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.
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
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.
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.
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.
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 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.
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
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 \(\frac{\pi}{2}\) .
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 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.
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 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.
Assertion (A): Vibration of polyatomic molecules is not simple harmonic motion.
Reason (R): The vibrations are superposition of SHMs of different frequency.
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
Vibration of polyatomic molecules involves multiple normal modes, each with a different frequency. The total vibration is a superposition of these individual SHMs.
This complex, multi-frequency nature means the overall motion is not a single SHM. Both A and R are true, and R explains A.
Assertion (A): If the amplitude of a simple harmonic oscillator is doubled, its total energy also becomes doubled.
Reason (R): In harmonic oscillation, the total energy is directly proportional to the amplitude of vibration.
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 total energy of an SHM is `\(E = \frac{1}{2}kA^2\)`. If amplitude `\(A\)` is doubled, energy becomes `\(E' = \frac{1}{2}k(2A)^2 = 4E\)`. So A is false.
Reason R states energy is directly proportional to amplitude, which is also false (it's proportional to `\(A^2\)`). Both are false.
Assertion (A): For a system executing SHM, the mechanical energy remains constant.
Reason (R): In SHM, kinetic energy and potential energy vary periodically with double the frequency of 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
For an ideal SHM, mechanical energy is conserved (A is true). Kinetic energy `\(KE = \frac{1}{2}m\omega^2A^2\cos^2(\omega t)\)` and potential energy `\(PE = \frac{1}{2}k A^2\sin^2(\omega t)\)` vary with `\(2\omega\)`, double the SHM frequency (R is true).
However, R describes the variation of KE/PE, not the reason for conservation of total mechanical energy. Thus, R does not explain A.