Question

In a simple harmonic motion, ________.

• A. the velocity is constant
• B. the motion is periodic
• C. the acceleration is directly proportional to velocity
• D. the acceleration is along the direction of displacement
• E. the motion must be along a straight line

Hint:

All SHM is periodic, but not all periodic motion is SHM.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Simple Harmonic Motion (SHM) is a specific type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. We need to evaluate the given statements based on the defining characteristics of SHM.
Step 2: Key Formula or Approach
The defining equation of SHM is:
\[ F = -kx \] where \(F\) is the restoring force, \(x\) is the displacement from the equilibrium position, and \(k\) is a positive constant.
Since \(F = ma\), the acceleration \(a\) is given by:
\[ a = -\frac{k}{m}x = -\omega^2 x \] where \(\omega = \sqrt{k/m}\) is the angular frequency.
The velocity in SHM is given by \(v = \pm \omega \sqrt{A^2 - x^2}\), where A is the amplitude.
Step 3: Detailed Explanation
Let's analyze each option:
(A) the velocity is constant: Incorrect. The velocity continuously changes, being maximum at the equilibrium position (\(x=0\)) and zero at the extreme positions (\(x=\pm A\)).
(B) the motion is periodic: Correct. By definition, SHM is a type of periodic motion, meaning it repeats itself over a regular time interval (the period, \(T = 2\pi/\omega\)). All simple harmonic motions are periodic, but not all periodic motions are simple harmonic.
(C) the acceleration is directly proportional to velocity: Incorrect. The acceleration is directly proportional to the negative of the displacement (\(a \propto -x\)).
(D) the acceleration is along the direction of displacement: Incorrect. The negative sign in \(a = -\omega^2 x\) indicates that the acceleration is always directed opposite to the displacement vector. It always points towards the equilibrium position.
(E) the motion must be along a straight line: Incorrect. While linear SHM occurs along a straight line, SHM can also be angular (like a torsional pendulum). The most general statement is that it is periodic.
Step 4: Final Answer
A defining characteristic of simple harmonic motion is that it is periodic.