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.