Angular Motion Engineering Science N4
Chapter 4 Angular Motion
4.1 Introduction
Angular
displacement unlike linear displacement is displacement that is defined by number
of rotations or number of revolutions per unit second about a fixed point or
center (shaft).
The
diagrams below illustrate the fact that the radius or diameter of the rotating
component is considered when there is a conversion of angular velocity into
linear velocity.
4.2 Formulas and Terminology
Angular
Velocity: This is the angular
displacement travelled per unit second. It has a symbol, w1.The unit
for this is radians per second (rad/s). Constant
Angular Velocity: The intial and final
angular velocity in radian per second or number of revolutions per second are
equal,(w1 =w2 ). Therefore, there is no angular
acceleration ( α=(w2 -w1) ÷ t=0 rad/s2).
Angular
Acceleration: This is the change in angular velocity per unit second. The
symbol is w2 Angular acceleration has a unit of radian per (second)2(rad/s2).
Radian: π radian = 180 degrees (π = 3.142 radian). The
circumference of a circle is πd (the product of π and its diameter(C=
πd)). The
radian is the ratio of the circumference of a circle and the diameter of that
same circle.
Angular Displacement: The symbol for angular displacement is Ɵ. This is the displacement of the rotating part or component. This displacement is usually given in most problems in terms of number of revolutions per second or number of revolution per minute. The relationship between the number of revolution per second or the number of revolution per minute and the angular displacement is as shown below. Number of Revolutions = Number of Revolutions/second X Time (seconds) Number of Revolutions = Number of Revolutions/minute X Time (Minutes)X 60 seconds Work Done: This the work done in Joules in terms of the number of revolutions per second or number of radians per second.
Work Done = 2πNTt
t= time taken (seconds)
Work done= TƟ
since Ɵ= angular displacement in radians
and Ɵ= 2πNt
Power: This the work done in Joules per unit second when a wheel or shaft has an angular velocity of N of revolutions per second.
Power,
P = 2πNT
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Table 4.1 Table of Angular
Motion Formulas |
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Linear Motion |
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Angular Motion |
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1. |
v=u+at |
1. |
ω 2= ω 1+αt |
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2. |
2as=v2-u2 |
2. |
2αƟ= ω 22 – ω 2 1 |
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3. |
s= ut+0.5at2 |
3. |
Ɵ= ω 1t + 0.5αt2 |
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4. |
vave= 0.5(v+u) |
4. |
ω ave=0.5(ω 2+ ω 1) |
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5. |
S= tvave |
5. |
Ɵ=t ω ave |
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6. |
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6. |
P= 2πNT |
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7. |
ω=
2πN
(Radians) |
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8. |
P= T Ɵ |
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9. |
P= T t ω ave |
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10. |
T= FR |
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v=final velocity |
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ω 2=final angular |
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velocity |
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u=initial velocity |
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ω 1=initial angular |
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velocity |
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vave= average |
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ω ave= average |
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velocity |
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angular velocity |
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a= acceleration |
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a= angular |
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acceleration |
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t= time taken |
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t= time taken |
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S= linear |
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s= angular |
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displacement |
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displacement |
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T= Torque |
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ω=
2πN |
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Average angular velocity: During acceleration or deceleration, the number of revolutions increases or decreases respectively. It is, thus important to calculate the average velocity.
Constant Angular Velocity: Uniform
or constant velocity is velocity that does not increase or decrease. There is,
therefore, no acceleration nor deceleration.
4.3 Linear Velocity and Angular Velocity.
Angular Velocity: The formula that states the relationship between the linear velocity of a wheel and its corresponding angular velocity is as stated below.
v=rω or
v=ωr
ω= radians per second
1 revolution per second =2πr
2 revolutions per second =2πr
2 revolutions per second =πd
3 revolutions per second =3πd
4 revolutions per second =4πd
5 revolutions per second =5πd
n revolutions per second =πdN
Therefore, v=πdN
2X Radius = Diameter
N= number of revolutions per second
See the diagrams below.
