Question

Which of the following statement represents the acceleration of a particle executing simple harmonic motion? (a is acceleration and x is the displacement from mean position).
1. a = 10x
2. a = -2x²
3. a = -50x
4. a = 6 x³

Answer 1

Correct Answer - Option 3 : a = -50x

CONCEPT:

• Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.

F α -x

Where F = force and x = the displacement from equilibrium.

• For a simple harmonic motion equation of acceleration

\(\overrightarrow{a}=-ω^2\overrightarrow{x}\)

where a is the acceleration ω is the angular frequency and x is the displacement.

EXPLANATION:

The equation of displacement in SHM is given by:

x = A sinωt .........(i)

differentiate eq (i) with respect to time t

v = Aω cosωt

differentiate it with respect to time t

a = -Aω² sinωt

a = -ω² A sinωt

a = -ω² x

\(\overrightarrow{a}=-ω^2\overrightarrow{x}\)

• So the equation acceleration in SHM can be represented when

1. acceleration is proportional to one power of displacement
2. acceleration is in the opposite direction of the displacement i.e. there is a negative sign in the equation.

• The above two conditions are fulfilled only in option 3.
• Hence the correct answer is option 3.