Classical Mechanics
# Work

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

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 \(\alpha +\beta +\mu \).

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?

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.

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