Question

If a force of \( (6x^2 - 4x) \, \text{N} \) acts on a body of mass 10 kg, then the work to be done by the force in displacing the body from \( x = 2 \, \text{m} \) to \( x = 4 \, \text{m} \) is

• A. \(22 \operatorname{J} \)
• B. \(44 \operatorname{J} \)
• C. \(66 \operatorname{J} \)
• D. \(88 \operatorname{J} \)

Hint:

For work done by a variable force \(F(x)\) over a displacement from \(x_1\) to \(x_2\), the key is to use integration. The formula is \(W = \int_{x_1}^{x_2} F(x) dx\). Remember to perform the integration correctly and then evaluate the definite integral using the given limits. The mass of the body is often extraneous information if only work done by the force (not related to kinetic energy change involving mass) is asked.

Answer 1

The Correct Option is D

Step 1: Understand the formula for work done by a variable force.
When a force \(F\) acting on a body is not constant but varies with position \(x\), the work done (\(W\)) by this force in displacing the body from an initial position \(x_1\) to a final position \(x_2\) is given by the definite integral: \[ W = \int_{x_1}^{x_2} F(x) dx \] Step 2: Identify the given force function and displacement limits.
The force acting on the body is \(F(x) = (6x^2 - 4x) \operatorname{N}\).
The initial position is \(x_1 = 2 \operatorname{m}\).
The final position is \(x_2 = 4 \operatorname{m}\).
The mass of the body (10 kg) is provided but is not required to calculate the work done by the force.
Step 3: Set up and evaluate the integral for work done.
Substitute the force function and the limits of integration into the work formula: \[ W = \int_{2}^{4} (6x^2 - 4x) dx \] Now, integrate the expression with respect to \(x\): \[ W = \left[6 \cdot \frac{x^{2+1}}{2+1} - 4 \cdot \frac{x^{1+1}}{1+1}\right]_{2}^{4} \] \[ W = \left[6 \cdot \frac{x^3}{3} - 4 \cdot \frac{x^2}{2}\right]_{2}^{4} \] \[ W = \left[2x^3 - 2x^2\right]_{2}^{4} \] Now, evaluate the definite integral by substituting the upper limit and subtracting the value obtained by substituting the lower limit: \[ W = \left(2(4)^3 - 2(4)^2\right) - \left(2(2)^3 - 2(2)^2\right) \] Calculate the values: \[ W = (2(64) - 2(16)) - (2(8) - 2(4)) \] \[ W = (128 - 32) - (16 - 8) \] \[ W = (96) - (8) \] \[ W = 88 \operatorname{J} \] Step 4: State the final answer.
The work done by the force in displacing the body from \(x = 2 \operatorname{m}\) to \(x = 4 \operatorname{m}\) is 88 J.
The final answer is $\boxed{88 \operatorname{J}}$.