Assertion (A): For large angle in simple pendulum \(T > 2\pi \sqrt{\frac{l}{g}}\)
Reason (R): \(sin \theta < \theta\), if the restoring force. \(mg sin \theta\) is replaced by \(mg \theta\), this amounts to effective reduction in g for large angle, hence an increase in T.
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 increases for large amplitudes compared to the small angle approximation. Reason (R) is true: For large angles, \(sin \theta < \theta\). This makes the actual restoring force smaller than the linear approximation, effectively reducing the 'g' and thereby increasing the period \(T\). (R) correctly explains (A).
Assertion (A): For a physical pendulum period of oscillation is maximum about an axis passes through centre of mass.
Reason (R): A physical pendulum is in neutral equilibrium about centre of mass.
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 false. If a physical pendulum is pivoted at its center of mass, it will be in neutral equilibrium and will not oscillate, so there is no period. Reason (R) is true. A body pivoted at its center of mass is indeed in neutral equilibrium. Thus, (A) is false and (R) is true.
Assertion (A): Two particles are in SHM with same time period, same amplitude, same position and same speed are in the same phase.
Reason (R): Phase of particle depends on position and speed of 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
The position of a particle in SHM is given by \(x = A cos(omega t + phi)\) and velocity by \(v = -Aomega sin(omega t + phi)\). Given same amplitude \(A\), time period \(T\) (thus \(omega\)), position \(x\), and speed \(v\), the phase \(phi\) must be the same. Thus, A and R are true and R is the correct explanation of A.
Assertion (A): Time period of partially immersed spring block system is less than full immersed spring block system.
Reason (R): Time period of spring system is independent of changing values of g.
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 a partially immersed block, the effective spring constant is \(k_{eff} = k + \rho_l A g\), leading to \(T_p = 2\pi \sqrt{m/(k + \rho_l A g)}\). For a fully immersed block, \(k_{eff} = k\), so \(T_f = 2\pi \sqrt{m/k}\). Since \(k + \rho_l A g > k\), \(T_p < T_f\). So A is true. The time period of a simple spring-mass system \(T = 2\pi \sqrt{m/k}\) is independent of \(g\). So R is true. However, R does not explain A because the change in period is due to effective spring constant, not general independence from \(g\).
Assertion (A): In forced oscillations, the steady state motion of the particle (after natural oscillations die out) is SHM whose frequency is the frequency of the driving frequency \(\omega_d\), not the natural frequency \(\omega\) of the particle.
Reason (R): In forced oscillation \(\omega_d\) should be greater than natural frequency \(\omega\) 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
In forced oscillations, the system eventually settles into oscillating at the frequency of the driving force, \(\omega_d\). So Assertion (A) is true. The driving frequency \(\omega_d\) can be any value (less than, equal to, or greater than) compared to the natural frequency \(\omega\). So Reason (R) is false.
Assertion (A): For a physical pendulum if distance of point of suspension from centre of mass increases time period first decreases then increases.
Reason (R): For a physical pendulum there is some distance from centre of mass at which frequency of oscillation is maximum.
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 physical pendulum is \(T = 2pi sqrt{(I_{CM} + mL^2)/(mgL)}\). Analyzing this function, \(T\) has a minimum value at \(L = sqrt{I_{CM}/m}\). This minimum time period corresponds to a maximum frequency. Thus, as \(L\) increases, \(T\) first decreases to a minimum and then increases. Both Assertion (A) and Reason (R) are true, and R correctly explains A.
Assertion (A): If a pendulum clock is taken to a mountain top, its time period decreases.
Reason (R): Value of acceleration due to gravity is more at heights.
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 simple pendulum is \(T = 2\pi \sqrt{L/g}\). At a mountain top, the altitude increases, causing the acceleration due to gravity \(g\) to decrease. A decrease in \(g\) leads to an increase in \(T\). Thus, Assertion (A) is false. Reason (R) is also false as \(g\) decreases, not increases, at higher altitudes.
Assertion (A): A small body suspended by a light spring performing SHM. When the entire system is immersed in a nonviscous liquid period of oscillation does not change.
Reason (R): The angular frequency of oscillation of the particle does not change.
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 period of a spring-mass system is given by \( T = 2\pi \sqrt{\frac{m}{k}} \). Immersion in a nonviscous liquid does not change the mass \( m \) or the spring constant \( k \). Hence, the period \( T \) remains unchanged. So (A) is true. Angular frequency is \( \omega = 2\pi / T \), so if \( T \) does not change, \( \omega \) also does not change. So (R) is true and explains (A).
Assertion (A): A simple pendulum is attached on a roof of a elevator. Time period of SHM is \( T \) when elevator is at rest. Time period of SHM must be greater than \( T \) if elevator start moving upward.
Reason (R): Time period of simple pendulum does not depend on acceleration due to gravity.
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 simple pendulum is \( T = 2\pi \sqrt{\frac{L}{g}} \). If the elevator accelerates upward with \( a \), the effective gravity becomes \( g_{eff} = g + a \). The new period is \( T' = 2\pi \sqrt{\frac{L}{g+a}} \). Since \( g+a > g \), then \( T' < T \). So (A) is false. The time period *does* depend on gravity, so (R) is false. Both (A) and (R) are false.