Fig.
4.1 Formula, v=πdN
4.4 Converting Number of Revolutions per seconds into Radians per seconds.
Angular Velocity: 3600= 1 revolution
1 revolution/s =2π radian/s
2 revolutions/s=2X2π radian/s
3 revolutions/s=3X2π radians/s
4 revolutions/s=4X2π radians/s
5 revolutions/s=5X2π radians/s
10 revolutions/s=10X2π radians/s
100 revolutions/s=100X2π radians/s
Therefore, n revolutions/s = 2πn radians/s
The
greater the diameter of a wheel the greater the circumferential velocity. The
circumferential velocity is a term used
for a point on a wheel or rotating component on a machine converted into linear
velocity.
The
concept of circumferential velocity may be clearly understood from the point of
perspective of Fig*****. This diagram illustrates the conversion of the number
of revolution of the tyre into linear motion in m/s.
Fig.
4.2 Wheel of diameter 0.35m rotating clockwise at 85 revolutions per second
Fig.
4.3 Wheel of diameter 0.7m rotating clockwise at 85 revolutions per second.
Fig.
4.4 Wheel of diameter 2.1m rotating clockwise at 85 revolutions per second.
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Table 4.2 |
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Three wheels of different
diameters, 0.35m, 0.7m and 2.1m rotating at 85rev/minute (Fig. 4.2
-4.4) |
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Angular Velocity |
Circumferential |
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Velocity, v= πdn |
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1. |
85 rev/ minute |
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2. |
85 rev/ minute |
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3. |
85 rev/ minute |
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Angular Velocity |
Angular Velocity |
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Rev/second |
Radian/second |
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Radian/s=2πXN rev/s |
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1. |
85 rev/ minute |
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2. |
85 rev/ minute |
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3. |
85 rev/ minute |
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4.5 Linear acceleration and angular
acceleration.
The
formula a=rα is similar to the formula v=ωr. The first formula is the
product of radius and angular acceleration. The second formula is the product
of radius and angular velocity.
4.6 Average angular velocity.
The
formula for average angular velocity is as stated from in the table able.
Assuming that a wheel has a constant angular velocity and the initial and final
angular velocity are 1000 revolutions per minute and 1000 revolutions per
minute, the average angular velocity will be 1000 revolutions per minute.
This
is not the case for an acceleration of the wheel from 500 revolutions per
minute to 850 revolutions per minute. The average angular velocity is
calculated thus: 1/2(500+800)= 650 revolutions per minute.
4.7 Torque.
The formulas for Torque
are as stated in the table above in terms of Power, radius of the wheel and
work done. The work done by a rotating wheel or body about a shaft is the
product of the number of revolutions per minute or second and the time duration
in seconds.
The
power developed by the wheel is actually work done by the wheel or rotating
body in Joules per unit second (Joules/second).
4.8 Circumferential Velocity and Power developed.
The table below shows the relationship between Torque, Work
done and Power. The development of power in motor vehicles is based on the
rotation of a shaft such that the number of revolutions per minute or number of
revolutions per seconds are converted into linear motion, v= πdn.
The
greater the number of revolutions per unit second the greater the power of the
sports car, bakkie or conventional car. This is the most basic reason why
people prefer to buy a German car.
The unit of Power
is the Joule per unit second. Conversion of this Power into Work done is based
on multiplying with the time duration in seconds for which that shaft or tyre
rotates while accelerating from initial angular velocity to final angular velocity.
The
reverse is the case when converting Work Done into Power. Divide the Work Done
in Joules by the time duration in seconds.
Since
the calculation of angular displacement is based on initial, final and average
angular velocities, the calculation of power is also based on a corresponding
instantaneous initial, instantaneous final and average power.
The
calculation of a problem given in terms of a final and initial angular
displacement in radians per second means that the terms 2πN1 and 2πN2 in the Power
formula P= 2πNT are equal to the initial and final values in radian per
second.
