Question

A solid sphere of uniform density has a mass of 3.0 × 104 kg and a radius of 1.0 m. What is the magnitude of the gravitational force due to the sphere on a particle of mass 9.3 kg located at a distance of (a) 1.4 m and (b) 0.21 m from the center of the sphere? (c) Write a general expression for the magnitude of the gravitational force on the particle at a distance r ≤ 1.0 m from the center of the sphere.

Answer 1

Answer:

a) Fg = 9.495x10⁻⁶N

b) Fg = 3.908x10⁻⁶N

c)

$F_{g} =\frac{Gm_{1}m_{2}R }{r^{3} }$

Explanation:

Given:

m₁ = mass = 3x10⁴kg

r = radius = 1 m

m₂ = 9.3 kg

Questions:

a) What is the magnitude of the gravitational force due to the sphere located at R = 1.4 m, Fg = ?

b) What is the magnitude of the gravitational force due to the sphere located at R= 0.21 m, Fg = ?

c) Write a general expression for the magnitude of the gravitational force on the particle at a distance r ≤ 1.0 m from the center of the sphere.

a) Since R > r, the equation for the gravitational force is:

$F_{g} =\frac{Gm_{1}m_{2} }{R^{2} }$

Here,

G = gravitational constant = 6.67x10⁻¹¹m³/s² kg

Substituting values:

$F_{g} =\frac{6.67x10^{-11}*3x10^{4}*9.3 }{1.4^{2} } =9.495x10^{-6} N$

b) Since R < r, the equation for the gravitational force is:

$F_{g} =\frac{Gm_{1}m_{2}R }{r^{3} } =\frac{6.67x10^{-11}*3x10^{4}*9.3*0.21 }{1^{3} } =3.908x10^{-6} N$

c) The general expression for the magnitude of the gravitational force on the particle at a distance r ≤ 1.0 is the same to b)

$F_{g} =\frac{Gm_{1}m_{2} R}{r^{3} }$