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Chapter 6: Problem 760

A shell of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) has point mass \(\mathrm{m}\) placed at a distance \(r\) from its center. The gravitational potential energy \(\mathrm{U}(\mathrm{r})-\mathrm{v}\) will be

Short Answer

Expert verified
The gravitational potential energy \(U(r)\) of a point mass \(m\) located at a distance \(r\) from the center of a shell of mass \(M\) and radius \(R\) is given by the formula: \(U(r) = -\dfrac{G m M}{r}\), where G is the gravitational constant.

Step by step solution

01

Write down the formula for potential energy

To calculate the gravitational potential energy, we need to use the formula: \(U(r) = -\dfrac{G m M}{r}\) Where U(r) is the potential energy as a function of r, G is the gravitational constant (approximately \(6.674 \times 10^{-11}\) N(m/kg)²), m is the given point mass, M is the mass of the shell, and r is the distance from the center of the shell.
02

Compute U(r)-v

The exercise asks to compute \(U(r)-v\). But there isn't information about the value of "v" anywhere in the exercise. We'll assume that this is an error and that the desired expression is only U(r). So, we need to find the gravitational potential energy at distance r from the center of the shell. By substituting the given values into the potential energy formula, we get: \(U(r) = -\dfrac{G m M}{r}\) Which is the potential energy as a function of the distance from the center of the shell.

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