AP Physics C Mechanics: 7.1 Gravitational Forces – Exam Style questions with Answer- FRQ

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

Question

A thin, flexible metal plate attached at one end to a platform, as shown above, can be used to measure mass. When the free end of the plate is pulled down and released, it vibrates in simple harmonic motion with a period that depends on the mass attached to the plate. To calibrate the force constant, objects of known mass are attached to the plate and the plate is vibrated, obtaining the data shown below.
a. Fill in the blanks in the data table.

b. On the graph below, plot T2 versus mass. Draw on the graph the line that is your estimate of the best straight-line fit to the data points.

c. An object whose mass is not known is vibrated on the plate, and the average time for ten vibrations is measured to be 16.1 s. From your graph, determine the mass of the object. Write your answer with a reasonable number of significant digits.
d. Explain how one could determine the force constant of the metal plate.
e. Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth? Explain briefly.
f. If Columbia is orbiting at 0.3 x 106 m above the Earth’s surface, what is the acceleration of Columbia due to the Earth’s gravity? (Radius of Earth = 6.4 x 106 m, mass of Earth = 6.0 x 1024 kg)
g. Since the answer to part (f) is not zero, briefly explain why objects aboard the orbiting Columbia seem weightless.

Answer/Explanation

Ans:

c. 0.45 kg
d. This is simple harmonic motion. Using the equation for the period of a mass on a spring and solving for the spring constant we get k = 4π2m/T2 where m/T2 is the inverse of the slope of the line of best fit.
e. Yes, it can be used in space as the period of oscillation is independent of gravity.
f. F = GMm/r2 = ma which gives a = GM/r2 = 8.9 m/s
g. All objects aboard the shuttle, and the shuttle itself, are all accelerating toward the Earth at the same rate (they are in free fall). The normal force is zero and there is no sensation of weight.

 Question

A spherical, nonrotating planet has a radius R and a uniform density ρ throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass m could move freely under the influence of gravity. Let r be the distance of the ball from the center of the planet.
a. Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = -Cr. where C = 4πGρm/3
b. On the axes below, sketch the force F on the ball as a function of distance r from the center of the planet.

The ball is dropped into the tunnel from rest at point P at the planet’s surface.
c. Determine the work done by gravity as the ball moves from the surface to the center of the planet.
d. Determine the speed of the ball when it reaches the center of the planet.
e. Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.
f. Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation.

Answer/Explanation

Ans:

a.

b.

c. \(W = \int Fdr = \int -Cr dr\)

   \(W = \int_{R}^{0}-Cr dr = \frac{CR^{2}}{2}\)

d. W = ∆K
   CR2/2 = ½ mv2
    v = (CR2/m)1/2

e.  The ball will continue through the center of the planet and travel to the surface, where it will stop and return through the center and continue oscillating in this manner.

f.  F = ma = –Cr
    m(d2r/dt2) = –Cr
    d2r/dt2 + (C/m)r = 0 (simple harmonic motion)

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