Question

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\). (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(\mathrm{b}\) (C) \(c\) (D) \(\mathrm{d}\) Statement \(-1:\) For a particle executing SHM, the amplitude and phase is decided by its initial position and initial velocity. Statement \(-2:\) In a SHM, the amplitude and phase is dependent on the restoring force. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)

Short answer

(c) Statement $-1$ is true, statement $-2$ is false

Step-by-step solution

Analyzing Statement 1

In Simple Harmonic Motion (SHM), the equation of position is given by: \[x(t) = A\cos(\omega t + \phi)\] Here, A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase. Initial position (\(x(0)\)) and initial velocity (\(\dot{x}(0)\)) are given by: \(x(0) = A\cos(\phi)\) \(\dot{x}(0) = - A\omega\sin(\phi)\) With this information, we can find the amplitude A and phase φ, which indicates that statement 1 is true.

Analyzing Statement 2

The restoring force in SHM is given by: \(F = -kx\) Here, k is the spring constant and x is the position. The restoring force determines the angular frequency ω since: \(\omega^2 = \frac{k}{m}\), where m is the mass of the particle. However, the amplitude A and the phase φ directly depend on the initial conditions like position and velocity, not on the restoring force. Therefore, statement 2 is false.

Checking if Statement 2 correctly explains Statement 1

Since statement 2 is false, it cannot be a correct explanation for statement 1.

Choosing the correct option

We have found that statement 1 is true, and statement 2 is false. Therefore, the correct option is (c).