Question

In a simple harmonic motion (SHM), the acceleration is proportional to:

• A. Velocity
• B. Displacement and opposite in direction
• C. Time
• D. Square of velocity

Hint:

The fundamental equation \(a = -\omega^2 x\) is the defining mathematical characteristic of all Simple Harmonic Motion systems. If acceleration is not directly proportional to the negative of displacement, the motion is not simple harmonic.

Answer 1

The Correct Option is B

Concept: Simple Harmonic Motion (SHM) is a specific type of periodic oscillatory motion where an object moves back and forth about a central equilibrium position. The defining dynamic condition of SHM is that the restoring force acting on the body is directly proportional to its displacement from that equilibrium point and points in the opposite direction.

Step 1: Linking the restoring force to acceleration using Newton's Second Law.
By definition, Hooke's Law for an object in simple harmonic motion is: \[ F = -kx \] where \(F\) is the restoring force, \(k\) is a positive proportionality constant, and \(x\) represents the displacement from the equilibrium origin. According to Newton's Second Law of Motion, force is equal to mass times acceleration (\(F = ma\)). Equating these expressions: \[ ma = -kx \]

Step 2: Isolating the acceleration term.
Dividing both sides of the equation by the mass \(m\): \[ a = -\left(\frac{k}{m}\right)x \] We define the constant ratio of the spring stiffness to mass as the square of the natural angular frequency (\(\omega^2 = \frac{k}{m}\)). Substituting this into the acceleration equation gives: \[ a = -\omega^2 x \]

Step 3: Analyzing the final proportionality relationship.
Let us analyze the components of this fundamental differential relation:
• Since \(\omega^2\) is a positive constant, the magnitude of the acceleration \(a\) is directly proportional to the magnitude of the displacement \(x\): \[ |a| \propto |x| \]
• The negative sign (\(-\)) shows that the acceleration vector points in the opposite direction of the displacement vector. When the object is pulled to the right (\(x > 0\)), its acceleration pulls it back toward the left (\(a < 0\)). Therefore, the acceleration is proportional to the displacement and opposite in direction, matching option (2).