# 13.7 Einstein's theory of gravity  (Page 7/19)

 Page 10 / 19

Using the technique shown in Satellite Orbits and Energy , show that two masses ${m}_{1}$ and ${m}_{2}$ in circular orbits about their common center of mass, will have total energy $E=K+E={K}_{1}+{K}_{2}-\frac{G{m}_{1}{m}_{2}}{{r}^{}}=-\frac{G{m}_{1}{m}_{2}}{2{r}^{}}$ . We have shown the kinetic energy of both masses explicitly. ( Hint: The masses orbit at radii ${r}_{1}$ and ${r}_{2}$ , respectively, where $r={r}_{1}+{r}_{2}$ . Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)

Given the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, ${v}_{p}$ , is given by ${v}_{p}=\sqrt{\frac{2G{M}_{\text{Sun}}}{\left(q+p\right)}\phantom{\rule{0.2em}{0ex}}\frac{q}{p}}$ . ( Hint: Use conservation of angular momentum to relate ${v}_{p}$ and ${v}_{q}$ , and then substitute into the conservation of energy equation.)

Substitute directly into the energy equation using $p{v}_{p}=q{v}_{q}$ from conservation of angular momentum, and solve for ${v}_{p}$ .

Comet P/1999 R1 has a perihelion of 0.0570 AU and aphelion of 4.99 AU. Using the results of the previous problem, find its speed at aphelion. ( Hint: The expression is for the perihelion. Use symmetry to rewrite the expression for aphelion.)

## Challenge problems

A tunnel is dug through the center of a perfectly spherical and airless planet of radius R . Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.

$g=\frac{4}{3}\phantom{\rule{0.1em}{0ex}}G\rho \pi r\to F=mg=\left[\frac{4}{3}\phantom{\rule{0.1em}{0ex}}Gm\rho \pi \right]\phantom{\rule{0.1em}{0ex}}r$ , and from $F=m\phantom{\rule{0.1em}{0ex}}\frac{{d}^{2}r}{d{t}^{2}}$ , we get $\frac{{d}^{2}r}{d{t}^{2}}=\left[\frac{4}{3}\phantom{\rule{0.1em}{0ex}}G\rho \pi \right]\phantom{\rule{0.1em}{0ex}}r$ where the first term is ${\omega }^{2}$ . Then $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{3}{4G\rho \pi }}$ and if we substitute $\rho =\frac{M}{4\text{/}3\pi {R}^{3}}$ , we get the same expression as for the period of orbit R .

Following the technique used in Gravitation Near Earth’s Surface , find the value of g as a function of the radius r from the center of a spherical shell planet of constant density $\rho$ with inner and outer radii ${R}_{\text{in}}$ and ${R}_{\text{out}}$ . Find g for both ${R}_{\text{in}} and for $r<{R}_{\text{in}}$ . Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.

Show that the areal velocity for a circular orbit of radius r about a mass M is $\frac{\text{Δ}A}{\text{Δ}t}=\frac{1}{2}\sqrt{GMr}$ . Does your expression give the correct value for Earth’s areal velocity about the Sun?

Using the mass of the Sun and Earth’s orbital radius, the equation gives $2.24\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{15}{\text{m}}^{2}\text{/s}$ . The value of $\pi {R}_{\text{ES}}^{2}\text{/}\left(1\phantom{\rule{0.2em}{0ex}}\text{year}\right)$ gives the same value.

Show that the period of orbit for two masses, ${m}_{1}$ and ${m}_{2}$ , in circular orbits of radii ${r}_{1}$ and ${r}_{2}$ , respectively, about their common center-of-mass, is given by $T=2\pi \sqrt{\frac{{r}^{3}}{G\left({m}_{1}+{m}_{2}\right)}}\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}r={r}_{1}+{r}_{2}$ . ( Hint: The masses orbit at radii ${r}_{1}$ and ${r}_{2}$ , respectively where $r={r}_{1}+{r}_{2}$ . Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expressions for the kinetic energy.)

Show that for small changes in height h , such that $h\text{<}\phantom{\rule{0.2em}{0ex}}\text{<}{\text{R}}_{\text{E}}$ , [link] reduces to the expression $\text{Δ}U=m\text{g}h$ .

$\text{Δ}U={U}_{f}-{U}_{i}=-\frac{G{M}_{\text{E}}m}{{r}_{f}}+\frac{G{M}_{\text{E}}m}{{r}_{i}}=G{M}_{\text{E}}m\left(\frac{{r}_{f}-{r}_{i}}{{r}_{f}{r}_{i}}\right)$ where $h={r}_{f}-{r}_{i}$ . If $h\text{<}\phantom{\rule{0.2em}{0ex}}\text{<}{\text{R}}_{\text{E}}$ , then ${r}_{f}{r}_{i}\approx {R}_{\text{E}}^{2}$ , and upon substitution, we have

$\text{Δ}U=G{M}_{\text{E}}m\left(\frac{h}{{R}_{\text{E}}^{2}}\right)=m\left(\frac{G{M}_{\text{E}}}{{R}_{\text{E}}^{2}}\right)h$ where we recognize the expression with the parenthesis as the definition of g .

Using [link] , carefully sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass, m . Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal acceleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle $\epsilon$ , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. $\text{tan}\left(\lambda +\epsilon \right)=\frac{g}{\left(g-{\omega }^{2}{R}_{\text{E}}\right)}\text{tan}\lambda$ , where $\omega$ is the angular velocity of Earth.

(a) Show that tidal force on a small object of mass m , defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitation at a distance R from M , is given by ${F}_{\text{tidal}}=\frac{2GMm}{{R}^{3}}\text{Δ}r$ where $\text{Δ}r$ is the distance between the near and far side and $\text{Δ}r\text{<}\phantom{\rule{0.2em}{0ex}}\text{<}R$ . (b) Assume you are falling feet first into the black hole at the center of our galaxy. It has mass of 4 million solar masses. What would be the difference between the force at your head and your feet at the Schwarzschild radius (event horizon)? Assume your feet and head each have mass 5.0 kg and are 2.0 m apart. Would you survive passing through the event horizon?

a. Find the difference in force,
${F}_{\text{tidal}}==\frac{2GMm}{{R}^{3}}\text{Δ}r$ ;
b. For the case given, using the Schwarzschild radius from a previous problem, we have a tidal force of $9.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{N}$ . This won’t even be noticed!

Find the Hohmann transfer velocities, $\text{Δ}{v}_{\text{EllipseEarth}}^{}$ and $\text{Δ}{v}_{\text{EllipseMars}}^{}$ , needed for a trip to Mars. Use [link] to find the circular orbital velocities for Earth and Mars. Using [link] and the total energy of the ellipse (with semi-major axis a ), given by $E=-\frac{Gm{M}_{\text{s}}}{2{a}^{}}$ , find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, $\text{Δ}v$ , at each point is the velocity boost or transfer velocity needed.

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
definition of inertia
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
position, velocity and acceleration of vector
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
I need the solving for this question
philip
is the eye the same like the camera
I can't understand
Suraia
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
classical mechanics?
Victor
principle of superposition?
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus