Alpha School of Academic Propensity
Engineering Science

Angular Motion
Call 0977 127 054 or 0968226870 or 0950868535 Email: - ddk2012@gmail.com Box 71528.Ndola

1.0 Definition: Motion will ever be motion except the point to be established is, what is being covered during that motion, in case of linear motion which we are all familiar with at this time what is being covered is distance (in meters or kilometers) whilst in angular motion what is being covered are angles (in degrees, radians or revolutions) hence the name angular motion. This is motion in which an angle (θ in radians) is swept through a time (t in seconds) see fig 1.

As a particle moves from point P to point Q it covers a distance s and as it sweeps an angle θ, through a time t seconds.

2.0 Units The radian is defined as the angle subtended by an arc whose length is equal to the radius of the circle.

Q

sr

1r
O r P

3. Relationship between the degree and the radian Taking ratios of similar shapes,
Arc length PQ ( s  r ) Angle POQ  Circumference of circle Total angle in acircle r 1 radian i.e.  2r 360o  2 radians  360o  Half revolution is  radians  180o and  90o 2 1 radian  57.2958o  57.3o one right angle is



4.0 Arc length: Let s be the arc length subtending an angle θ radians at the centre of a circle Then arc length s  radians  Circumference Totalangle in a circle

i.e.

s 2r



 radians 2 radians
{s 

 s  r ( in radians)


360 o

 2r if  is in deg rees )

5.0 Relationship between Linear motion and angular motion. Motion will ever be motion except the point to be established is, what is being covered during that motion, in case of linear motion which we are all familiar with at this time what is being covered is distance (in meters or kilometers) whilst in angular motion what is being covered are angles (in degrees, radians or revolutions) hence the name angular motion.

Taking advantage of the equations of motions which we derived last semester, building on those we can simply compare and end up with the following table overleaf.

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By Daka Koloto Aggrey. Well seasoned Tutor, Let God arise and his enemies be scattered.

TYPE OF MOTION
DESCRIPTION
time displacement Initial Velocity Final Velocity LINEAR Symbol (Units) abrev'n t seconds ANGULAR Symbol (Units) abrev'n t seconds

s u v a

m m/s m/s

θ (radians, revs) rads ω 1 (rad/sec) rad/s ω 2 (rad/sec)
2

rad/s

Acceleration
Equations 1 2 3 4

m/s

α (rad/sec )

2

rad/s

2

s = ut + ½ at 2 s = ½(u + v)t v = u +at v 2 = u 2 + 2as

θ = ω1 + ½ αt2 θ = ½ (ω1 +ω2)t ω2 = ω1 + αt ω22 = ω12 + 2αθ

6.0 EXAMINATION TYPE QUESTIONS Question 1 A Car starts from rest and accelerates to a speed of 20m/s in 5 seconds, it moves with ths velocity for 12 seconds and then comes to rest in the next two seconds. The diameter of the car wheel is 70cm. Determine:i. ii. iii. iv. Question 2 The rotor shaft of a generator at Kariba North Bank Power Station has a diameter of 800mm, and moment of inertia of 1200kg/m2, when the gate to the penstock is open water hits the guide vanes which are tangentially connected to the rotor shaft which attains a speed of 3000r.p.m in 7 seconds. Determine:i. ii. iii. the maximum angular speed attained by the rotor. the acceleration of the rotor in rad/sec2. the magnitude of the force imposed on the shaft by the water. the acceleration of the car in m/s2. the acceleration of the wheel of the car in rad/s2. the retardation of the wheel of the car in rad/s2. the total number of revs made by the wheel during this journey.

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By Daka Koloto Aggrey. Well seasoned Tutor, Let God arise and his enemies be scattered.

Question 3 A DVD player turntable rotates at 66rpm. What is the linear speed of the lenses sensor: (a) the beginning and (b) the end of the recording? (The distances of the lenses sensor from the turntable axis are 15cm and 7cm respectively, at the two positions). Question 4 a) The angular speed of a two-speed drilling machine is increased from 1, 200 rev/min to 3,000 rev/min in 12 seconds. (a) What is its angular acceleration, assuming it to be uniform? (b) How many revolutions does the rotor make during this time? b) A wheel has a constant angular acceleration of 3 rad/s2. In a 4-second interval, it turns through an angle of 120 radians. Assuming the wheel started from rest, how long had it been in motion from the start of this 4-second interval? Question 5 a) The diameter of the wheel for the Mazhandu bus is 0.56m. Determine the distance covered by a point on the tyre tread when the wheel has turned through 7500. b) As this bus enters the Ndola city it goes round a roundabout and it covers a distance of 40m whilst making a 90° turn, determine the diameter of the roundabout. Question 5 a) A wheel rotating at 490 rev/min accelerates uniformly at 0.5 rad/s2 for 30s. Find: i. ii. iii. the angular velocity after 30s the pulley’s peripheral speed if its diameter is 20 cm the angle turned through by the wheel during this time.

b) An electric motor is rotating at 900 rev/min and when it is switched off it is brought to rest with uniform retardation in 420 revolutions. Find the time taken to bring the motor to rest and the angular retardation. c) If the fan blades make 30 complete revolutions in 15s, what is the rotating speed in rev/min and in rad/s? What is the angle turned through in radians by a point on the tip of one fan blade in 7seconds?

Whether it is the coconut that crushes on the egg, or the egg that crushes on the coconut, it’s the egg to suffer. D2k June 2013

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By Daka Koloto Aggrey. Well seasoned Tutor, Let God arise and his enemies be scattered.