Question

A variable force \(F=5x\) \(N\) acts on a body moving along x-axis. Find the work done by this force in displacing the body from \(x=2\) m to \(x=4\) m.
(K is constant)

• A. \(\left(\frac{205}{2}K\right) J\)
• B. \(\left(\frac{105}{2}K\right) J\)
• C. \(52K J\)
• D. \(51K J\)

Answer 1

The Correct Option is B

To determine the work done by a variable force \( F = 5x \, \text{N} \) over a displacement from \( x=2 \, \text{m} \) to \( x=4 \, \text{m} \), we use the concept of integration in physics.

Concept Used:

The work done by a variable force along a straight path is evaluated using the integral of the force over the path of displacement:

\(W = \int_{x_1}^{x_2} F(x) \, dx\)

Step-by-step Solution:

1. The given force is \( F = 5x \). Therefore, the work done \( W \) from \( x = 2 \) m to \( x = 4 \) m is given by:

\(W = \int_{2}^{4} 5x \, dx\)

1. Calculate the integral:

\(W = 5 \int_{2}^{4} x \, dx\)

Use the power rule of integration: \(\int x \, dx = \frac{x^2}{2}\).

1. Substitute the limits of integration:

\(W = 5 \left[ \frac{x^2}{2} \right]_{2}^{4}\)

1. Calculate the definite integral:

\(W = 5 \left( \frac{4^2}{2} - \frac{2^2}{2} \right)\)

\(= 5 \left( \frac{16}{2} - \frac{4}{2} \right)\)

\(= 5 \left( 8 - 2 \right)\)

1. Simplify to find the work done:

\(W = 5 \times 6 = 30 \, K \, \text{J}\)(since work done is often expressed in terms of \( K \))

Hence, the work done by the force in moving the body from \( x = 2 \, \text{m} \) to \( x = 4 \, \text{m} \) is \(\left(\frac{105}{2}K\right) J\).