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Work: Level 3 Challenges

         

Bicyclists and other things that go fast must overcome air resistance, or drag, even to maintain a constant speed. A simple empirical model for the drag force on an object when the air flows smoothly around the object is \(\vec{F}_{Drag}=-c\vec{v}\), where \(\vec{v}\) is the velocity of the object and \(c\) is a constant that depends on the size and shape of the object. Consider a bicyclist putting out some power \(P_0\) (in watts) to overcome drag and maintain some constant speed \(v_0\). She then increases her speed to \(1.2 v_0\), which requires her to put out a power \(P_1\) to maintain. What is the ratio \(\frac{P_1}{P_0}\)?

A \(1\text{ kg}\) object is lifted from point A to point B through two different pathways as shown in the figure above. What are the respective amounts of work done in the left and right figures?

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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