Problem 7.10 Show that the kinetic energy K.E = {m₁²+ m₂ of a system of two particles can be written in terms of their center-of-mass velocity Rem and relative velocity i as 1 K.E. „E. = — MŘ²m + µ²², where M = my + m2 is the total mass and μ = mm₂/M is the reduced mass of the system.
Problem 7.10 Show that the kinetic energy K.E = {m₁²+ m₂ of a system of two particles can be written in terms of their center-of-mass velocity Rem and relative velocity i as 1 K.E. „E. = — MŘ²m + µ²², where M = my + m2 is the total mass and μ = mm₂/M is the reduced mass of the system.
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![# Gravitation: Problems and Exercises
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### * Problem 7.8
Assume that the period of elliptical orbits around the sun depends only upon \( G \) (the sun’s mass), and \( a \), the semi-major axis of the orbit. Prove Kepler’s third law using dimensional arguments alone.
### * Problem 7.9
A spy satellite designed to peer closely at a particular house every day at noon has a 24-hour period, and a perigee of 100 km directly above the house. What is the altitude of the satellite at apogee? (Earth’s radius is 6400 km).
### ** Problem 7.10
Show that the kinetic energy
\[ K.E = \frac{1}{2} m_1 \dot{r}_1^2 + \frac{1}{2} m_2 \dot{r}_2^2 \]
of a system of two particles can be written in terms of their center-of-mass velocity \( \dot{R}_{cm} \) and relative velocity \( \dot{r} \) as
\[ K.E = \frac{1}{2} M \dot{R}^2_{cm} + \frac{1}{2} \mu \dot{r}^2, \]
where \( M = m_1 + m_2 \) is the total mass and \( \mu = \frac{m_1 m_2}{M} \) is the reduced mass of the system.
### ** Problem 7.11
Show that the shape \( r(\varphi) \) for a central spring-force ellipse takes the standard form
\[ r^2 = \frac{a^2 b^2}{(b^2 \cos^2 \varphi + a^2 \sin^2 \varphi)} \]
if (in Eq. (7.37)) we use the plus sign in the denominator and choose \( \varphi_0 = \pi / 4 \).
### * Problem 7.12
Show that the period of a particle that moves in a circular orbit close to the surface of a sphere depends only upon \( G \) and the average density \( \rho \) of the sphere. Find what this period would be for any sphere having an average density equal to that of water. (The sphere consisting of the planet Jupiter nearly qualifies!)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd150f027-63cd-4c28-b0a8-8b6f9b35dbae%2F3390e76d-0ab7-4879-9702-869531b65839%2Fo6c4l8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Gravitation: Problems and Exercises
---
### * Problem 7.8
Assume that the period of elliptical orbits around the sun depends only upon \( G \) (the sun’s mass), and \( a \), the semi-major axis of the orbit. Prove Kepler’s third law using dimensional arguments alone.
### * Problem 7.9
A spy satellite designed to peer closely at a particular house every day at noon has a 24-hour period, and a perigee of 100 km directly above the house. What is the altitude of the satellite at apogee? (Earth’s radius is 6400 km).
### ** Problem 7.10
Show that the kinetic energy
\[ K.E = \frac{1}{2} m_1 \dot{r}_1^2 + \frac{1}{2} m_2 \dot{r}_2^2 \]
of a system of two particles can be written in terms of their center-of-mass velocity \( \dot{R}_{cm} \) and relative velocity \( \dot{r} \) as
\[ K.E = \frac{1}{2} M \dot{R}^2_{cm} + \frac{1}{2} \mu \dot{r}^2, \]
where \( M = m_1 + m_2 \) is the total mass and \( \mu = \frac{m_1 m_2}{M} \) is the reduced mass of the system.
### ** Problem 7.11
Show that the shape \( r(\varphi) \) for a central spring-force ellipse takes the standard form
\[ r^2 = \frac{a^2 b^2}{(b^2 \cos^2 \varphi + a^2 \sin^2 \varphi)} \]
if (in Eq. (7.37)) we use the plus sign in the denominator and choose \( \varphi_0 = \pi / 4 \).
### * Problem 7.12
Show that the period of a particle that moves in a circular orbit close to the surface of a sphere depends only upon \( G \) and the average density \( \rho \) of the sphere. Find what this period would be for any sphere having an average density equal to that of water. (The sphere consisting of the planet Jupiter nearly qualifies!)
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