Problem 14 - Master Resource Book in JEE Main Physics

Question

A position-dependent force \(F=3 x^{2}-2 x+7\) acts on a body of mass \(7 \mathrm{~kg}\) and displaces it from \(x=0 \mathrm{~m}\) to \(x=5 \mathrm{~m}\). The work done on the body is \(x^{\prime}\) joule. If both \(F\) and \(x\) are measured in SI units, the value of \(x^{\prime}\) is (a) 135 (b) 235 (c) 335 (d) 935

Step-by-Step Solution

Answer

The work done is 135 Joules, so the answer is (a) 135.

Step 1: Understanding the Work Done

The work done by a variable force along a straight line is calculated using the integral: \[ W = \int_{x_1}^{x_2} F(x) \, dx \] where \( F(x) = 3x^2 - 2x + 7 \), \( x_1 = 0 \), and \( x_2 = 5 \). We will evaluate this integral to find the work done.

Step 2: Setting Up the Integral

Set up the integral based on the given force function and limits of integration: \[ W = \int_{0}^{5} (3x^2 - 2x + 7) \, dx \] Identify each term that you need to integrate separately.

Step 3: Integrating the Force Function

Integrate each term separately:1. \( \int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} = x^3 \)2. \( \int -2x \, dx = -2 \cdot \frac{x^2}{2} = -x^2 \)3. \( \int 7 \, dx = 7x \)Combine these results to form:\[ W = \left[ x^3 - x^2 + 7x \right]_{0}^{5} \]

Step 4: Evaluating the Integral at Limits

Evaluate the expression \( x^3 - x^2 + 7x \) from \( x=0 \) to \( x=5 \):\[ W = \left( 5^3 - 5^2 + 7 \times 5 \right) - \left( 0^3 - 0^2 + 7 \times 0 \right) \]Simplify:\[ W = (125 - 25 + 35) - 0 = 135 \]

Step 5: Conclusion

Thus, the work done \( x' \) is 135 Joules.

Key Concepts

Integration of Force Function

When dealing with a position-dependent force, it's crucial to understand how to calculate the work done on an object over a specific path. For such cases, we employ the concept of integration. This means we take the force function, which depends on the position, and integrate it over the range of motion. This gives us the total work done by the force as the object moves along the path. In mathematical terms, integration essentially involves adding up an infinite series of infinitely small amounts, resulting in a total value for the work done over the specified path.

In our current example, the force function is given as \( F(x) = 3x^2 - 2x + 7 \). We need to calculate the work done as this force moves an object from \( x = 0 \) to \( x = 5 \). To find the total work done \( W \), we perform the integration as follows:

• Identify the force function \( F(x) \).
• Set up the integral with the limits of motion, from \( x_1 = 0 \) to \( x_2 = 5 \).
• Integrate each term in the function individually.

In this example, integrating \( 3x^2 \), \( -2x \), and \( 7 \) individually helps us reach the work travel equation. The integration results are then evaluated at the limits to calculate the work done.

Position-Dependent Force

Position-dependent forces are forces whose magnitude and/or direction vary depending on the position. Unlike constant forces, position-dependent forces require more sophisticated mathematical treatments, especially when calculating work done. In our scenario, the force function \( F(x) = 3x^2 - 2x + 7 \) changes as the object moves along the x-axis.

It's important to notice the terms "position-dependent" and "variable force"; they tell us that the force is not constant. The function involves powers of the variable \( x \), so each term contributes to the force at different rates depending on the value of \( x \).

The dominance of each term in the force equation shifts as \( x \) changes:

• For small values of \( x \), the constant term (\( 7 \)) is significant.
• As \( x \) increases, the second-degree term (\( 3x^2 \)) becomes more dominant due to the "squared" factor.

This non-linear behavior of the force is why integration is essential. By integrating across the range of interest, we account for all these variations and calculate the total work done accurately.

Calculation of Work Done

To find the work done by a position-dependent force, such as \( F(x) = 3x^2 - 2x + 7 \), over a distance, we perform integration over the given interval. This approach provides the exact work done as opposed to an approximation that might arise from simplifying assumptions.

The integration process involves evaluating the integral function, post-integration, at the specified limits. Plugging in these limit values into the integrated function helps us calculate the net work done

• Calculate the integral of the force function individually: \( W = \left[ x^3 - x^2 + 7x \right]_0^5 \).
• Evaluate this function at the upper and lower limits: plug \( x = 5 \) and \( x = 0 \) into the function.
• Subtract the result at the lower limit from the result at the upper limit.

In our example, evaluating \( x^3 - x^2 + 7x \) from \( x = 0 \) to \( x = 5 \) gives us \( W = 135 \) Joules. This shows the total energy transferred by the force moving the object over that distance.