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A spherical object of mass 150kg is orbiting the Earth. The distance between the centre of the object and the centre of the Earth is 25,000m. What is the kinetic energy of the object?

This is an example of circular motion caused by a centripetal force. In this case, the centripetal force is the gravitational force between the Earth and the object. Linking the equations for circular motion (F = mv^2/r) and for the gravitational force between two masses (F = GMm/r^2) can give us the information requested in the question.

The force terms (F) in both equations should be equated, as it is the gravitational force in the second equation that is causing the circular motion described in the first. This means we can write mv^2/r = GMm/r^2 (where m is the mass of the orbiting object and M is the mass of the orbited object, in this case the Earth, which is 5.98*10^24 kg. G is the gravitational constant and r is the distance between the masses). Cancelling the m and r terms gives:

v^2 = GM/r

It should be noted that v here gives the speed of the object - the velocity in the direction of the tangent of the orbit.

The equation for the kinetic energy of an object is given by:

KE = 1/2 * mv^2

Since we have an equation that tells us v^2, we simply need to multiply that equation by m/2 to complete the question.

KE = 1/2 * GMm/r

(remember, M and m are different - m is the mass of the smaller object in this case)

Now it is just a case of plugging in the values to get your final answer.

KE = (1/2 * 6.67*10^-11 * 5.98*10^24 * 150)/25000 = 1.2*10^12 Joules.

Done!

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