Question

1. 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 \t radius
(a) ½m \t r
(b) m \t ½r
(c) m \t r
(d) m \t 2r
(e) 2m \t r

1. 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 \t (b) ii only \t (c) i and iii only \t (d) ii and iii only \t (e) i, ii, and iii

Explanation:

Problem 4

Step1: Derive the formula for satellite speed

For a satellite in circular orbit, the gravitational force provides the centripetal force. So, \(F_{grav}=F_{centripetal}\). The gravitational force is \(F_{grav}=G\frac{Mm}{r^{2}}\) (where \(M\) is the mass of the massive object, \(m\) is the mass of the satellite, \(G\) is the gravitational constant, and \(r\) is the orbital radius), and the centripetal force is \(F_{centripetal}=m\frac{v^{2}}{r}\). Equating them: \(G\frac{Mm}{r^{2}}=m\frac{v^{2}}{r}\). Canceling out \(m\) from both sides, we get \(v = \sqrt{\frac{GM}{r}}\). Here, \(M\) is constant (since the central object is the same for all satellites).

Step2: Calculate the speed for each option

• Option A: \(v_A=\sqrt{\frac{GM}{R}}\)
• Option B: \(v_B=\sqrt{\frac{GM}{\frac{1}{2}R}}=\sqrt{2\frac{GM}{R}}\)
• Option C: \(v_C=\sqrt{\frac{GM}{R}}\)
• Option D: \(v_D=\sqrt{\frac{GM}{2R}}\)
• Option E: \(v_E=\sqrt{\frac{GM}{R}}\)

Step1: Check statement I

The relationship between linear speed \(v\) and angular speed \(\omega\) is \(v = r\omega\). Rearranging for \(\omega\), we get \(\omega=\frac{v}{R}\). So, statement I is True.

Step2: Check statement II

The satellite is in circular motion. The tangential acceleration \(a_t=\frac{dv}{dt}\). Since the speed \(v\) is constant, \(a_t = 0\). So, statement II is True.

Step3: Check statement III

The centripetal acceleration \(a_c=\frac{v^{2}}{R}\). The magnitude of the centripetal acceleration is constant (because \(v\) and \(R\) are constant), but the direction of the centripetal acceleration is always changing (towards the center of the circle). However, the problem asks about the magnitude. So, statement III is True.

Answer:

B. \(m\), \(\frac{1}{2}R\)

Problem 5