Example 4.1
Three angle grinders, A, B and C, of diameters as stated below in Table 4.3 accelerate from an angular velocity of 0 revolutions per minute to 1000 revolutions per minute in 2 minutes. Calculate:
a) the angular acceleration of each one of the grinders.
b) the average angular velocity
c)
the circumferential velocity of grinders A, B and C.
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Table 4.3 |
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Angle Grinder |
diameter |
Circumferential |
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Velocity (v=πdn) |
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1. |
A |
125mm |
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2. |
B |
175mm |
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3. |
C |
180mm |
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d) the angular displacement during 2 minutes.
e) the number of revolutions.
Solution:
a) Final Angular velocity, w2 = 1000 rev/min
Initial Angular Velocity, w1= 0 rev/min
Convert 1000 revolutions per minute into radians per seconds and substitute into the formula below.
1000
revolutions per minute
w2=w1+αt
α= (w2- w1)÷t
α
Alternatively, substitute into the formula by
converting the difference of the angular velocity into radian/ seconds as shown
below.
w2=
α = 0.8727 rad/s2
b) average angular velocity wave=(w2+w1)÷2
wave= 0.5(104.7198 +0) rad/s = 52.3599 rad/s
The concept of average angular velocity may be understood from as follows:
The angular velocity at 50 seconds is greater than the angular velocity at 40 seconds.
The angular velocity at 60 seconds is greater than the angular velocity at 50 seconds.
The angular velocity keeps increasing until it reaches a maximum velocity of 1000 rev/minute.
d) Angular displacement in 2 minutes
Ɵ= 52.3599 rad/s X 2minutes X 60 seconds = 6283.188 rad
Alternatively, the the following formulas may be applied:
i) 2αƟ= w22 –w2 1
Ɵ= [(104.7198rad/s)2 –(0)2] ÷ (2 X 0.8727 rad/s2)
= 6283.rad
Ɵ= (w22 –w2 1)÷ 2α ii) Ɵ= w1t + 0.5αt2
= (0 rad/s X t) + 0.5(0.8727 rad/s2X (2min X 60sec)2
= 6283.188 rad
e) Conversion
of number of revolutions per seconds into radian per second is carried by
multiplying by 2π radian. This is the reverse case when converting from
rad/s to number of revolution per second.
Number of revolutions per second = 1000.0405
revolutions
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Table 4.4 |
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Angular Velocity |
Angular Velocity |
Time taken (sec) |
Angular Acceleration |
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Average Angular Velocity |
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Angular Displacement
(Radian) |
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Initial w1 |
Final w2 |
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a. |
rest |
2π rad/s |
30 sec |
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b. |
rest |
2π rad/s |
15 sec |
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c. |
rest |
2π rad/s |
10 sec |
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d. |
rest |
2 rev/s |
5 sec |
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e. |
60rev/min |
180rev/min |
2 min |
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f. |
60rev/min |
180rev/min |
1 min |
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g. |
rest |
80rev/min |
30 sec |
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h. |
100rev/s |
20rev/s |
1 min |
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i. |
πrad/s |
5 πrad/s |
60 sec |
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j. |
1 rev/s |
4rev/s |
45 sec |
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k. |
rest |
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30 sec |
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50 revolutions
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l. |
rest |
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15 sec |
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20 revolutions |
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m. |
rest |
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10 sec |
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n. |
rest |
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5 sec |
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***** |
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o. |
60rev/min |
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1 min |
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****** |
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p. |
rest |
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30 sec |
7 rev |
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q. |
100rev/s |
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1 min |
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r. |
πrad/s |
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60 sec |
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s. |
1 rev/s |
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45 sec |
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t. |
3π rad/s |
0 rad/s |
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u. |
2π rad/s |
2π rad/s |
30sec |
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v. |
20rev/s |
20rev/s |
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w. |
1000 rev/min |
1000
rev/min |
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x. |
4rev/s |
4rev/s |
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y. |
1 |
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Example 4.2
Assuming that one of the angle grinders in example 1, angle grinder A, accelerates from an angular velocity of 120 revolutions per minute to 1000 revolutions per minute in 2 minutes. Calculate:
a) the angular acceleration of the grinder.
b) the average angular velocity.
d) the angular displacement during 2 minutes.
e) the number of revolutions.
f)
State the angle grinder that accelerates faster. Give reasons for your answer.
