Question

ap physics c, unit 3b, 2-d forces and gravitation, test v2 name: lucy luo
e 1 the two spheres pictured above have equal densities and are subject only to their mutual gravitational attraction. which of the following quantities must have the same magnitude for both spheres?
(a) acceleration (b) velocity (c) kinetic energy (d) displacement from the center of mass (e) gravitational force
a 2 a newly discovered planet, \cosmo,\ has a mass that is 4 times the mass of the earth. the radius of the earth is rₑ. the gravitational field strength at the surface of cosmo is equal to that at the surface of the earth if the radius of cosmo is equal to
(a) ½rₑ (b) rₑ (c) 2rₑ (d) √rₑ (e) rₑ²
d 3. the radius of the earth is approximately 6,000 kilometers. the acceleration of an astronaut in a perfectly circular orbit 300 kilometers above the earth would be most nearly
(a) 0 m/s² (b) 0.05 m/s² (c) 5 m/s² (d) 9 m/s² (e) 11 m/s²
e 4. each of five satellites makes a circular orbit about an object that is much more massive than any of the satellites. the mass and orbital radius of each satellite are given below. which satellite has the greatest speed?
mass radius
(a) ½m r
(b) m ½r
(c) m r
(d) m 2r
(e) 2m r
d 5. a satellite of mass m moves in a circular orbit of radius r with constant speed v. true statements about this satellite include which of the following?
i. its angular speed is v/r.
ii. its tangential acceleration is zero.
iii. the magnitude of its centripetal acceleration is constant.
(a) i only (b) ii only (c) i and iii only (d) ii and iii only (e) i, ii, and iii

Explanation:

Step1: Analyze gravitational force

According to Newton's law of universal gravitation, the gravitational force \(F = G\frac{m_1m_2}{r^{2}}\). For two spheres subject to mutual gravitational attraction, by Newton's third law, the gravitational force that one sphere exerts on the other is equal in magnitude and opposite in direction. So the gravitational force has the same magnitude for both spheres.

Step2: Analyze acceleration

Acceleration \(a=\frac{F}{m}\). Since the masses of the two spheres (from \(m =
ho V=
ho\frac{4}{3}\pi r^{3}\), different volumes as different sizes) are different, even if \(F\) is the same (from step 1), \(a\) is different (\(a\propto\frac{1}{m}\)).

Step3: Analyze velocity

Velocity \(v = v_0+at\). Since \(a\) (from step 2) and the time of interaction (assuming they start from rest) are related, and \(a\) is different, \(v\) is different.

Step4: Analyze kinetic energy

Kinetic energy \(K=\frac{1}{2}mv^{2}\). Since \(m\) and \(v\) (from step 3) are different, \(K\) is different.

Step5: Analyze displacement

Displacement \(x=\frac{1}{2}at^{2}\). Since \(a\) (from step 2) is different, \(x\) is different.

Answer:

E. Gravitational force