### Topic 4 Circular Motion

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Topic 4
Circular Motion
Contents
4.1 Introduction and Angular Displacement . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Angular Displacement and Radians . . . . . . . . . . . . . . . . . . . .
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4.2 Angular Velocity, Acceleration and Periodic Time . . . . . . . . . . . . . . . . .
4.3 Angular Motion and Kinematic Relations . . . . . . . . . . . . . . . . . . . . . .
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4.4 Angular Velocity and Tangential Speed . . . . . . . . . . . . . . . . . . . . . . .
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4.5 Tutorial Topic 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Prerequisite knowledge
• Understanding of vectors and scalars.
• Ability to apply kinematic relationships to linear motion.
• Some background in elementary Physics.
• Familiarity with SI system of units.
Learning Objectives
By the end of this topic, you should be able to:
• Convert angles from degrees into radians.
• Explain angular displacement.
• Define angular displacement, velocity and acceleration.
• Derive angular kinematic relationships.
• Solve problems for angular displacement, velocity and acceleration.
• Define the relationship between angular velocity and tangential speed.
• Solve increasingly complex problems associated with angular motion.
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TOPIC 4. CIRCULAR MOTION
4.1
Introduction and Angular Displacement
The material presented in this section is quite similar to the material developed in Topic
2 where kinematic relations were derived for problems involving linear motion; similar
relations will now be derived for circular or angular motion.
In Topic 5 Newton’s Laws will then be applied to angular motion. However, before
starting it is instructive to check which symbols and units will be used for angular motion
and how these correspond to linear motion:
Quantity
Linear
Kinematic
Symbols
Displacement
s
Velocity
v or u
a
Acceleration
Linear
Kinematic
Units
m
s-1
m
m s-2
Angular
Kinematic
Symbols
Angular
Kinematic
Units
θ
ω or ω o
α
Angular Displacement θ is the angle subtended by a segment of a circle’s
circumference at its centre. However, this angle is not expressed in degrees but in
"radians"; see later for a full definition and conversion factors.
Angular Velocity is defined as rate of change of angular displacement with respect to
time. The final angular velocity in equations is denoted by ω, the initial angular velocity
by ω o and in SI units they are measured in (rad s-1 ).
Angular Acceleration is defined as rate of change of angular velocity with respect to
time, in equations this is denoted by α and in SI units it is measured in (rad s-2 ).
4.1.1
The radian is a derived SI unit - the radian is in fact a distance divided by a distance, so
that the radian is just a number with no units at all - it is said to be "dimensionless":
• Velocity and acceleration are often written as (rad s-1 and rad s-2 ).
• However, when "cancelling" units these must be written as (s-1 and s-2 )
respectively. If this step is ignored then "rad", which is after all dimensionless,
will remain uncancelled in the final check for dimensional consistency.
TOPIC 4. CIRCULAR MOTION
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Take a circle of radius r. The angle θ in radians is defined as that angle which subtends
an arc of length s at the circumference of the circle divided by the radius.
- “S” is the length of the arc subtended by
angle “θ” while “r” is the radius:
S
- The angle “θ” in radians is then defined as:
θ
r
…………………...…..(4.1)
In order to find out how many radians are needed to subtend the entire circumference
of a circle, simply apply the definition in the following way:
θ=
s
r
But for 360◦ the circumference is
s = 2πr
Substitute this value into (4.1) to get
θ=
2π r
r
Thus 2π radians is equal to 360◦ which leads to the following conversion:
θ in degrees
× 2π .................(4.2)
360
Whenever radians need to be converted into degrees, (4.2) may be rearranged to give
θ in degrees =
× 360.................(4.3)
2π
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TOPIC 4. CIRCULAR MOTION
4.2
Angular Velocity, Acceleration and Periodic Time
The angular velocity ω is the rate of angular displacement per unit time - diagram (a) is
t = 0; diagram (b) is some later time:
θ
(a)
t=0
(b)
Some Later Time
The sign convention is that angular velocity and angular displacement are positive
anticlockwise and negative clockwise. The two definitions of instantaneous linear and
instantaneous angular velocity are given side-by-side below:
Quantity
Velocity
Linear
Definition
ds
v=
dt
Linear
Units
(m s-1 )
Angular
Definition
dθ
ω=
dt
Angular
Units
If the acceleration is constant then the velocity-time graph will plot as a straight line
and the value of the average angular velocity will equal its instantaneous value. The
derivations that follow are based on this assumption.
