A point particle is acted upon by a restoring force –kx3 . The time period of oscillation is T when the amplitude is A. The time period for an amplitude 2A will be :

Options

A. `T`
B. `T//2`
C. `2T`
D. `4T`

Explanation

To solve the problem, we need to determine the time period of oscillation for a point particle acted upon by a restoring force proportional to the cube of the displacement, specifically \( F = -kx^3 \). We are given that the time period is \( T \) when the amplitude is \( A \), and we need to find the time period when the amplitude is \( 2A \).

### Step-by-Step Solution:

1. **Understanding the Force**: The restoring force is given by \( F = -kx^3 \). This indicates that the force is non-linear with respect to displacement \( x \).

2. **Finding the Time Period**: The time period \( T \) for oscillations can be expressed in terms of the amplitude \( A \). For a restoring force of the form \( F = -kx^n \), the time period \( T \) can be related to the amplitude \( A \) using dimensional analysis or other methods.

3. **Dimensional Analysis**:
- The force has dimensions of \( [F] = MLT^{-2} \).
- The term \( kx^3 \) implies that \( k \) must have dimensions such that \( k \) has dimensions of \( [k] = \frac{MLT^{-2}}{L^3} = \frac{M}{L^2T^2} \).
- The time period \( T \) can be expressed as a function of the amplitude \( A \) and the constant \( k \).

4. **Expressing Time Period in Terms of Amplitude**:
- We can express the time period as \( T \propto \frac{1}{\sqrt{A}} \) for small oscillations. Thus, we can write:
\[ T \propto A^{\frac{1}{2}} \]
- This implies that if the amplitude changes, the time period will change accordingly.

5. **Calculating the New Time Period**:
- Let the initial amplitude \( A_1 = A \) and the corresponding time period \( T_1 = T \).
- When the amplitude is doubled, \( A_2 = 2A \). We can express the new time period \( T_2 \) as:
\[ T_2 \propto (2A)^{\frac{1}{2}} = \sqrt{2} \cdot A^{\frac{1}{2}} \]
- Since \( T \propto A^{\frac{1}{2}} \), we can relate \( T_2 \) to \( T \):
\[ T_2 = T \cdot \sqrt{2} \]

6. **Final Result**: Thus, the time period for an amplitude of \( 2A \) is:
\[ T_2 = T \sqrt{2} \]