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Chapter 6: Problem 6

Does the gravitational force of the Sun do work on a planet in a circular orbit? On a comet in an elliptical orbit? Explain.

Short Answer

Expert verified
No, the gravitational force of the Sun doesn't do work on a planet in a circular orbit or on a comet in an elliptical orbit. Over a complete orbit, the displacement is perpendicular to the gravitational force in each case, resulting in zero work done.

Step by step solution

01

Work in a circular orbit

When a planet is in a circular orbit, the displacement at any point is always perpendicular to the direction of the gravitational force, i.e., the angle \(\theta = 90^{\circ}\). When we substitute this angle in our work formula, cos(\(\theta\)) becomes zero. So, in the calculation \(W = F \cdot d \cdot cos \theta\), the value of cos(\(\theta\)) being zero means the Work done \(W\) is zero.
02

Work in an elliptical orbit

For a comet in an elliptical orbit, the orbit is not strictly perpendicular to the gravitational pull. However, it's important to note that in gravitational systems, the force is central - it's always directed towards the center. This results in a perpendicular direction to the tangent of motion at every point, which is similar to the circular case. Thus, the work done over a complete orbit is still zero.

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Most popular questions from this chapter

A particle moves from the origin to the point \(x=3 \mathrm{m}, y=6 \mathrm{m}\) along the curve \(y=a x^{2}-b x,\) where \(a=2 \mathrm{m}^{-1}\) and \(b=4 .\) It's subject to a force \(c x y \hat{\imath}+d \hat{\jmath},\) where \(c=10 \mathrm{N} / \mathrm{m}^{2}\) and \(d=15 \mathrm{N}\) Calculate the work done by the force.

You're an engineer for a company that makes bungee-jump cords, and you're asked to develop a formula for the work involved in stretching cords to double their length. Your cords have forcedistance relations described by \(F=-\left(k x+b x^{2}+c x^{3}+d x^{4}\right)\) where \(k, b, c,\) and \(d\) are constants. (a) Given a cord with unstretched length \(L_{0},\) what's your formula? (b) Evaluate the work done in doubling the stretch of a 10 -m cord with \(k=420 \mathrm{N} / \mathrm{m}\) \(b=-86 \mathrm{N} / \mathrm{m}^{2}, c=12 \mathrm{N} / \mathrm{m}^{3},\) and \(d=-0.50 \mathrm{N} / \mathrm{m}^{4}\)

A force pointing in the \(x\) -direction is given by \(F=F_{0}\left(x / x_{0}\right)\) where \(F_{0}\) and \(x_{0}\) are constants and \(x\) is position. Find an expression for the work done by this force as it acts on an object moving from \(x=0\) to \(x=x_{0}\)

A typical car battery stores about \(1 \mathrm{kW}\). h of energy. What's its power output if it drains completely in (a) 1 minute, (b) 1 hour, and (c) 1 day?

The force exerted by a rubber band is given approximately by $$F=F_{0}\left[\frac{L_{0}-x}{L_{0}}-\frac{L_{0}^{2}}{\left(L_{0}+x\right)^{2}}\right]$$ where \(L_{0}\) is the unstretched length, \(x\) is the stretch, and \(F_{0}\) is a constant. Find the work needed to stretch the rubber band a distance \(x\)
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