The angular velocity ω may also be expressed in terms of the periodic time T using the
following relation:
T =
2π
.................(4.4)
ω
Where
ω = angular velocity (rad s-1 )
T = periodic time (s)
In the same way that linear acceleration is defined as rate of change of linear velocity
per unit time, so angular acceleration is defined as rate of change of angular velocity per
unit time - see table that follows:
TOPIC 4. CIRCULAR MOTION
Quantity
Acceleration
Linear
Definition
dv
a=
dt
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Linear
Units
(m s-2 )
Angular
Definition
dω
α=
dt
Angular
Units
The above expression is for instantaneous angular acceleration - if acceleration is
constant then, as said before, the velocity-time graph will plot as a straight line, so that
the average and instantaneous angular acceleration will be the same.
The angular acceleration may be positive or negative, irrespective of whether motion is
in a clockwise or an anticlockwise direction - the sign really depends on whether ω is
increasing or decreasing in the specified direction of rotation.
Example : 4.2.1
Problem:
Convert 45◦ , 90◦ , 180◦ and 360◦ into radians
Solution:
Equation (4.2) provides the necessary conversion
θ in degrees
45
π
× 2π =
360
360
4
90
π
360
2
180
360
360
360
..........................................
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TOPIC 4. CIRCULAR MOTION
Example : 4.2.2
Problem:
Calculate the number of degrees in 1 rad
Solution:
Equation (4.3) provides the necessary conversion:
θ in degrees =
1
× 360 =
× 360 = 57.3o
2π
2π
..........................................
Example : 4.2.3
Problem:
The crankshaft of a car engine revolves clockwise at 5,000 r.p.m (revolutions per
minute) calculate the periodic time and the angular velocity (hint - each revolution of
the crankshaft sweeps out one complete cycle or 2π radians of angular displacement).
Solution:
First calculate the periodic time by taking the reciprocal of revolutions per minute (r.p.m).
This gives (min revolution-1 ), or just (min) because "revolutions" are dimensionless.
Then convert minutes into seconds. The result is the periodic time (s) as follows:
T =
1
(min)
5000
60 s
1 min
= 0.012 s
Equation (4.4) can then be used to convert periodic time into angular velocity - if rotation
is clockwise the sign for angular velocity is taken to be negative:
ω=
2π
2π
=−
T
0.012
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TOPIC 4. CIRCULAR MOTION
Example : 4.2.4
Problem:
Repeat the above for a device moving anticlockwise at 5000 r.p.m.
Solution:
1
(min)
T =
5000
60 s
1 min
= 0.012 s
and,
ω=
2π
2π
=+
T
0.012
The angular velocity sign is now positive due to anticlockwise direction of rotation.
..........................................
Standard Physics textbooks provide a wide variety of worked examples based on solving
practical problems. [Cutnell and Johnson]
These problems are often framed around common day-to-day experience which helps
improve understanding and increase interest.
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TOPIC 4. CIRCULAR MOTION
4.3
Angular Motion and Kinematic Relations
The exact definition of angular acceleration α is the rate of change of angular velocity ω
- thus, instantaneous angular acceleration is given by
α=
dω
.................(A)
dt
However, if angular acceleration α is constant then the velocity-time diagram will have a
constant slope and equation (A) reduces to equation (4.4) as follows:
α =
Change in Angular Velocity
Elapsed Time
If ω o is initial angular velocity and ω is final angular velocity and, if the time taken for the
velocity to change is t (s), then the angular acceleration α (rad s-2 ) may be written as
α=
ω − ωo
t
∴ ω = ωo + α t .................(4.5)
Average angular velocity ω̄ is the angular displacement of a body θ divided by the
elapsed time t and is given by
Average Ang Velocity =
Ang Disp
Elapsed Time
This may be expressed mathematically as follows:
∴ ω̄ =
θ
t
Or, in terms of final angular velocity ω and initial angular velocity ω o as
ω̄ =
ω + ωo
2
Eliminating the average velocity ω̄ between last two equations leads to
TOPIC 4. CIRCULAR MOTION
θ=
ω + ωo
2
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t
Substitute equation (4.5) into the above gives
θ=
ωo + αt + ωo
2
t
1
∴ θ = ωo t + α t2 .................(4.6)
2
Squaring both sides of equation (4.5) yields
ω 2 = ωo 2 + 2ωo α t + α2 t2
∴
ω2
= ωo
2
1
+ 2α ωo t + α t2
2
Now substitute equation (4.6) into the above leads to the final result
ω 2 = ωo 2 + 2αθ .................(4.7)
Equations (4.5), (4.6) and (4.7) are the three angular kinematic relationships - all very
similar to the linear kinematic relationships derived before.
