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).

Key concepts

Understanding Amplitude in SHM

In simple harmonic motion (SHM), the amplitude represents the maximum displacement from the equilibrium position. This means how far the particle can move from its central resting state. The amplitude is a fundamental property of the oscillation, indicating how "big" or intense the motion is. Think of it as the height of a wave.
Knowing the amplitude helps us understand the energy in the system since energy is proportional to the square of the amplitude. A higher amplitude results in greater energy stored in the system.
In SHM, the amplitude is determined by the initial conditions of the system, particularly the initial position and velocity of the particle. When the initial velocity is higher, or the initial displacement is further from the equilibrium, the amplitude will be larger. It is crucial to grasp that the amplitude in SHM doesn’t rely on the restoring force, but rather on these specific initial conditions.

Exploring the Phase in Simple Harmonic Motion

The phase (\(\phi\)) in simple harmonic motion defines the initial angle of the oscillation at time zero. In equations like \(x(t) = A\cos(\omega t + \phi)\), the phase shift \(\phi\) tells us where in its cycle the particle begins. This affects whether the motion starts at an extreme position, at the equilibrium, or somewhere in between.
The phase can be thought of as a "starting point reset". Altering \(\phi\) shifts the whole motion forward or backward in time. It's essentially the clock that marks when the oscillation begins.
A key takeaway is that phase, similar to amplitude, depends on initial position and velocity. So while phase significantly determines the particle's motion path, it is not tied to the restoring force. Instead, it maps out the timeline of the particle based on where and how fast it starts its journey.

The Role of Restoring Force in SHM

The restoring force in simple harmonic motion is the force that pulls the particle back towards its equilibrium or central position. This force is crucial in enabling periodic motion, ensuring that the system oscillates rather than drifting away.
This force is characterized by its dependency on the negative displacement, represented as \(F = -kx\), where \(k\) is the spring constant and \(x\) is the displacement. The negative sign signifies that the force is always directed opposite to the displacement.
While the restoring force determines the angular frequency \(\omega\) (since \(\omega^2 = \frac{k}{m}\), where \(m\) is the mass), it does not affect the amplitude or phase. The role of the restoring force is purely about maintaining the oscillation at a specific rate, irrespective of how the motion initially starts. Therefore, it's fundamental in defining the rhythm and consistency of SHM, rather than the starting parameters like amplitude and phase.