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Classical Mechanics

# 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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