**5 Gravitational force Weight – Problems and Solutions**

1. What is the force of gravity acting on an object at the Earth’s surface? Earth’s mass = 5.98 x 10^{24} kg, object’s mass = 1000 kg, the radius of the Earth is 6.38 x 10^{6} m.

__Known :__

Earth’s mass (m_{E}) = 5.98 x 10^{24} kg

Object’s mass (m_{o}) = 1000 kg

The radius of the Earth (r_{E}) = 6.38 x 10^{6} m

Universal constant (G) = 6.67 x 10^{-11} N m^{2} / kg^{2}

Acceleration due to gravity (g) = 9.8 m/s^{2}

__Wanted :__ the force of gravity

__Solution :__

*w = weight, F = force, G = universal constant, m*_{E}* = **Earth’s mass**, m*_{o}* = o**bject’s mass**, r = the distance between the Earth’s center and object.*

The object is at the surface of the Earth, so r = the radius of the Earth

*Object’s weight (Newton’s second law of motion) :*

w = m g

w = (1000 kg)(9.8 m/s^{2})

w = 9800 N

2. What is the force of gravity acting on an object at 10,000 meters above the Earth’s surface ? Earth’s mass = 5.98 x 10^{24} kg, object’s mass = 1000 kg, the radius of the Earth is 6.38 x 10^{6} m.

__Solution :__

3. The weight of a spacecraft is w. If D = Earth’s diameter, determine the weight of spacecraft when the spacecraft is at 2D above the Earth’s surface.

__Known :__

D = Earth’s diameter,

R = the radius of the Earth

1 D = 2 R, 2 D = 4 R

__Wanted:__ spacecraft’s weight at 2D above the Earth’s surface?

__Solution :__

4. The ratio of the mass of the planet A and planet B is 2 : 3, while the ratio of the radius of the planet A and planet B is 1 : 2. If the weight of an object on planet A is w, what is the weight of the object on the planet B.

__Known :__

Mass of planet A (m_{A}) = 2

Mass of planet B (m_{B}) = 3

Radius of the planet A (r_{A}) = 1

Radius of planet B (r_{B}) = 2

Mass of object = m

Object’s weight on planet A = w

__Wanted:__ Object’s weight on planet B

__Solution :__

The equation of the force of weight from Newton’s law of gravity :

*w = weight, G = gravitational constant, M = mass of planet, m = mass of object, r = the distance between object and planet. If the object on the planet surface the r = radius of the planet.*

Object’s weight on planet A :

Object’s weight on planet B :

Object’s mass is same so that substitute m with w/2G.

5. A rocket with the weight of w launched vertically from the surface of the Earth. D is the diameter of the Earth. When rocket at the height of 0.5 D above the surface of the Earth, then what is the weight of the rocket.

__Known :__

Rocket’s weight = w

The radius of Earth = distance from the center of Earth = R

Diameter of Earth = D = 2R

__Wanted:__ Weight of rocket when the rocket at the height of 0.5 D above the Earth surface.

__Solution :__

0.5 D = 0.5 (2R) = R

The distance of rocket from the center of Earth = radius of Earth + radius of a rocket from the surface of Earth = R + R = 2R

Weight is the force of gravity that acts on an object. The force of gravity (F) is inversely proportional to the square of the distance from the center of the earth (R) so that the weight is inversely proportional to the square of the distance.

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