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.