Cavendish and the Value of G
Isaac Newton's law of universal gravitation proposed that the gravitational attraction between any two object is directly proportional to the product of their masses and inversely proportional to the distance between their centers. In equation form, this is often expressed as follows:
The constant of proportionality in this equation is G - the universal gravitation constant. The value of G was not experimentally determined until nearly a century later by Lord Henry Cavendish using a torsion balance.
Cavendish's apparatus for experimentally determining the value of G involved a light, rigid rod which was 6-feet long. Two small metal spheres were attached to the ends of the rod and the rod was suspended by a wire. When the long rod becomes twisted, the torsion of the wire begins to exert a torsional force which is proportional to the angle of rotation of the rod. Cavendish had calibrated his instrument to determine the relationship between the angle of rotation and the amount of torsional force. A diagram of the apparatus is shown below.
Cavendish then brought two large lead spheres near the smaller spheres attached to the rod. Since all masses attract, the large spheres exerted a gravitational force upon the smaller spheres and twisted the rod a measurable amount. Once the torsional force balanced the gravitational force, the rod and spheres came to rest and Cavendish was able to determine the gravitational force of attraction between the masses. By measuring m1, m2, d and Fgrav, the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10-11 N m2/kg2. Today, the currently accepted value is 6.67259 x 10-11 N m2/kg2.
The value of G is an extremely small numerical value. Its smallness accounts for the fact that the force of gravitational attraction is only appreciable for objects with large mass. While two students will indeed exert gravitational forces upon each other, these forces are too small to be noticeable. Yet if one of the students is replaced with a planet, then the gravitational force between the other student and the planet becomes noticeable.
Question:
Suppose that you have a mass of 70 kg (equivalent to a 154-pound person). How much mass would another object have to have in order for your body and the object to attract each other with a force of 1-Newton when separated by 10 meters?
m = 2.14 x 10^10 kg
Use the equation Fgrav = G*m1*m2/d^2 where
m1=70 kg, d=10 m and G=6.67x10^-11.
Substitute and solve.
(Note that the object is equivalent to approximately
23 million ton object!! It takes a large mass to have
a significant gravitational force.)