The Open Door Web Site

Mechanics

Universal Gravitation

Newton suggested (in 1687) that every object attracts every other object with a force called which he called gravitation.

He proposed what is now known as Newton’s Law of Universal Gravitation:

Every body attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass.
F a m1m2 F a 1/r²
F a m1m2/r²

and the constant of proportionality, G, is called the universal gravitational constant.

The value of G (found by experiment) is 6·7 × 10-11 Nm²kg-2 which explains why gravitational forces are negligible unless the mass of (at least one of) the bodies concerned is very large.

Relation between the Acceleration due to Gravity and the Universal Constant of Gravitation

If a body is allowed to fall freely, near the earth’s surface, it will fall with acceleration, g.

From Newton’s second law we have:

where m is the mass of the body and F is the force of gravity acting on it.

Also,

where R is the radius of the earth and M is the mass of the earth.

Combining these two equations shows us how the acceleration due to gravity depends on the value of the universal constant of gravitation.

 

It should be noted that, to obtain the above relation, we have assumed that the "m" which appears in Newton’s second law of motion is the same quantity as the "m" which appears in the universal gravitation law.

In other words we have assumed the equivalence of 
inertial and gravitational mass.

Experiments show that this assumption is justified but it is not understood why inertial and gravitational masses are the same.

Acceleration due to Gravity on different Planets

When an object is thrown upwards, the maximum height it reaches depends on:

i) the initial velocity, u
ii) the acceleration due to gravity, g.

Consider an object which is thrown vertically upwards with an initial speed, u.

v² = u² + 2gs

If we consider the instant at which the object reaches its maximum height, v = 0,
so, we have

u² = -2gs

If objects are thrown upwards on different planets but each time with the same initial speed, we can say, gs = a constant.

In other words the height the object will reach is inversely proportional to the acceleration due to gravity.

So, when comparing different planets

g1s1 = g2s2

x

Example
 A person can jump 1·5m on the earth. How high could the person jump on a planet having the twice the mass of the earth and twice the radius of the earth?

 

© David Hoult 2008