Universal Gravitation

​Mass vs Weight

• ​Mass is an inherent quality of objects.

• It is the amount of matter present in an object.

• Whether on Earth, in space, or on the moon, this amount remains the same.

• Weight, on the other hand, is the force acting on an object that has mass.

• This force is the pull of gravity.

• Weight is not a property of the object; rather, it is the pull exerted by the surface on which the object exists.

• Therefore, objects will have different weights on different planets.

Calculating Weight

• ​Recall that Newton's Second Law states the sum of the forces acting on an object is equal to the mass of the object times its acceleration.

• Since weight is a force, we can say that: F_g = mg

• Here F_g is the wieght, m is the mass, and g is the acceleration due to gravity.

• On a planet then, the weight vector is pointed towards the center of the planet.

• While there are many units used to measure weight in society, since it is a force, we use the units of newtons (N) in physics.

• So for example, a 2.0 kg bar would be said to weigh 19.6 N on Earth, since g = -9.8 m/s²

Universal Law of Gravitation

• We know that gravity is a field force, meaning that it doesn't need contact to transmit the force.

• We also know that the acceleration from gravity is different on different planets.

• What we haven't mentioned yet is that the acceleration of gravity is different depending on how far we are from the planet's center of mass.

• The table shows the values for different distances from the Earth's surface.

Inverse Square Law

• Based on the motion of the planets, Newton proposed that the force of gravity (F_g) follows an inverse square law with respect to the distance between two bodies.

• In other words: F_g ∝ 1/r²

• This means that if we double the distance between two objects, then the gravitational force decreases by a factor of four.

• Newton also realized that gravity's pull on an object is not just influenced by its distance from Earth's center, its mass also plays a role.

• The greater the mass of the object, the greater the pull of gravity.

• Gravity is directly proportional to the mass of the object.

Universal Law of Gravitation

• Putting all of this together, we can express gravity as: F_g = G(m_E⋅m_obj)/r².

• Where m_E is the mass of the Earth, m_obj is the mass of the object, and G is the universal gravitational constant.

• The value of G was found by Lord Henry Cavendish almost a century after Newton's death.

• He found that the value of the gravitational force between two objects of 1 kilogram each, kept 1 meter apart, would equal just 0.0000000000667 newtons.

• Thus, the value of G is 6.673 × 10^-11 newtons⋅meters²/kilograms².

• From the value of this constant you can see that the gravitational force is insignificant for most day–to–day objects you encounter.