Solution:
a) Convert 120 revolutions per minute into radians per seconds and substitute into the formula below.
w2=w1+αt
120
revolutions per minute =
α= (w2- w1)÷t
w2=
α = 0.7679 rad/s2
b) average angular velocity wave= 0.5(w2+w1)
wave=
0.5(12.5664 + 104.7198 ) rad/s = 58.6431
rad/s
d) the angular displacement during 2 minutes.
Ɵ= 58.6431 rad/s X 2minutes X 60 seconds
= 7037.172 rad
Alternatively, the the following formulas may be applied:
i) 2αƟ= w22 –w2 1
Ɵ= [(104.7198rad/s)2 –(12.5664)2] ÷ (2 X 0.7679 rad/s2)
= 7037.584 rad
Ɵ= (w22 –w2 1)÷ 2α
ii) Ɵ= w1t + 0.5αt2
= (12.5664 rad/s X 2 X60) + 0.5(0.7679 rad/s2.X (2X 60sec)2
= 7036.848 rad
e) the number of revolutions
7037.172 rad ÷ 2π = 1120 revolutions
f)
The angle grinder in example A accelerates faster than the one in example 2.
The change in velocity per unit second in example 1 is greater. α = 0.8727 rad/s2
is greater than α =
0.7679 rad/s2
Example 4.3 A wheel of diameter 1m has
an initial angular velocity of 0 rev/minute and final angular velocity of 2π rad/s in 30 seconds.
Calculate:
a) the
average angular velocity in radian per second. b) the angular
velocity in radian/s2.
c) the angular displacement in radian. d) the angular
displacement in revolutions. (Refer to
Table 4.4 a)
Example 4.4
A wheel of diameter 1m accelerates from 60 rev/min to 180 rev/min in 2 minutes. Calculate:
a) the initial and final velocities in radian per second
b) the angular acceleration in radian/s2.
c) the average angular velocity in radian per second.
d) the angular displacement in radian.
e)
the angular displacement in revolutions. (Refer to Table 4.4 )
Example 4.5
A wheel of diameter 2m has an initial angular velocity of 2π rev/s and a final angular velocity of 2π rev/s in 30 seconds. Calculate:
a) the initial and final velocities in radian per second
b) the angular acceleration in radian/s2.
c) the average angular velocity in radian per second.
d) the angular displacement in radian.
e)
the angular displacement in revolutions. (Refer to
Table 4.4 )
Example 4.6
An angle grinder accelerates from rest to a final velocity in 30 seconds. It makes 50 revolutions. Calculate:
a) final angular velocity in radian per second
b) the angular acceleration in radian/s2.
c) the average angular velocity in radian per second.
d) the angular displacement in radian.
(Refer to Table 4.4 )
a) Final Angular velocity, w2 = 1000 rev/min
From the Table ,
Ɵ=twave
wave=0.5(w2+w1)
Ɵ= 50 revolutions
Substituting: 50 revolutions = 30 seconds X wave
wave = 1.667 revolutions/s
Make w2 subject of the formula based on wave=0.5(w2+w1)
w2= 2wave - w1
w2= 2(1.667 rev) – 0
w2= 3.334 revolutions/s
b) w2=w1+αt
α = (w2-w1)/t
Substituting:
α = (3.334-0)/30 seconds
α = 0.11113 radian/s
c) From a above:
wave = 1.667 revolutions/s X 2π
wave = 10.4754 radian/s
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