Question

An object moves in a quadrant of radius R from point A to B such that A & B lie on two ends of the quadrant & the force applied is always directed towards B. Find the work done.

Answer 1

An object moves in a quadrant of radius R from point A to B such that A & B lie on two ends of the quadrant & the force applied is always directed towards B. Find the work done.

Solution:

Formula for work:

$$A = \int_{A}^{B} \vec{F} * \overrightarrow{ds} = \int_{A}^{B} F * ds * \cos \alpha * d\alpha(1)$$ $$ds = R * \alpha$$

For a small angle alpha:

$$\alpha \approx \sin \alpha$$ $$ds = R \sin \alpha \quad (2)$$ $$(2) \text{ in } (1):$$ $$\begin{aligned} A &= \int_{A}^{B} FR \sin \alpha * \cos \alpha \, d\alpha = \int_{A}^{B} FR \cos \alpha \, d(\cos \alpha) = \frac{FR \cos^{2} \alpha}{2} \Big|_{A}^{B} = \\ &= \frac{FR \cos^{2} \alpha}{2} \Big|_{A}^{B} = \frac{FR \cos^{2} 0}{2} - \frac{FR \cos^{2} 45^{o}}{2} = \\ &= \frac{FR}{2} - \frac{FR}{4} = \frac{FR}{4} \quad (F - force, R - radius) \end{aligned}$$

Answer: $A = \frac{FR}{4}$