# Gravitational Forces among the Four Objects

1.Three solid spheres of lead, each of mass 9.8 kg, are located at three corners of a square with side lengths of 50 cm. A small object is released at the forth corner. Considering only the gravitational forces among the four objects, determine the magnitude and direction of the acceleration of the smaller object when it is released.

2.A spy satellite is in circular orbit around the earth and makes four revolutions every day.

a)How high above the earth's surface is the satellite?

b) Calculate the satellite's speed?

3.A hand gun fires a 12.0 g bullet at a speed of 400 m/s.

What is its kinetic energy?

At what speed must a motorcycle of mass 173 kg move to have the same kinetic energy as the bullet? Express your answer in km/h.

4. A dinner plate of mass 510 g is pushed 60 cm along a dining table by a constant force of 3.0 N directed 22° below the horizontal. If the coefficient of kinetic friction between the plate and the table's surface is 0.44, determine the work done on the plate by

the applied force

the force of gravity

the normal force

the force of kinetic friction

5. A 15 kg block slides along a horizontal frictionless surface 3.0 m above the ground at a constant speed of 2.0 m/s. The block then slides down an incline that makes an angle of 35° with the horizontal and has a coefficient of kinetic friction equal to 0.30. After reaching the end of the incline, the block continues sliding horizontally across the frictionless ground. Calculate the kinetic energy of the block as it slides

a) along the upper surface.

b) along the ground.

#### Solution Preview

1.

Three solid spheres of lead, each of mass 9.8 kg, are located at three corners of a square with side lengths of 50 cm. A small object is released at the forth corner. Considering only the gravitational forces among the four objects, determine the magnitude and direction of the acceleration of the smaller object when it is released.

Let mass be located at position and mass m locates at position

The vector of the attractive gravitational force applied by M on m is given by:

(1.1)

Where G is the universal gravitational constant and is the magnitude of teh diatance between M and

Hrnce, according to Newton's second law, the acceleration of m due to M will be:

(1.2)

If we have N such masses, then the total acceleration vector is simply the sum of all these vectors:

(1.3)

In our case, the system looks like this:

The side of the square is a, hence the mass m is located at:

(1.4)

The three other masses are located at:

(1.5)

In our case all the three masses are the same, hence we can write:

(1.6)

We can use a table:

i

1

2

3

Together:

(1.7)

The magnitude of the acceleration is:

(1.8)

The direction is the direction of the vector which is simply a vector that makes an angle of ...

#### Solution Summary

The gravitational forces among the four objects are determined.