Topic 6: Circular motion
and gravitation
See the guide for this topic.
6.1 – Circular motion
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Period, frequency, angular displacement and
angular velocity
Uniform circular motion refers to circular motion at constant speed.
In a uniform circular motion, speed is constant while (angular) velocity and (angular)
acceleration are constantly changing.
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While the magnitude of its velocity remains constant, the direction of its velocity is
constantly changing.
The acceleration causing this change in velocity is always directed towards the
center of the circular path.
The period is the time taken for the object to complete one full circle and is usually
calculated in seconds. The frequency can be calculated by 1/period and is usually
measured in Hz.
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Centripetal force
Centripetal force is the corresponding force (resultant force) which causes the
centripetal acceleration.
Properties:
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Direction: Pointing towards the center of the circle / perpendicular to the
instantaneous velocity
Magnitude:
Work done by centripetal force = 0
Centripetal force is not a type of force; rather, it is just the name we give to the net
force causing a circular motion. For example:
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Centripetal acceleration
The acceleration which gives rise to a circular motion is called the centripetal
acceleration. Its magnitude is given by
It is directed towards the center of the circular motion and is perpendicular to the
instantaneous velocity of the object.
6.2 – Newton’s law of gravitation
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Newton’s law of gravitation
The gravitational force between two objects can be calculated using Newton’s universal
law of gravitation
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Gravitational field strength
The gravitational field strength at a point is the force per unit mass experienced by a
test mass at that point.
The gravitational field strength (g) due to an object is given by
Gravitational field strength at the surface of a planet
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The gravitational field strength at the surface of a planet can be calculated by using
the equation for gravitational field strength and substituting M and r by the mass
and the radius of the planet respectively.
If we calculate the gravitational field strength at the surface of the Each using the
mass and the radius of the Earth, we would obtain the value 9.81m/s^2, which is
equal to the acceleration due to gravity on the surface of the Earth.
Different planets have different radii and masses. Consequently, different planets
have different gravitational field strengths.
Characteristics of Circular Motion
For an object in circular motion we define:
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The period, T, as the time taken to complete one revolution (seconds,
s)
The frequency, f, as the number of revolutions per second (hertz, Hz)
Remember1Hz=1s−11Hz=1s−1, or one revolution per second.
These two variables are related through f=1/Tf=1/T.
Note that we often encounter different units when discussing the circular
motion. For example, think of the rpm, (revolutions per minute) of engines,
and the periods of planets in the Solar System expressed in years.
Angular velocity vs linear speed
Angular velocity: Change in angle per second. (Rad s-1) Symbol = ωω
Linear speed: Change in distance of a point on the circumference per
second. (ms-1) Symbol = V
An object spinning can have a linear speed based on the distance a point
on the circumference or an angular speed based on the angle.
Since the circumference is given by l=θ×rl=θ×r The linear velocity
is V=ωrV=ωr
What are the characteristics of circular motion?
Uniform circular motion: Object with circular trajectory and constant
angular velocity.
Circular motion is the type of motion that is undergone when a body curves
around some 'point'. What's important here is that the distance between
the body and the 'point' remains constant during the curve. It is very
common in our world. For example, when a car goes around a corner, it
is undergoing circular motion. Or when Earth orbits the sun, or when the
moon orbits the Earth. These are all examples of circular motion. (They are
actually elliptical orbits in real life but ignore this for this topic)
Centripetal force
When things move in circular motion they are changing direction
constantly. From Newton's first law we know that if an object changes
direction there is an external force present, therefore we know that there
is force is being applied which causes the circular motion. The name
of this force is a centripetal force. Centripetal force always points
towards the center which is intuitive if you think about how you would
keep an object moving in a circle.
F = ma, therefore, F = mv^2 / r or and acceleration formula times the mass
Towards the center of the circle
There are many different types of force: drag, friction, gravitational, normal,
tension, magnetic, electric, and other pushes and pulls between objects.
Any of these forces can be centripetal.
All that 'centripetal' means is that the force is acting towards the center of
the circle: it is a label for the particular force involved.
It is not yet another type of force to add to the list above. Hence, the
centripetal force does not appear on a free body diagram, which only
includes the physical forces acting on the object.
Centripetal acceleration
Centripetal acceleration: The acceleration the centripetal force causes
towards the center.
The formula for centripetal acceleration: a=v2ra=v2r
Always towards the center of the circle
There is no intuitive proof for this it is just obvious from experimental data.
This relation means that the centripetal force is connected to your speed
and your radius. You may not realize this but you will have seen this in reallife situations before. For example, if you turn a corner at high speeds a
sharp corner (a small radius) is much harder than a wide turn (large
radius) this is because the wider the turn the smaller the centripetal
acceleration.
If you are given the period instead of the velocity, you can use a different
equation (Both equations given in the formula book): a=4π2rT2a=4π2rT2
With so many formulae linking various variables, uniform circular motion is
a popular area in which examiners check your proportional reasoning in
both Paper 1 and Paper 2. Use the IB Physics data booklet and pay close
attention when rearranging and substituting.
Banking
Banking is when the slope an object moving