The linear and angular kinematic relations are summarised in tabular form below:
Linear Kinematic Relations
s
v̄ =
t
v = u + at
1
s = ut + at2
2
v 2 = u2 + 2as
Angular Kinematic Relations
θ
t
ω = ωo + αt
1
θ = ω o t + α t2
2
ω 2 = ωo 2 + 2αθ
ω̄ =
If the linear relations are memorised, then the angular relations can be written down by
inspection; just substitute ω for v, ω o for u, θ for s and α for a.
The basic assumption underlying the entire derivation is that angular acceleration is
constant - this leads to the angular velocity versus time plot being a straight line with
constant slope. The instantaneous velocity can then be replaced with a simple arithmetic
average velocity.
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TOPIC 4. CIRCULAR MOTION
Example : 4.3
Problem:
The crankshaft of a car engine is idling at 1,000 r.p.m (clockwise when viewed from the
front), if it takes 1.5 s for the engine to stop rotating after being switched off calculate the
periodic time, the initial angular velocity and the angular acceleration.
Solution:
The periodic time is the time taken for a single revolution of the crankshaft, but the
crankshaft is revolving at 1,000 r.p.m. Remember revolutions are dimensionless.
1
(min)
T =
1000
60 s
1 min
= 0.06 s
The angular velocity is linked to the periodic time through equation (4.4) - remember the
sign convention calls for a negative value if rotation is clockwise, so that
ω=
2π
2π
=−
T
0.06
∴ ω = −104.72 rad s−1
Now re-arrange equation (4.5) to find the acceleration of the crankshaft, remember the
final angular velocity must be ω = 0 rad s-1 .
α=
ω − ωo
0 − (−104.72)
=
t
1.5
It is interesting to try and make sense of this - the above result means that each second
the angular velocity changes by +69.81 rad s-1 .
• The initial angular velocity was -104.72 rad s-1 .
• However, after 1.5 seconds the angular velocity of the crankshaft must change by
+69.81 x 1.5 = +104.72 rad s-1 .
• Thus, the final angular velocity must be zero.
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TOPIC 4. CIRCULAR MOTION
4.4
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Angular Velocity and Tangential Speed
The next question is how the angular velocity ω relates to the tangential speed. The
tangential speed v (m s-1 ) is the speed with which the object is moving around in a circle:
v (m s-1)
r
S
- The angular velocity (rad s-1) is the
rate with which the angle at the
centre of a circle is being swept out.
θ
- The tangen"al speed is the linear
velocity (m s-1) that an object has at
any distance from the centre.
If an object is moving around in a circle and the cable were to break then the object
would fly off in a straight line, tangent to the circle, at the tangential speed.
To derive the relation between ω and v start with definition or θ the angular displacement
as follows:
θ=
s
r
Now divide both sides by t, the time take to sweep out an angle θ, to get
s
θ
=
.................(A)
t
rt
But the average angular velocity (rad s-1 ) is given by
ω=
θ
.................(B)
t
And, average tangential velocity (m s-1 ) is
v=
s
.................(C)
t
Substituting (B) and (C) into (A) leads to
ω=
v
r
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TOPIC 4. CIRCULAR MOTION
Re-arrange this expression to get the relationship between the angular velocity ω and
the tangential speed v as follows:
v = rω .................(4.8)
In a similar way the relationship between the angular acceleration α and the tangential
acceleration a is given below:
a = rα .................(4.9)
Example : 4.4
Problem:
A stone shot is loaded into a sling and the sling is rotated anticlockwise around a
person’s head at a rate of 3 revolutions per second. If the sling is 1.5 m in length
calculate the following:
a) The angular velocity of the stone (rad s-1 ).
b) The tangential speed with which the stone leaves the sling (m s-1 ) and (km/hr).
c) Repeat for a sling of 2.5 m in length.
Solution:
a) The average angular velocity, in an anticlockwise direction, is simply
ω=
θ
3 × 2π
=
t
1
b) The tangential speed is given by equation (4.8) as follows:
v = rω = 1.5 × 18.85 = 28.28 m s−1 (101.8 km/hr)
c) Repeat for longer sling - notice, ω is the same but not v, since sling is longer
ω=
θ
3 × 2π
=
t
1
v = rω = 2.5 × 18.85 = 47.13 m s−1 (169.7 km/hr)
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TOPIC 4. CIRCULAR MOTION
4.5
Tutorial Topic 4
1. As an exercise in degree-radian conversion determine the following:
c) Convert π/6 radians into degrees.
d) Convert 1.5 π radians into degrees.
2. A car engine is turning at 3,500 r.p.m. in a clockwise direction, when viewed from
the front, determine the following:
a) Calculate the periodic time (s).
b) Calculate the angular velocity (rad s-1 ).
c) Repeat the above for a device operating anticlockwise.
