PHY 1151

Chapter 12, Gravity

D12.1 Calculate the mass of Jupiter, given that its moon Callisto has a mean orbital radius of 1.88 x 10⁶ km and an orbital period of 16 days, 16.54 hours.

The force of gravity provides the centripetal force to keep Callisto in its orbit.

F_g = G M_J m / r² = m v² / r = F_c
M_J =

We must find the linear speed of Callisto.

v =

T = 16 d, 16.54 h = [(16)(24) + 16.54 ] h = 400.54 h
T = 400.54 h [3600 s/h] = 1.44 x 10⁶ s

F_gravity = F_centripetal

F_gravity = G M_J m_C/r² = m_C v²/r = F_centripetal

G M_J m_C/r² = m_C v²/r

M_J = r v²/G

M_J = 1.9 x 10²⁷ kg

D12.2 Starting with the moon's period of 27.3 days, calculate the radius of its orbit.

The gravitational force between Earth and our moon provides the centripetal force,

F_g = = m = F_c
= m

Don’t try to solve for the radius immediately for we know the velocity only in terms of the radius,

v =
= =

= m = m

T = 27.3 da (24 h/da) (3600 s/h) = 2.36 x 10⁶ s
r³ =
r³ =
r³ = 5.627 x 10²⁵ m³
r = 3.83 x 10⁸ m
r = 3.83 x 10⁵ km

D12.3 The acceleration of a falling body near Earth’s surface, at a distance R from Earth’s center, is 9.80 m/s².

(a) Use a suitable proportion to calculate the acceleration toward Earth of a falling body that is 60 R from Earth’s center.
(b) Our moon is in an orbit of radius 60 R, with a period of revolution of 27.26 days. Show, as did Sir Isaac Newton, that the centripetal acceleration of the moon toward Earth agrees with your answer from part (a).

Earth’s radius is R = 6.38 x 10³ km.

F = G
m g = G
a(R) = g = G
a(60R) = G = G = g
a(60R) = g = (9.8 ) = 2.7 x 10^-3 = 2.7

a_c =
v = = = =
v = 1,020 m/s
a_c = = = = 2.7 x 10^-3 = 2.7

D12.4 What orbital radius should a weather satellite have if it is to have a period of 6.0 hours?

F_c = m = G m = F_g
= G
v = =
= = G
r ³ = G M_E T ²
r ³ = (6.67 x 10 ^{- 11}) ( 5.98 x 10 ²⁴) (6.0 h ) ²
r ³ = 4.71 x 10 ²¹

r = 1.68 x 10 ⁷ m = 1.68 x 10 ⁴ km = 16,800 km

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D12.5 From the data below, calculate the acceleration of free fall on the surface of

a) Jupiter,
b) Saturn, and
c) our Moon.

	mass	radius
Jupiter	1 900 x 10^24 kg	71 400 km
Saturn	561 x 10^24 kg	60 000 km
moon	0.0736 x 10^24 kg	1 740 km

F = m a = G

a = G

a)

b)

c)

 

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