating in air loses energy, so damped \( (C-1) \).
4. The paper rider flies off when the string is driven at its natural frequency, which is resonance \( (D-2) \).

Teacher's Note:
1. Identify each type of motion.

Answer: (c) a sin(ωt − kx)

Explanation:
1. A progressive wave is a function of the combination (\( \omegat \) − kx) or (\( \omegat \) + kx).
2. Only a sin(ωt − kx) has this form.

Teacher's Note:
1. Progressive wave: \( \omegat \) ∓ kx.

Answer: (d) zero

Explanation:
1. After one complete period the particle returns to its starting point.
2. The displacement is therefore zero (the distance covered is \( 4A \)).

Teacher's Note:
1. Displacement is zero after a period.

Answer: (c) on a circle

Explanation:
1. \( x = A \cos \omegat \) and \( y = A \sin \omegat \) give \( x^{2} + y^{2} = A^{2} \).
2. The particle moves on a circle of radius \( A \).

Teacher's Note:
1. Eliminate \( t \).

Answer: (b) elasticity as well as inertia

Explanation:
1. A restoring (elastic) force brings the body back, and inertia carries it past the mean position.
2. Both are needed for oscillation.

Teacher's Note:
1. Elasticity plus inertia.

Answer: (d) \( \frac2b \)

Explanation:
1. The amplitude decays as \( e^{-bt/2} \).
2. It falls to \( 1/e \) when bτ/2 = 1, so τ = \( 2/b \).

Teacher's Note:
1. A = \( A_{0}e^{-bt/2} \).

Answer: (c) \( \omega\sqrt{a^2-y^2} \)

Explanation:
1. \( v = \omega\sqrt{a^{2} - y^{2}} \) from energy conservation or differentiation.

Teacher's Note:
1. \( v = \omega\sqrt{a^{2} - y^{2}} \).

Answer: (d) A is false, but R is true.

Explanation:
1. At the extremes the velocity is zero but the acceleration is maximum, so the assertion is false.
2. The reason is a true statement.

Teacher's Note:
1. a is maximum at the extremes.

Answer: (d) all of these

Explanation:
1. The drag coefficient depends on the viscosity of the medium and on the size and shape of the body.

Teacher's Note:
1. Drag depends on the medium and the body.

Answer: (c) 39 cm

Explanation:
1. Submerged depth \( h = 54 \times 650/900 = 39 \) cm.
2. The block oscillates with \( T \) = 2π√(ρ_b H/(ρ_l g)) = 2π√(h/g,) the same as a pendulum of length 39 cm.

Teacher's Note:
1. Equivalent length is the submerged depth.

Answer: (c) 1 : 4

Explanation:
1. \( E ∝ A^{2} \).
2. The ratio is 5² : \( 10^{2} = 1 \) : 4.

Teacher's Note:
1. \( E ∝ A^{2} \).

Answer: (d) \( \sqrt2\,T \)

Explanation:
1. \( T = 2\pi\sqrt{m/k} \) and the mass becomes \( 2M \).
2. The new period is \( \sqrt{2} T \).

Teacher's Note:
1. \( T ∝ \sqrt{m} \).

Answer: (c) displacement

Explanation:
1. \( a = -\omega^{2}x \).
2. It is proportional to the displacement.

Teacher's Note:
1. a \( ∝ -x \).

Answer: (d) It is never zero

Explanation:
1. The total energy is ½kA², which is constant and non-zero.
2. It is the sum of kinetic and potential energy at every instant.

Teacher's Note:
1. \( E \) = ½kA².

Answer: (c) II and III

Explanation:
1. \( T = 2\pi\sqrt{m/k} \) depends on \( m \) and \( k \) only.
2. It is independent of the amplitude.

Teacher's Note:
1. \( T = 2\pi\sqrt{m/k} \).

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. SHM is a to-and-fro periodic motion, so the assertion is true.
2. The velocity formula is also true, but it does not by itself explain why the motion is periodic.

Teacher's Note:
1. \( v = \omega\sqrt{A^{2} - x^{2}} \).

Answer: (a) acceleration = −k(x + a)

Explanation:
1. SHM needs acceleration proportional to the displacement from some mean position and directed towards it.
2. \( -k(x + a) \) has this form about \( x = -a \).

Teacher's Note:
1. \( a = -k(x - x_{0}) \).

Answer: (a) \( \frac16\ \text{s} \)

Explanation:
1. KE = 75% of \( E \) means PE = 25%, so \( x = A/2 \).
2. A \( \sin \omegat = A/2 \) gives \( \omegat = \pi/6 \), and with \( \omega = \pi \) rad/s, \( t = 1/6 s \).

Teacher's Note:
1. Find \( x \) from PE = \( E/4 \).

Answer: (c) oscillatory

Explanation:
1. The particle falls through the tunnel and rises to the same height on the other side, repeating the journey.
2. The motion is periodic and oscillatory, but not simple harmonic because the force above the surface is not proportional to the displacement.

Teacher's Note:
1. Outside the earth the force is not linear.

Answer: (d) frequency

Explanation:
1. The frequency (and the time period) depend only on the system and do not change during the motion.
2. The other quantities vary with position.

Teacher's Note:
1. \( f \) is constant.

Answer: (b) II only

Explanation:
1. SHM needs a force proportional to the displacement and directed towards the mean position, that is a restoring force.
2. Acceleration merely proportional to displacement could also point away from the mean position.

Teacher's Note:
1. Direction matters.

Answer: (b) the frequency of the driving force

Explanation:
1. In the steady state the system oscillates at the frequency of the external driving force.

Teacher's Note:
1. Steady state follows the driver.

Answer: (d) The graph between the length of the pendulum and the time period is a parabola

Explanation:
1. \( T ∝ \sqrt{l} \), so \( l \) = gT²/4π² is a parabola in \( T \).
2. For small angles \( T \) is independent of amplitude, and a spring watch does not depend on \( g \).

Teacher's Note:
1. \( l ∝ T^{2} \).

Answer: (a) independent of x

Explanation:
1. The total energy ½kA² is constant.
2. It does not depend on the displacement \( x \).

Teacher's Note:
1. \( E \) is constant.

Answer: (b) \( 3^3 \)

Explanation:
1. The amplitude falls to 1/3 every \( 2 s \).
2. After \( 6 s \) it is \( (1/3)^{3} = 1/27 \) of the original, so \( n = 3^{3} \).

Teacher's Note:
1. Three intervals of \( 2 s \).