Classical Mechanics
# Work

A rectangular box of length \(\frac{2}{3} \text{m}\) is initially traveling at \(2 \text{ m/s},\) with its entire length over the smooth (perfectly frictionless) blue surface (shown above). The box gradually moves onto a rough surface (gray) and stops the instant that its entire length is positioned within the rough region.

Determine the coefficient of kinetic friction.

\(\)

**Details and Assumptions:**

- \(g = 10\text{ m/s}^2.\)
- The pressure at the bottom of the box is always uniform over its area.

A \(1.2\text{ kg}\) mass is projected down a rough semi-circular vertical track of radius \(2.0 \text{ m}\), as shown in the diagram below. The speed of the mass at point \(A\) is \(3.2\text{ m/s}\), and that at point \(B\) is \(6.0 \text{ m/s}\).

How much work is done on the mass between \(A\) and \(B\) by the force of friction?

The gravitational acceleration is \(g=9.8\text{ m/s}^2.\)

Two forces \(\left(6\hat{i} + 2\hat{j} - 3\hat{k}\right) \si{\newton}\) and \(\left(5\hat{i} - 3\hat{j} + 7\hat{k}\right) \si{\newton}\) act on an object that moves on a frictionless surface and, in doing so, displace it from \(\left(2\hat{i} + 3\hat{j} - 5\hat{k}\right) \si{\metre}\) to \(\left(-3\hat{i} - 3\hat{j} + 4\hat{k}\right) \si{\metre}\).

Find the magnitude of the work done on the object, \(W\) in joules.

A locomotive of mass \(m\) starts moving so that its velocity varies according to the law \(v=k \sqrt{s}\), where \(k\) is a constant and \(s\) is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first \(t\) seconds, after the beginning of motion.

- If \( W=\large\frac { m{ k }^{ \alpha }{ t }^{ \beta } }{ \mu } \), find \(\alpha +\beta +\mu \).

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