Asolid sphere of uniform density has a mass of 4.4 × 104 kg and a radius of 1.9 m. what is the magnitude of the gravitational force due to the sphere on a particle of mass 8.3 kg located at a distance of (a) 3.7 m and (b) 0.41 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 ≤ 1.9 m from the center of the sphere.

(a)

Here we want to find the gravitational force exerted on the particle at a distance of 3.7 m from the center of the sphere. Since the radius of the sphere is 1.9 m, we are outside the sphere, so we can use Newton's law of gravitation:

where

G is the gravitational constant

m = 8.3 kg is the mass of the particle

is the mass of the sphere

r = 3.7 m is the distance

Substituting into the formula, we find

(b)

Here we want to find the gravitational force exerted on the particle at a distance of 0.41 m from the center of the sphere. Since the radius of the sphere is 1.9 m, this time we are inside the sphere, so the formula for the gravitational force is different:

where

G is the gravitational constant

m = 8.3 kg is the mass of the particle

is the mass of the sphere

r = 0.41 m is the distance from the centre of the sphere

R = 1.9 m is the radius of the sphere

Substituting numbers into the formula, we find

(c)

The magnitude of the gravitational force on the particle when located inside the sphere can be found starting from Newton's law of gravitation:

(1)

where the only difference compared to the standard law is that M' is not the total mass of the sphere, but only the amount of mass of the sphere enclosed by the spherical surface of radius r centered in the center of the sphere.

The mass enclosed is

(2)

where is the density of the sphere and V' is the enclosed volume. We can rewrite the density of the sphere as ratio between mass of the sphere (M) and volume of the sphere:

(3)

where R is the radius of the sphere.

Substituting (3) into (2):

And substituting the last equation into (1), we find

which depends linearly on r.

d₁ = displacement of tern from arctic to antarctic = 11000 miles

d₂ = displacement of tern from antarctic to arctic =   - 11000 miles

d = total displacement for the journey

total displacement for the journey is given as

d = d₁ + d₂

d = 11000 + (-11000)

d = 0 miles

t₁ = time of travel from arctic to antarctic = 3 months = 3 x 30 days = 90 days

t₂ = time of travel from antarctic to arctic = 3 months = 3 x 30 days = 90 days

t = total time of travel

total time of travel is given as

t = t₁ + t₂

t = 90 + 90

t = 180 days

average velocity is given as

average velocity = total displacement/ total time

v = d/t

v = 0/180

v = 0 miles/day

a cold ice-tea.

pour in room temperature or cold water. cover the pitcher and refrigerate. steep white or green tea for 6 to 8 hours; steep black or oolong tea for 8 to 12 hours.jun 28, 2015