If maximum speed and maximum acceleration of a particle executing SHM is found to be 5 cm/s and \( 50\pi \, \text{cm/s}^2 \) respectively, then its time period will be
1. 5 s
2. 2 s
3. \( \frac{1}{5} \, \text{s} \)
4. \( \frac{1}{10} \, \text{s} \)
View Answer
Using \( v_{\text{max}} = A \omega = 5 \) and \( a_{\text{max}} = A \omega^2 = 50\pi \), we get \( \omega = \frac{a_{\text{max}}}{v_{\text{max}}} = \frac{50\pi}{5} = 10\pi \, \text{rad/s} \). The time period is \( T = \frac{2\pi}{\omega} = \frac{2\pi}{10\pi} = \frac{1}{5} \, \text{s} \).
The differential equation for a particle executing S.H.M. is given by \( \frac{d^2 y}{dt^2} + 4y = 0 \), where symbols have their usual meaning. The angular velocity of the particle is given by
1. \( 4\text{ rad/s} \)
2. \( 3\text{ rad/s} \)
3. \( 2\text{ rad/s} \)
4. \( 4pi\text{ rad/s} \)
View Answer
The standard differential equation of S.H.M. is \( \frac{d^2 y}{dt^2} + \omega^2 y = 0 \). By comparison, \( \omega^2 = 4 ⇒ \omega = 2\text{ rad/s} \).
Assertion (A): A hole were drilled through the centre of earth and a ball is dropped into the hole at one end, it will not get out of other end of the hole.
Reason (R): Ball will execute simple harmonic motion inside the hole.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Concept of gravity inside earth and SHM. Inside Earth, \(F = -k r\). A ball dropped in a hole through the Earth's center oscillates in SHM. It reaches the other end with zero velocity and turns back, thus not 'getting out' (escaping) the hole.
Both A and R are true, and R explains A.
A vertical spring block system is made to oscillate.
Assertion (A): Its time period on earth is more than that on the moon.
Reason (R): Its extension on moon (in equilibrium) is more than that on the earth.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
The time period of a vertical spring-block system is \(T = 2\pi \sqrt{\frac{m}{k}}\), which is independent of gravity (g). So (A) is false. The equilibrium extension is \(x_{eq} = \frac{mg}{k}\). Since \(g_{moon} < g_{earth}\), then \(x_{eq,moon} < x_{eq,earth}\). So (R) is also false.
Assertion (A): Total mechanical energy in SHM is conserved.
Reason (R): Kinetic energy of SHM at mean position is equal to potential energy at ends for a particle moving in SHM.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
In ideal SHM, total mechanical energy is conserved because the restoring force is conservative. So (A) is true. At the mean position, \(KE_{max} = \frac{1}{2}m(A\omega)^2\), and at the ends, \(PE_{max} = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2\). Thus, \(KE_{mean} = PE_{ends}\). So (R) is true. However, (R) describes a consequence of energy conservation, not the fundamental reason for it.
Assertion (A): A SHM may be assumed as composition of many SHM’s.
Reason (R): Superposition of many SHM’s (along same line) of same frequency will be a SHM.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) states that an SHM can be viewed as a composition of multiple SHMs. Reason (R) states that the superposition of multiple SHMs along the same line and with the same frequency results in another SHM.
Both statements are true, and (R) provides the explanation for how (A) can be possible.
Assertion (A): Displacement-time equation of a particle moving along \(x\)-axis is \(x = 4 + 6 sin\omega t\). Under this situation, motion of particle is not simple harmonic.
Reason (R): \(\frac{d^2x}{dt^2}\) for the given equation is not proportional to \(-x\).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For SHM, \(a = -\omega^2 (x-x_0)\). Given \(x = 4 + 6 sin\omega t\), the equilibrium position is \(x_0=4\). The acceleration is \(\frac{d^2x}{dt^2} = -\omega^2 (x-4)\). If SHM is strictly defined as \(a \propto -x\) (equilibrium at origin), then (A) is true. (R) is also true as \(\frac{d^2x}{dt^2}\) is proportional to \(-(x-4)\), not \(-x\). (R) explains (A).
Assertion (A): For a particle performing SHM, its speed decreases as it goes away from the mean position.
Reason (R): In SHM, the acceleration is always opposite to the velocity of the particle.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
As a particle in SHM moves from the mean to extreme position, its speed decreases as the restoring force opposes motion. So (A) is true. Acceleration is always directed towards the equilibrium. When moving towards equilibrium, velocity and acceleration are in the same direction, so (R) is false.
Assertion (A): We can assume damped oscillation to be approximately periodic motion for small damping
Reason (R): Small damping means \( \frac{b}{\sqrt{km}} \ll 1 \)
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true: For small damping, the amplitude decays slowly, and the frequency is nearly constant, making the motion approximately periodic. Reason (R) is true: Small damping is characterized by a small damping factor \(b\) relative to \(\sqrt{km}\), specifically \( \frac{b}{\sqrt{km}} \ll 1\) (or \( \zeta ll 1\)). This condition directly ensures the motion is approximately periodic.
Assertion (A): When a simple pendulum is made to oscillate on the surface of moon, its time period increases.
Reason (R): Gravity at moon is less than gravity at earth.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true: The period of a simple pendulum is inversely proportional to the square root of 'g' (\(T = 2pi sqrt{l/g}\)). Reason (R) is true: Gravity on the moon is significantly less than on Earth. Since 'g' decreases, the period 'T' increases, making (R) the correct explanation for (A).