Gravity and Inverse Square Relationships NIS

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Defined as ‘φ’ or work done measure in Joules

mass kg

Work done on a unit of mass in a gravitation field by bringing that mass from infinity

Allows for easier accounting of work and energy in a field where force varies with distance

Has the units of Joules per kilogram

Is a scalar quantity

At infinity Potential is zero, therefore Potential is always negative

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V=ϕ=-GM/r Potential becomes Zero at Infinity

How do we get this expression?

Slide 12

Advanced Students: Integration will give the are under the curve which is work done on the mass

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Work done by the mass= -mΔV

And work done by the mass is

force times distance moved so

mgΔr = -mΔV

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Combining equations and calculus for ‘g’ gives the general formula of ‘V’

Two equations for ‘g’ and

Combining

Integrating and solving for V

Slide 15

Slide 16

Escape Velocity

Slide 17

Deriving Escape Velocity

We can calculate the energy necessary to escape earth's gravity well completely.

Gravitational Potential (Φ):

There G is the universal gravitational constant; M is the mass of the earth and r is the distance from the center of the earth.

We want to find the difference in potential of an object at infinity (i.e., it has escaped earth forever) and at the surface of the earth. Using r0 as the radius of the earth can write this difference as

Since the 1/∞ term will go to 0 we find the potential needed to escape earth is

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Deriving Escape Velocity:

Gravitational potential energy is the same as gravitational potential per unit mass. The speed you would need to have enough energy to escape earth's gravity well is called escape velocity To find this number we set the potential energy equal to kinetic energy.

The mass of the object m cancels out as expected because the escape velocity should be the same for all objects. Solving for v we get

Substituting our escape potential we get

Plugging in numbers we find the escape velocity to be 11,181 m/s or about 25,011 mph.

Slide 19

Geostationary Satellites