Assertion (A): Amplitude of SHM \(x = 4sin^2\omega t + 2cos^2\omega t + 2sin\omega t cos\omega t\) is \(sqrt{2}\).
Reason (R): Angular frequency of given equation is \(2\omega\).
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 expression \(x = 4sin^2\omega t + 2cos^2\omega t + 2sin\omega t cos\omega t\) simplifies to \(x = 3 + sin(2\omega t) - cos(2\omega t)\). The oscillatory part is \(sin(2\omega t) - cos(2\omega t)\). Assertion (A) is true, its amplitude is \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\). Reason (R) is true, the angular frequency is \(2\omega\). However, the angular frequency does not explain the specific amplitude value.
Assertion (A): In SHM acceleration leads displacement by phase \(\pi\).
Reason (R): In SHM velocity leads displacement by phase \(\pi/2\).
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
Assertion (A) is true. If displacement \(x = Asin(\omega t)\), then acceleration \(a = -A\omega^2sin(\omega t) = A\omega^2sin(\omega t + \pi)\). Reason (R) is true. Velocity \(v = A\omega cos(\omega t) = A\omega sin(\omega t + \pi/2)\). Both statements are true, but the phase relationship of velocity with displacement does not explain the phase relationship of acceleration with displacement directly; they are separate facts of SHM.
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.
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 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): 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): 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.
The differential equation of motion of a particle executing SHM is \(\frac{d^2y}{dt^2} + Ky = 0\) where K is positive constant. The time period of the oscillation is given by
1. \(\frac{2\pi}{K}\)
2. \(2\pi K\)
3. \(\frac{2\pi}{\sqrt{K}}\)
4. \(2\pi\sqrt{K}\)
View Answer
Comparing with the standard equation of SHM, \(\frac{d^2y}{dt^2} + \omega^2 y = 0\), we get \(\omega = \sqrt{K}\). Therefore, the time period is \(T = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{K}}\).
The displacement of a harmonic oscillator is given by \(x = \alpha \sin\omega t + \beta \cos\omega t\). The amplitude of the oscillation is
1. \(\alpha + \beta\)
2. \(\alpha^2 + \beta^2\)
3. \(\sqrt{\alpha^2 + \beta^2}\)
4. \(\sqrt{\alpha^2 + \beta^2 + 2\alpha\beta}\)
View Answer
Since the two perpendicular components have a phase difference of \(\frac{\pi}{2}\), the net amplitude is \(A = \sqrt{\alpha^2 + \beta^2}\).
A body is vibrating with SHM of amplitude \(15\text{ cm}\) and frequency \(4\text{ Hz}\). The maximum value of acceleration is
1. \(10.65\text{ m s}^{-2}\)
2. \(80.52\text{ m s}^{-2}\)
3. \(94.65\text{ m s}^{-2}\)
4. \(68.52\text{ m s}^{-2}\)
View Answer
The maximum acceleration is given by \(a_{\text{max}} = \omega^2 A = (2\pi f)^2 A\). Substituting \(f = 4\text{ Hz}\) and \(A = 0.15\text{ m}\) gives \(a_{\text{max}} = 4\pi^2 (16)(0.15) \approx 94.65\text{ m/s}^2\).