Question

A body of mass 3 kg is under a constant force which causes a displacement s in meters in it, given by the relation s = (\(\frac{1}{3}\)) t², where t is in s. Work done by the force in 2 s is :

• A. (\(\frac{17}{3}\))J
• B. (\(\frac{3}{8}\))J
• C. (\(\frac{8}{3}\))J
• D. (\(\frac{3}{17}\))J

Answer 1

The Correct Option is C

To find the work done by the force in 2 seconds, we need to first understand the relationship between displacement, velocity, acceleration, and force.

1. Given the displacement as a function of time:

   \( s = \frac{1}{3}t^2 \)

2. Velocity is the derivative of displacement with respect to time:

   \( v = \frac{ds}{dt} = \frac{d}{dt} \left(\frac{1}{3}t^2\right) = \frac{2}{3}t \)

3. Acceleration is the derivative of velocity with respect to time:

   \( a = \frac{dv}{dt} = \frac{d}{dt} \left(\frac{2}{3}t\right) = \frac{2}{3} \)

4. Using Newton's second law, the force \( F \) is given by:

   \( F = ma = 3 \times \frac{2}{3} = 2 \, \text{N} \)

5. Work done \( W \) by the force in moving the body a distance \( s \) is:

   \( W = F \cdot s \)

6. Calculate the displacement at \( t = 2 \) s:

   \( s(2) = \frac{1}{3} \times 2^2 = \frac{1}{3} \times 4 = \frac{4}{3} \, \text{m} \)

7. Substitute in the formula for work done:

   \( W = 2 \times \frac{4}{3} = \frac{8}{3} \, \text{J} \)

Therefore, the work done by the force in 2 seconds is \(\frac{8}{3}\) Joules.

The correct answer is: \(\frac{8}{3}\) J.