Optional Unit V: Applications of Kinematics and Dynamics
E. Universal Gravitation

Key Concepts

A field is a region in space where one object can exert an influence on another object at a distance.

A force is a push or pull on an object.

A force is applied by one field acting on another similar field.

A force applied to an object has a tendency to change the shape or the motion of an object.

There are four important forces in nature: the force of gravity, the electromagnetic force, the weak nuclear force, and the strong nuclear force. (Scientists are currently looking for a possible fifth force.)

Force is a vector quantity.

The SI unit of force is the newton (N).

In fundamental units: 1 N = 1 kg m/s²

Gravitational field strength is the force acting on a 1 kg mass. It is measured in N/kg.

Mass (m) depends on the amount of matter in an object. The SI fundamental unit for mass is the kilogram.

The mass of an object is independent of gravitational field strength.

Mass can be determined on an equal arm balance, by making comparisons against standard masses. At the same place in an external field, two objects with the same mass will have the same weight.

Weight is the force that a gravitational field exerts on an object. The terms weight, force of gravity, and gravitational force are sometimes used synonymously.

Weight is a vector quantity.

At the surface of the Earth an object's weight acts downward, towards the centre of the Earth.

Changes in altitude and latitude affect the gravitational field strength on the surface of the Earth.

Changes in the composition of the Earth's crust affect the gravitational field strength.

The SI unit for weight is the newton (N). The weight of an object can be determined using:

Gravitational mass can be derived by determining the weight of an object within a known gravitational field.

The weight of an object depends on its location with respect to one or more celestial bodies.

The force of gravity involves the gravitational fields of objects acting upon one another.

A spring balance may be used to measure weight. A force equilibrium is established between the spring tension and the force of gravity.

The force of gravity between two masses varies directly with the product of the masses and inversely with the square of the distance between their centres of mass.

The gravitational forces two objects experience are equal in magnitude, but act in opposing directions.

At a given separation:

F = m₁m₂

If the masses remain constant, and if b (y intercept) = 0 (e.g., F=k(1/d²) + b), then:

or

Newton's Law of Universal Gravitation can be expressed as:

where F is the attractive force between m₁ and m₂ (in N), m₁ and m₂ are the masses in kg, and d is the distance between the masses, in metres.

The value of , the gravitational constant, can be expressed as:

6.67 x 10^-11 Nm²/kg².

The numerical value of depends on the fundamental units used.

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: field, force, mass, weight, gravitational field strength.

2. State the correct SI units and symbols for force, mass, and weight.

3. Describe the effects that a force can have when acting on an object.

4. Identify important forces found in nature.

5. Explain that force is a vector quantity.

6. Describe the difference between mass and weight.

7. Describe and compare how mass and weight can be determined.

8. Demonstrate an understanding of a gravitational field.

9. Describe methods of determining the gravitational field strength or gravitational mass at some point in space.

10. Describe why slight variations in gravitational field strength are found on different places on the Earth.

11. Compare the weight of a given object in different locations in space and on different celestial bodies.

12. Solve problems relating to gravitational force.

13. Interpret direct and inverse square law relationships, as illustrated by Newton's Law of Universal Gravitation.

14. Determine the gravitational force on an object at various distances, expressed in multiples of Earth radii, from the centre of the Earth.

15. Explain how the gravitational mass of an object within a known gravitational field can be derived.

Teaching Suggestions, Activities and Demonstrations

1. When considering laws for the first time in physics, it might be worth identifying what a law is and how it differs from a theory. Students could brainstorm their ideas and create posters or bulletin board displays to visually show the difference between a law and a theory.

2. Derivations may be of some interest to a few students, but should not be pursued in too much depth. In particular, derivations of such things as Newton's Law of Universal Gravitation from Kepler's Laws go well beyond the scope of the course for most students. As a challenge activity for some more advanced students though, this may be one way of adapting the material in the course to suit their needs.