Question

Which one of the following statements is true for the speed 'v' and the acceleration 'a' of a particle executing simple harmonic motion

• A. Value of a is zero, whatever may be the value of 'v'
• B. When 'v' is zero, a is zero
• C. When 'v' is maximum, a is zero
• D. When 'v' is maximum, a is maximum

Answer 1

The Correct Option is C

To determine which statement about the speed 'v' and acceleration 'a' of a particle executing simple harmonic motion (SHM) is true, we need to recall some key characteristics of SHM.

In simple harmonic motion, the acceleration 'a' is always directed towards the mean position and is given by:

a = -\omega^2 x

where \omega is the angular frequency, and 'x' is the displacement from the mean position. The negative sign indicates that acceleration is always directed towards the center.

Similarly, the velocity 'v' in SHM is maximum at the mean position and zero at the extreme positions. This is because at the mean position, the potential energy is zero (all energy is kinetic), and at the extremes, the velocity is zero because the entire energy is potential.

Let's analyze each option:

1. Value of 'a' is zero, whatever may be the value of 'v'.

   This is incorrect. The value of acceleration 'a' depends on the displacement 'x' (i.e., it is a = -\omega^2 x), and is not zero unless 'x' is zero (at mean position). So, 'a' cannot be zero for all 'v'.

2. When 'v' is zero, 'a' is zero.

   This is incorrect. When 'v' is zero, the particle is at an extreme point, and at this position, acceleration 'a' is maximum (not zero) as it tries to restore the particle to the mean position.

3. When 'v' is maximum, 'a' is zero.

   This is correct. When 'v' is maximum, the particle is at the mean position, and the displacement 'x' is zero. Therefore, acceleration 'a' is zero, fulfilling the condition a = -\omega^2 x = 0.

4. When 'v' is maximum, 'a' is maximum.

   This is incorrect. As explained, when 'v' is maximum, acceleration 'a' is actually zero because the particle is at the mean position.

Thus, the correct statement is: When 'v' is maximum, 'a' is zero.