Problem 7.10 Please. # Gravitation: Problems and Exercises---### * Problem 7.8Assume 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 7.9A 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 7.10Show 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 7.11Show 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 7.12Show 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!)

Answered: Image /qna-images/answer/3390e76d-0ab7-4879-9702-869531b65839.jpg

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.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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