1. GRAVITATION
Prepared by fRancis Chong

2. GRAVITATIONAL FIELD
It is a region in space that surrounds a mass, where its influence can be observed.
In this region a force will be exerted on a mass that is placed in the field.

3. Field lines are drawn such that
GRAVITATIONAL FIELD
Field lines are drawn such that
- The tangent to the field represents the direction of g.
- The number of field lines per unit cross-sectional area is proportional to the magnitude of g.
Earth

4. NEWTON’S LAW OF GRAVITATION
The force of attraction between two given point masses is directly proportional to the product of their masses and inversely proportional to the square of their distance apart.
Sometimes written as
To indicate that it is an attractive force.

5. NEWTON’S LAW OF GRAVITATION
Example
Find the gravitational force exerted by Earth on a point mass at a distance h away from the Earth’s surface.
Solution
Recall rE h

6. GRAVITATIONAL FIELD STRENGTH
It is defined as the gravitational force per unit mass acting on a small mass placed at a point.

7. GRAVITATIONAL FIELD STRENGTH
Consider M r m
From Newton’s Law of Gravitation,
Also,

8. GRAVITATIONAL FIELD STRENGTH
Variation of g with distance from a point mass
Taken from

9. GRAVITATIONAL FIELD STRENGTH
Variation of g with distance from the centre of a uniform spherical mass of radius, R r m M
Taken from

10. GRAVITATIONAL FIELD STRENGTH
Variation of g on a line joining the centres of two point masses
If m1 > m2 then
Taken from

11. GRAVITATIONAL FIELD STRENGTH
This means that we can find a point between two masses where their combined field strength is zero. d
Moon
Earth
At this point, the net resultant gravitational force exerted by both masses is zero. i.e. both the Earth and Moon are exerting the same but opposite forces at p.
Taken from

12. GRAVITATIONAL POTENTIAL
The Gravitational Potential is defined as the work done by an external agent in bringing a unit mass from infinity to its present location.
Unit of Φ = J kg-1

13. GRAVITATIONAL POTENTIAL
The expression is negative because of the way we define Gravitational potential.
The zero of the potential is taken to be when the mass is at infinity.
Gravitational force is an attractive force.
Therefore to bring a unit mass from infinity to its present location, an external agent must be present to do negative work.

14. GRAVITATIONAL POTENTIAL ENERGY
The Gravitational Potential energy is defined as the work done by an external agent in bringing a mass from infinity to its present location.
Unit of U = J

15. Tips for solving Orbital / Satellite problems
We always equate or

16. ESCAPE VELOCITY
It is possible to find a velocity of a projected mass such that it can escape the Earth’s gravitational field and reach infinity.
At infinity, GPE is zero.
Also if the mass is just able to reach infinity, KE should be zero.
Therefore total energy E = PE + KE = 0
By conservation of energy, total energy of the mass should be the same throughout.

17. ESCAPE VELOCITY
Therefore, any object with total energy = 0 will be albe to reach infinity.
This is known as the escape velocity.

18. In the figure, ABC is a triangle in which CA = CB = r.
Practice 1
In the figure, ABC is a triangle in which CA = CB = r.
Equal masses m are situated at A and B.
(a) In which direction relative to the x and y directions does the resultant field at C act? A B C x y

19. (b) Find the gravitational potential at C.
Practice
(b) Find the gravitational potential at C. A B C x y

20. Practice 2
At what height above the Earth’s surface will g = 9.81 ms-1 rounded off to 2 d.p. change?