3. A car engine is turning at 2,000 r.p.m. in a clockwise direction. If the engine’s
rotational speed is increased linearly (that is in a uniform manner) to 4,000 r.p.m,
over a 2 second period of time, determine the following:
a) Calculate the angular acceleration (rad s-2 ).
b) Repeat the calculation but now decreasing from 4,000 r.p.m to 2,000 r.p.m.
again over a 2 second period.
4. An agitator paddle is rotating clockwise at a rate of one complete revolution per
second. If the rotational speed were to be uniformly increased to 1.5 revolutions
per second, over a five second period, determine the following:
a) The initial angular velocity (rad s-1 ).
b) The final angular velocity (rad s-1 ).
c) The angular acceleration (rad s-2 ).
5. An agitator paddle, 2 m in diameter, is rotating clockwise at a rate of two complete
revolutions per second. If the rotational speed were to be uniformly decreased to
0.5 revolutions per second, over a five second period, determine the following:
a) The initial angular velocity (rad s-1 ).
b) The final angular velocity (rad s-1 ).
c) The angular acceleration (rad s-2 ).
d) The initial and final tangential speed at the agitator tip (m s-1 ).
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TOPIC 4. CIRCULAR MOTION
6. A car is initially at rest waiting for traffic lights to change. Once the green
light comes on is accelerates smoothly at a constant rate of 0.5 m s-1 (linear
acceleration) over a 15 second period of time. If the radius of the wheel and
tyre is 0.35 m and there is no slippage between the tyre and the road determine
the following, over the entire 15 seconds:
a) Calculate the angular acceleration of the wheel (rad s-2 ).
b) The total angular displacement of the wheel.
c) The final angular velocity of the wheel.
d) The tangential speed at the tyre.
e) The linear velocity of the car.
7. The angular velocity of a centrifuge rotor increases from 100 rad s-1 to 500 rad s-1
in 10 seconds, given this information determine the following:
a) Calculate the total angle through which the rotor turns during this period of
time.
b) Calculate the angular acceleration.
8. Consider an edible circular pie, now cut slices from the pie such that the arc length
of each slice is equal to the radius. After all the pieces have been eaten calculate
the apex angle (rad) of the remaining slice.
9. A ship’s propeller starts from rest and accelerates at 2 x 10-3 rad s-2 for 2,000
seconds. For the next 30,000 seconds it rotates at a constant angular velocity.
Finally it decelerates at 1 x 10-3 rad s-2 until it slows down to an angular velocity of
1 rad s-1 . Calculate the total angular displacement of the propeller over this period
of time.
4.6
Bibliography
1. Cutnell, John D. and Johnson, Kenneth W. 2012. Introduction to Physics. 9th ed.
Singapore; Wiley.
GLOSSARY
Glossary
Angular Acceleration
Angular acceleration (rad s-2 ) is the rate of change of angular velocity. The angular
acceleration may be positive or negative depending on direction of rotation and
whether angular velocity is increasing or decreasing.
Angular Displacement
In the case of rotary (angular) motion, angular displacement (rad) is the angle,
measured in radians, swept out by an object as it circles around some point this point being the centre of the circle. The rotation may either be clockwise or
anticlockwise, so that angular displacement will have a sign.
Angular Kinematic Relationships
These expressions provide all the relationships between angular displacement
"θ", angular velocity (initial "ω o " and final "ω"), angular acceleration "α" and time
"t" needed to solve angular motion problems - the assumption underlying their
derivation is one of constant angular acceleration, which results in a linear angular
velocity time relationship. Thus, simple arithmetic average angular velocities may
be used. They are the angular analogues of the linear kinematic relations
Angular Velocity
Angular velocity (rad s-1 ) is the rate of change of angular displacement. The
angular velocity may be positive or negative depending on direction of rotation.
Periodic Time
Periodic time (s) is the time taken to sweep out one complete revolution of a circle.
The periodic time has no sign associated with its value.
Angles at the centre of a circle may be expressed either in degrees or radians.
Any angle at the centre of a circle, radius "r", will subtend an arc of length "s" at
its circumference. The angle in radians is the ratio of "s" divided by "r". Radians
in SI units are written as (rad). It is a dimensionless quantity being a distance
divided by a distance - when cancelling units it may be struck out, because it has
no dimensions.
Sign Convention
Angular displacement and velocity are positive in an anticlockwise direction and
negative in a clockwise direction. For angular acceleration the sign may be positive
or negative in either direction depending on whether angular velocity is increasing
or decreasing.
Tangential Speed
Tangential speed is the linear speed (m s-1 ) that an object has at any distance from
the centre of the circle around which it is circling. At constant angular velocity the
tangential speed is directly proportional to the distance - that is, the distance from
the point of interest to the centre of the circle.