Unit 4: Waves, Fields and
Nuclear Energy
UNIFORM CIRCULAR MOTION
Angular Measure
The S.I. unit of angle is the radian (rad).
r
s
θ
s = θ in radians
_
r
r
Definition: one radian is the angle θ, above, when the arc length s is equal to the radius r.
2
360
1
57. 3
180
or
1
90
0.017
Angular frequency is the number of revolutions per second (or sometimes revs per minute, r.p.m.)
The angular velocity or angular speed (ω) is defined as the angle turned through per second, (in radians per
second, rad s‐1).
An object moving in a circular path still has a linear speed (v). It is a measure of the distance moved around
the circle per second, (in metres per second, m s‐1).
Relationship between v and ω
The diagram above shows that
and since
it follows that
or
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BARNARD CASTLE SCHOOL DEPARTMENT OF PHYSICS ‐ ADVANCED LEVEL
Unit 4: Waves, Fields and
Nuclear Energy
UNIFORM CIRCULAR MOTION
Centripetal Acceleration
An object travelling in a circular path at a constant speed is still accelerating. This is because its direction is
changing all the time. Acceleration is the rate of change of velocity and velocity is a vector combining speed
and direction. If either of these changes, the object is accelerating.
The direction of this acceleration is towards the centre of the circle in which the object moves. It is a centre‐
seeking or centripetal acceleration.
v2
∆θ
v2
v1
r
∆θ
v1
s
v2 - v1 = ∆v
Consider a small time ∆t in which an object travelling in a circular path moves through a small angle ∆θ.
v1 and v2 are the velocity vectors of the object at the beginning and end of the short time interval. The
magnitudes of v1 and v2 are equal. We shall call it v.
Since ∆θ is so small,
Δ
Δ
(This follows from s/r = θ, since v2 – v1 is small enough to be considered as an arc).
Centripetal acceleration
But we have seen that
So,
or
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BARNARD CASTLE SCHOOL DEPARTMENT OF PHYSICS ‐ ADVANCED LEVEL
Unit 4: Waves, Fields and
Nuclear Energy
UNIFORM CIRCULAR MOTION
Centripetal Force
Using Newton’s Second Law of motion (force = mass x acceleration) it is easy to write expressions for the
force on an object moving in a circular path:
or
This force, which acts towards the centre of the circle, is known as a centripetal force. This is not a separate
type of force. Something must provide a centripetal force, for example, if you were to swing a conker around
on a string, the centripetal force is provided by the tension force in the string.
(We shall not use the term “centrifugal force”. This is simply the force set up as a reaction to the centripetal
force, according to Newton’s Third Law. When you exert a centripetal force on the conker above, the conker
pulls back on your hand with an equal and opposite centrifugal force).
Examples of centripetal forces
(i)
A cyclist leaning as he rounds a bend
The frictional force between the tyres and the road is not sufficient to provide the centripetal force the
cyclist needs to turn sharp corners. He leans into the bend, using a component of his weight to provide a
greater centripetal force.
The component W cosθ provides the force between the wheel and the road
The component W sinθ provides the centripetal force and is directed towards the centre of the
circular path in which he will travel.
To travel in a tighter circle, he leans further over, increasing θ and therefore increasing the component
W sinθ. The drawback is that this decreases the component W cosθ and eventually the tyres may slip and
he will fall over.
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BARNARD CASTLE SCHOOL DEPARTMENT OF PHYSICS ‐ ADVANCED LEVEL
Unit 4: Waves, Fields and
Nuclear Energy
(ii)
UNIFORM CIRCULAR MOTION
An aircraft banking to turn
L cosθ
Lift L
L sinθ
The lift force L acts vertically upwards underneath the wings. When the aircraft banks, part of that lift is
diverted to provide the centripetal force needed to turn.
The component L cosθ balances the weight of the aircraft which acts vertically down
The component L sinθ provides the centripetal force and points towards the centre of the circular
path in which the aircraft travels.
The more the aircraft banks, the greater the centripetal force becomes but the smaller the component
L cosθ becomes and the aircraft will lose height. To avoid this, the engine thrust is increased to provide a
greater forward force and a greater lift.
The Gravity Force
The centripetal force needed to keep the Moon orbiting the Earth, the Earth moving around the Sun, the Sun
orbiting the centre of the Milky Way and all artificial satellites circling the Earth, is provided by gravity. The
centripetal force formula F = mv2/r shows that faster moving objects will orbit at greater heights. If a
satellite is slowed down, it will move down to a lower orbit.
Satellite and planetary orbits are discussed in the next section on gravitation.
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BARNARD CASTLE SCHOOL DEPARTMENT OF PHYSICS ‐ ADVANCED LEVEL