Question

Suppose a large spherical object, such as a planet, with radius $R$and mass $M$has a narrow tunnel passing diametrically through it. A particle of mass m is inside the tunnel at a distance $x\le R$from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius $r\le x$; there is no net gravitational force from the mass in the spherical shell with $r>x$.

$a$. Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of $x$, $R$, $m$, $M$, and any necessary constants.

$b$. You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with $SHM$. Suppose an intrepid astronaut exploring a $150$-km-diameter, $3.5\times {10}^{18}$kg asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?

Short answer

$\left(a\right)$The gravitation on the particle is $F=-G\frac{mM}{R^3}r$.

$\left(b\right)$The time taken by the astronaut to fall all the way through the asteroid and emerge on the opposite side is localid="1650304271788" $t=70\min$.

Step-by-step solution

The Principles.

$1$- Newton's Law of Universal Gravitation: The magnitude of the attractive gravitational force that an object with mass $m_1$exerts on an object with mass $m_2$separated by a center-to-center distance $r$is:

$F_{1\text{on}2}=G\frac{m_1{m}_2}{r^2}$

where $G=6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2$is known as the gravitational constant.

$2$- Hooke's Law: If any object causes a spring to stretch or compress, the spring exerts an elastic force on that object. If the object stretches the spring along the $x$-direction, the $x$-component of the force the spring exerts on the item is:

$F_{\mathrm{S}\text{on}\mathrm{O},x}=-kx$

where $k$is that the spring constant measured in newtons per meter and is a measure of the stiffness of the spring (or any elastic object), $x$is the distance that the object has been stretched/compressed (not the total length of the object). The elastic force exerted by the spring on the object points in a direction opposite to the direction it was stretched (or compressed), hence, the negative sing in front of $kx$. The object in turn exerts a force on the spring:

$F_{\mathrm{O}\text{on}\mathrm{S},x}=+kx$

$3$- The period of an oscillator in simple harmonic motion is given by

$T=2\pi \sqrt{\frac{m}{k}}$

Note that $T$does not depend upon on the amplitude but only on the mass $m$and the force constant $k$.

The given data.

The radius of the planet is: $R$.

The mass of the planet is: $M$.

A particle of mass $m$is inside a narrow tunnel passing diametrically through the planet at a distance $x\le R$from the center.

The net gravitational force on the particle is due entirely to the sphere of mass with radius $r\le x$.

The planet has uniform density.

The diameter of the asteroid is: $D=(150\mathrm{km})\left(\frac{1000\mathrm{m}}{1\mathrm{km}}\right)=150\times {10}^3\mathrm{m}$.

The radius of the asteroid is then: $R=\frac{D}{2}=75\times {10}^3\mathrm{m}$.

The mass of the asteroid is: $M=3.5\times {10}^{18}\mathrm{kg}$.

The asteroid has a tunnel through the center.

To find the required data.

In part $\left(a\right)$, we are asked to determine an expression for the gravitational force on the particle of mass $m$.

In part $\left(b\right)$, we are asked to determine the time taken by the astronaut to fall all the way through the asteroid and emerge on the other side.

Step 4  Expression for gravity force.

$\left(a\right)$Let the positive direction point outward from the center of the planet. The gravitational force on the object of mass $m$located a distance $r$from the center of the planet is found from Equation (1):

localid="1650302779928" $F=-G\frac{m{m}_{\mathrm{enc}}}{r^2}$

where the negative sing indicates that the gravitational force is directed inward to the center of the planet.

The mass localid="1650302784021" $m_{\text{enc}}$enclosed by a sphere of radius $r$is found in terms of the density of the planet as follows:

localid="1650302788444" $m_{\mathrm{enc}}=\rho V_{\mathrm{enc}}$

where localid="1650302792117" $V_{\text{enc}}$is that the volume of the sphere of radius $r$which is equal to the density of the planet which is equal to the full mass $M$of the planet divided by its volume $4/3\pi R^3$. So

$m_{\mathrm{enc}}=\left(\frac{M}{\frac{4}{3}\pi R^3}\right)\left(\frac{4}{3}\pi r^3\right)$

$=M\left(\frac{r^3}{R^3}\right)$

Substitute for $m_{\text{enc}}$into Equation (4):

$F=-G\frac{mM}{r^2}\left(\frac{r^3}{R^3}\right)$

 To seek out the time taken.

$\left(b\right)$Comparing Equation$\left(5\right)$on the gravitational force on the object to the Equation$\left(2\right)$for the restoring force of a spring, we see that the object of mass $m$undergoes simple harmonic motion with:

$k=G\frac{mM}{R^3}$

which has the samedimension asthe spring constant.
The time taken by the astronaut to fall all the way through the asteroid and emerge on the otherside is half the time of the astronaut's motion:

$t=\frac{T}{2}$

Substitute for $T$from Equation (3):

$t=\pi \sqrt{\frac{m}{k}}$

Substitute for $k$from Equation (6):

$t=\pi \sqrt{\frac{{\left(75\times {10}^3\mathrm{m}\right)}^3}{\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)\left(3.5\times {10}^{18}\mathrm{kg}\right)}}$

$=4223s$

$=\left(4223s\right)\left(\frac{1min}{60sec}\right)$

$=70min$.