Topic 6

Circular Motion and

Gravitation

LEARNING OBJECTIVES

1. Centripetal Force2. Newton’s Law of

Gravitation 3. Gravitational Field

Strength

Topic 6 “The Killers”

Always remember….the formulae for

centripetal force and gravitational fore are

equal when considering orbits.

ROOKIE MISTAKE!

1 Centripetal

Force

RadiansMeasuring angles in degrees

becomes less useful in advanced maths and physics,

because they are arbitrary.

Radians are used because they are multiples of π, which is a

natural number and the natural unit for trigonometric

functions.

The angle in radians, θ, is defined as the arc-length divided by the radius.

Radians

For a complete circle:

θ = arc-length = 2πr = 2π rads radius r

Any other angles are a fraction of 2π

Degrees and Radians

Degrees Radians

90∘

360∘45∘

180∘π/2

π

2ππ/4

Angular Displacement, θAngular displacement, θ, is the angle in radians (or

degrees) through which a point has been rotated about a specified axis.

Angular Displacement, θ

Linear VelocityLinear velocity (or tangential velocity) is the velocity of a point on a rotating object.

It is given by the equation:

v = 2πrt

Where r is the radius of the point and t is the time taken for one revolution.

Units of v are ms-1

Linear Velocity

The radius is smaller for the red point, but

the time taken for one complete revolution remains the same.

vred < vblue

Consider a rotating bicycle wheel:

Angular VelocityAngular velocity is the rate of change of angular displacement with respect to time

It is given by the equation:

Where θ is the angular displacement and t is the time taken.

Units of ω are rads-1

ω = θt

Angular Velocity

As the wheel rotates, the angle subtended by both red and blue points is the same,

with respect to time.

ωred = ωblue

Consider the rotating bicycle wheel again:

Linear and Angular Velocity

The relationship linking linear (tangential) and

angular velocity is:

v=ωr

Velocity Changes

When an object travels at a constant speed in circular

motion, the velocity is constantly changing due to

the constant change in direction.

If velocity is changing then the object is accelerating

Centripetal AccelerationAcceleration is defined as the rate of change of velocity.

Consider the change of direction of velocity below:

When an object is in circular motion, the acceleration of

the object is always directed

towards the centre of the circle.

ACTUAL

EXAMINER

FEEDBACK

“Candidates were unable to use a vector diagram to explain the need for a centripetal force in circular motion.”

Acceleration ExpressionsCentripetal Acceleration Angular Acceleration

measured in rads-2

a =ω 2ra = vr

2

measured in ms-2

Centripetal Force

Newton’s 2nd Law states:

Resultant force equals mass times

acceleration in the direction if the force.

An object in circular motion always experiences a force directed towards the centre of the circle.

Force Expressions

Measured in Newtons, N

F = ma = mvr

2

F = ma = mω 2r

Using Centripetal Acceleration

Using Angular Acceleration

1. Centripetal Force2. Newton’s Law of

Gravitation

Topic 6 “The Killers”

Gravitational Force

• Gravitational force acts at a distance and has an associated force field

• Newtons’ 3rd Law: • Earth exerts a gravitational force on the Moon. • Moon exerts an equal and opposite gravitational force on

the Earth. • Gravity is the weakest force and is only measurable with

large masses (e.g. planet sizes)

Gravitational Force

The size of the gravitational force is:

Newton’s Law of Gravitation, where:

ACTUAL

EXAMINER

FEEDBACK

“Candidates could not provide a statement to encompass Newton’s Law of Gravitation”

Gravitational Force

• The size of the gravitational force is: • Directly proportional to the product of the masses • Indirectly proportional to the square of the distance

between the masses • Assumption: masses have uniform density and the mass

is concentrated at the centre (i.e. point masses)

Orbits

The size of the gravitational force is EQUAL TO the centripetal force

Kepler’s Law

Not directly examined BUT… Recent paper has asked to prove:

WHAT?!?

Kepler’s LawNot directly examined BUT…

Recent paper has asked to prove:

Start Here:

Substitute into above:

rearrange and….

1. Centripetal Force2. Newton’s Law of

Gravitation 3. Gravitational Field

Strength

Topic 6 “The Killers”

Gravitational Field Strength

• A test mass is needed to define the strength of the gravitational field surrounding a large mass

• The test mass must have negligible gravitational effect i.e. very small mass

Gravitational field strength is the force per unit mass experienced by a small test mass:

Gravitational Field Strength

Gravitational field strength is the force per unit mass experienced by a small test mass:

Gravitational Field Lines

• Gravitational field strength is a vector • Gravity is always attractive • Find lines point towards the centre of the massive

object • Field lines are perpendicular to the surface • Lines are closer together near the surface - g is greater

near the surface. • Line are further apart as distance increases from the

surface - g decreases with distance.

“Special Relationship”Outside the spherical mass: • g is inversely proportional to the r2 (distance from centre of mass) Inside the spherical mass: • g is directly proportional to r from the centre of the mass

(assuming uniform density).

“Two Bodies”The syllabus says the students should be able to

determine the resultant gravitational field strength due to two bodies along a straight line.

Question

3m

3m 6m

Calculate the resultant gravitational field at P.