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.
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.
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): The graph between velocity and displacement for a harmonic oscillator is a parabola.
Reason (R): Velocity does change uniformly with displacement in harmonic motion.
For a harmonic oscillator, velocity \( v \) and displacement \( x \) are related by \( v = \omega \sqrt{A^2 - x^2} \). Squaring this gives \( v^2 = \omega^2 (A^2 - x^2) \), which is an equation of an ellipse, not a parabola. So (A) is false. Velocity does not change uniformly with displacement, hence (R) is also false. Thus, both A and R are false.
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.
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): The graph of potential energy and kinetic energy of a particle in SHM with respect to position is a parabola.
Reason (R): The potential energy and kinetic energy of a particle in SHM, do not vary linearly with position.
Potential energy in SHM is \( PE = \frac{1}{2} kx^2 \), which is a parabola. Kinetic energy is \( KE = \frac{1}{2} k(A^2 - x^2) \), also a parabola. So (A) is true. Since both are quadratic functions of position \( x \), they do not vary linearly. Thus, (R) is true and correctly 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.
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.
Assertion (A): Maximum potential energy in simple harmonic motion is equal to net mechanical energy.
Reason (R): Maximum kinetic energy in simple harmonic motion is equal to net mechanical energy.
In SHM, total mechanical energy \( E \) is conserved. At extreme positions (maximum displacement), kinetic energy is zero, so \( E = PE_{max} \). Thus (A) is true. At the equilibrium position, potential energy is zero, so \( E = KE_{max} \). Thus (R) is true. However, (R) does not explain (A); both are independent statements describing energy distribution in SHM.
Assertion (A): Sine and cosine functions are periodic functions.
Reason (R): Sinusoidal functions repeat its values after a definite interval of time.
Periodic functions like sine and cosine repeat their values over a fixed period. Reason (R) defines periodicity, which directly explains Assertion (A).
Thus, both are true, and R explains A.
Assertion (A): In SHM the velocity is maximum when the acceleration is minimum.
Reason (R): Displacement and velocity in SHM differ in phase by \(\frac{\pi}{2}\) .
In SHM, velocity is max at equilibrium (where displacement is zero), and acceleration is min (zero) at equilibrium. So A is true.
Displacement `\(x = A\sin(\omega t)\)` and velocity `\(v = A\omega\cos(\omega t)\)` differ in phase by \(\frac{\pi}{2}\). So R is true.
However, R explains phase relation, not why maximum velocity occurs at minimum acceleration. Hence, R does not explain A.
Assertion (A): The periodic time of a hard spring is less as compared to that of a soft spring.
Reason (R): The spring constant is large for hard spring.
The periodic time of a spring is given by `\(T = 2\pi\sqrt{\frac{m}{k}}\)`.
A hard spring has a large spring constant `\(k\)`, which means a smaller `\(T\)`. A soft spring has a small `\(k\)`, hence a larger `\(T\)`. Both A and R are true, and R explains A.