Question

A satellite of mass 100 kg is in circular orbit about planet Uzaz, which has a mass M and radius R, at an altitude (above the surface) h.

(a) (2 points) Express the linear speed v of the satellite in terms of the given variables and the universal gravitational constant.

(b) (3 points) Suppose that if h = 1 × 104 km, the period of the orbit is T = 11 hours. If Uzaz has a mass of 7.0 × 1024 kg, what does the satellite weigh on the surface of Uzaz?

You are an astronaut, and you have landed on planet Uzaz. You have in your research inventory: water (density 1000 kg/m3 , a beaker, nylon cord (linear mass density 0.0011 kg/m), and a frequency analyzer. Your task is to determine the density of an Uzazian rock you found that you were able to determine has a mass of 5.5 kg. On the surface of Uzaz, you perform the following experiment. You put water in the beaker, and suspend the rock from the end of the cord so the entire rock is just submerged in the water. The length of the cord is 0.75 m. While the rock is suspended, with the frequency analyzer, you measure the frequency of fundamental mode standing waves in the cord to be 92 Hz.

(c) What is the wavelength of the fundamental oscillations in the cord?

(d) Using (c), what is the speed with which those standing waves travel in the cord?

(e) Using (d), what is the tension in the cord?

(f) Using the results from the question (a) and (b), what is the weight of the rock?

(g) Using (e) and (f), determine the buoyant force exerted on the rock by the water.

(h) Using (g) and the results from question (a) and (b), determine the mass of water displaced by the rock.

(i) Use the density of water to determine the volume of the rock.

(j) Find the density of the rock.

(k) Suggest an elemental composition for the rock, based on your research.

(l) What fraction of the volume of Uzaz (assumed spherical) would, if it were comprised of the same material as your rock, account for the entire mass of Uzaz? What can you conclude about the relative abundance of your rock’s material on Uzaz?

Answer 1

We will use Mass of Planet "M", radius "R" , mass of satellite "m" and altitude above planet's surface "h" and linear speed of satellite "v"

Part a)

Gravitational force on satellite will be equal to centripetal force:

GMm / (R+h)² = mv²/(R+h)

v² = GM / (R+h)

v = SQRT [GM / (R+h)]

Part b):

Period of Orbit; T = 11 hr = 11*3600 = 39600 second

As per this, linear speed of satellite will be 2π (R+h) /T

We can compare this with speed derived in Part a)

SQRT [GM / (R+h)] = 2π (R+h) /T

GM / (R+h) = 4π² (R+h)² /T²

(R+h)³ = GMT² /(4π²)

We have

h = 1* 10⁴ km = 10⁷ m ; G = 6.67 * 10-11 Nm²/kg² ; M = 7 * 10²⁴ kg ; T = 39600 s

( R +h)³ = 6.67 * 10^-11 * 7 * 10²⁴ * (39600)² / (4π²) = 1.854 * 10²²

(R+h) = (1.854 * 10²²) ^1/3 = 2.65 * 10⁷ m

R = 2.65 * 10⁷ - h = 2.65 * 10⁷ - 1 * 10⁷ = 1.65 * 10⁷ m

Value of Gravitational acceleration g_u at Planet Uzaz surface will be GM/R²

g_u = GM/R² = 6.67 * 10^-11 * 7 * 10²⁴ / (1.65 * 10⁷)² = 1.71 m/s²

Weight of 100 kg Satellite at Uzaz planet = 100 * 1.71 = 171 N

Part c)

Length of Cord ; L = 0.75 m

For fundamental standing wave, wavelength; λ can be given as:

λ /2 = L

λ = 0.75 * 2 = 1.5 m

Part d)

Frequency of standing wave; n = 92 Hz

Speed of standing wave = nλ = 92 * 1.5 = 138 m/s