#### Transcript Circular Motion

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The circumference of a circle is 2πr. Since a radian is based
on an arc with length r, there are 2π radians in a full circle.
degrees
360
2π
180
π
90
π/2
r
r
2π
To convert from degrees into radians, multiply by
.
360
360
To convert from radians into degrees, multiply by
.
2π
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Degrees and radians in a circle
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Angular displacement and velocity
Angular displacement and angular velocity, like linear
displacement and linear velocity, are vector quantities, i.e.
quantities with a direction.
Displacement, x, is a distance with a direction. Linear
velocity, v, is a speed with direction. It is the rate of change of
displacement with time.
Δx
v=
Δt
Angular displacement, Δθ, is the angle moved through
relative to a specific axis.
Angular velocity, ω, is the rate of change of angular
–1.
The
units
of
ω
are
s
displacement with time.
Δθ
ω=
Δt
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Period of circular motion
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An equation for v
The distance covered by an object moving in a circle in
one period is the circumference of the circle, 2πr.
v
distance
speed =
time
therefore
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r
v=
2πr
T
An equation for ω
The angle an object moving in a circle moves through in
2π
Δθ
ω=
Δt
therefore
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2π
ω=
T
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Frequency
Frequency, f, is the number of rotations per second.
i.e.
Because
number of rotations
frequency =
time taken
time taken
period =
number of rotations
1
T=
f
and
1
f=
T
2π
Using ω =
, this gives ω = 2πf
T
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Matching equations
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Can you calculate v and ω?
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Centripetal acceleration
Any object moving in a circle has a changing velocity, even if
it is moving at a constant speed, because its direction is
changing. It is always moving along a tangent to the circle.
v
v
v
time
Acceleration is the rate of change of velocity.
Δv
a=
Δt
Therefore any object moving in a circle is
accelerating. This is called the centripetal acceleration.
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Deriving equations for a
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Centripetal force
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F = ma and direction of acceleration
According to Newton’s second law, force = mass × acceleration
(F = ma).
v
Acceleration and
force therefore act in
the same direction as
one another.
a
It can therefore now
be seen that the
direction of centripetal
acceleration, a, is
towards the centre of
the circle. This is perpendicular to the direction of the velocity, v.
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Deriving equations for F
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Matching centripetal forces
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Centripetal force and acceleration
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Effect of m, v and r on F
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