Question

A particle under force $F = -kx$. Then Motion is

• A. uniform
• B. simple harmonic
• C. circular
• D. random

Hint:

Any system with a linear restoring force ($F \propto -x$) undergoes simple harmonic motion.
The time period is given by $T = 2\pi\sqrt{\frac{m}{k}}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to identify the type of motion a particle undergoes when subjected to a restoring force proportional to its displacement ($F = -kx$).

Step 2: Key Formula or Approach:
According to Newton's Second Law of Motion:
\[ F = m a = m \frac{d^2x}{dt^2} \]
For the given force $F = -kx$:
\[ m \frac{d^2x}{dt^2} = -kx \implies \frac{d^2x}{dt^2} + \left(\frac{k}{m}\right)x = 0 \]

Step 3: Detailed Explanation:

• Let $\omega^2 = \frac{k}{m}$, where $\omega$ is the angular frequency of oscillation.

• The equation becomes:
\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \] This is the standard second-order linear differential equation that defines a Simple Harmonic Oscillator.

• The solution to this equation is sinusoidal:
\[ x(t) = A \sin(\omega t + \phi) \]

• This indicates that the particle oscillates periodically about its equilibrium position ($x = 0$), which is the definition of Simple Harmonic Motion (SHM).

Step 4: Final Answer:
The motion of the particle is simple harmonic.