Gravitational Force AP Physics C Mechanics FRQ – Exam Style Questions etc.
Gravitational Force AP Physics C Mechanics FRQ
Unit 2: Force and Translational Dynamics
Weightage : 20-15%
Question
Two stars each of mass M form a binary star system such that both stars move in the same circular orbit of radius R. The universal gravitational constant is G.
a. Use Newton’s laws of motion and gravitation to find an expression for the speed v of either star in terms of R, G, and M.
b. Express the total energy E of the binary star system in terms of R, G, and M.
Suppose instead, one of the stars had a mass 2M.
c. On the following diagram, show circular orbits for this star system.
d. Find the ratio of the speeds, v2M/vM.
Answer/Explanation
Ans:
a. Fg = Fc gives \(\frac{GMM}{(2R)^{2}}= \frac{Mv^{2}}{R}\) Solving for v gives \(v = \frac{1}{2}\sqrt{\frac{GM}{R}}\)
b. E = PE + KE = \(-\frac{GMM}{2R}+2\left ( \frac{1}{2}Mv^{2} \right )= – \frac{GMM}{2R}+2\left ( \frac{1}{2}M\left ( v = \frac{1}{2}\sqrt{\frac{GM}{R}} \right )^{2} \right ) = -\frac{GM^{2}}{4R}\)
c.
d. Fg2 = Fg1 = Fc
\(\frac{(2M){v_{2}}^{2}}{1/3d}=\frac{M{v_{1}}^{2}}{2/3d}\) gives v2 / v1 = 1/2