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Chapter 7: Problem 56

Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

Short Answer

Expert verified
The escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet, as shown by the derived equation \( v=\sqrt{2G\rho \pi} * r \), and thus \( v \propto r \).

Step by step solution

01

Identify Given Variables

In this problem, we know that the planet has a uniform density (\( \rho \)). We are also provided with the general formula for escape speed: \( v = \sqrt{2gr} \). The goal is to show this speed is directly proportional to the radius of the planet.
02

Calculate Gravity

First, calculate the gravity of the planet using its radius and mass. The mass of the planet is given by its volume, which is \( \rho \pi r^3 \) for a sphere. Therefore, gravity can be calculated with the formula \( g = \frac{GM}{r^2} = \frac{G\rho \pi r^3}{r^2} = G\rho \pi r \).
03

Substitute Gravity in Escape Speed Formula

Next, substitute the calculated gravity into the escape speed formula, which gives us \( v=\sqrt{2G\rho \pi r * r} = \sqrt{2G\rho \pi} * r \).
04

Interpret Result

The escape speed \( v = \sqrt{2G\rho \pi} * r has r bound up in the equation as \( \longrightarrow v \propto r \). This shows that the escape speed is directly proportional to the radius of the planet.

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Most popular questions from this chapter

A digital audio compact disc carries data along a continuous spiral track from the inner circumference of the disc to the outside edge. Each bit occupies \(0.6 \mu \mathrm{m}\) of the track. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of \(1.30 \mathrm{~m} / \mathrm{s}\). Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of \(2.30 \mathrm{~cm}\), and (b) at the end of the recording, where the spiral has a radius of \(5.80 \mathrm{~cm}\). (c) A full-length recording lasts for \(74 \mathrm{~min}, 33 \mathrm{~s}\). Find the average angular acceleration of the disc. (d) Assuming the acceleration is constant, find the total angular displacement of the disc as it plays. (e) Find the total length of the track.

A \(40.0\)-kg child takes a ride on a Ferris wheel that rotates four times each minute and has a diameter of \(18.0 \mathrm{~m}\). (a) What is the centripetal acceleration of the child? (b) What force (magnitude and direction) does the seat exert on the child at the lowest point of the ride? (c) What force does the seat exert on the child at the highest point of the ride? (d) What force does the seat exert on the child when the child is halfway between the top and bottom?

An athlete swings a \(5.00-\mathrm{kg}\) ball horizontally on the end of a rope. The ball moves in a circle of radius \(0.800 \mathrm{~m}\) at an angular speed of \(0.500 \mathrm{rev} / \mathrm{s}\). What are (a) the tangential speed of the ball and (b) its centripetal acceleration? (c) If the maximum tension the rope can withstand before breaking is \(100 \mathrm{~N}\), what is the maximum tangential speed the ball can have?

A certain light truck can go around a flat curve having a radius of \(150 \mathrm{~m}\) with a maximum speed of \(32.0 \mathrm{~m} / \mathrm{s}\). With what maximum speed can it go around a curve having a radius of \(75.0 \mathrm{~m}\) ?

A massless spring of constant \(k=78.4 \mathrm{~N} / \mathrm{m}\) is fixed on the left side of a level track. A block of mass \(m=\) \(0.50 \mathrm{~kg}\) is pressed against the spring and compresses it a distance \(d\), as in Figure P7.76. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius \(R=1.5 \mathrm{~m}\). The entire track and the loop-the-loop are frictionless, except for the section of track between points \(A\) and \(B\). Given that the coefficient of kinetic friction between the block and the track along \(A B\) is \(\mu_{k}=0.30\), and that the length of \(A B\) is \(2.5 \mathrm{~m}\), determine the minimum compression \(d\) of the spring that enables the block to just make it through the loop-the-loop at point C. Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.
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