Unit 4 mm9400 ver1.1 (2014)

Kinematics 4 - 1
Statics & Dynamics (MM9400) Version 1.1
4. KINEMATICS
Objectives: At the end of this unit, students should be able to:
 Understand uniformly accelerated linear motion
 Define:
(a) linear displacement
(b) linear velocity
(c) linear acceleration
 Sketch the velocity vs time graphs for motions
involving uniform accelerations and retardations.
 Derive formulae for uniformly accelerated motion.
 Solve problems involving uniformly accelerated
motion using formulae & velocity vs time graphs.
 Understand uniformly accelerated angular motion
 Define:
(a) the radian
(b) angular displacement
(c) angular velocity
(d) angular acceleration
 Solve problems involving rotational motion using
formulae & angular velocity vs time graphs.
 Understand the relation between linear and angular motion
 Derive formulae relating linear and angular motion.
 Solve problems involving linear motion linked to angular motion.
Linear Motion
Angular Motion
Linear and Angular Motion

4 - 2 Kinematics
Version 1.1 Statics & Dynamics (MM9400)
4.1 Introduction
Kinematics is the study of motion without concern for the forces causing the motion.
It involves quantities such as displacement, velocity, acceleration and time.
Types of motion considered
The common crank and piston mechanism found in engines involves motion of interest to the
engineer. Basically, two types of motion can be observed. They are:
Fig 4.1 Simple Slider Crank Mechanism
(a) Linear Motion
The piston C undergoes linear motion; each point on it moves in a straight line (Fig 4.1).
Other examples include moving trains and vehicles.
(b) Angular Motion
The pin B at the end of crank AB has angular motion (Fig. 4.1); every point on the crank
follows the path of a circle about a fixed reference point A. Other examples are found in
the turning of a pulley or motor-shaft, rotation of tyres about the car axle and spinning of
washing machine drums.
4.2 Relationship of Linear Displacement, Velocity and Acceleration
4.2.1 Linear Displacement
Linear displacement is defined as “change of position” and is measured by the straight-line
vector joining the initial to the final position. It has units of length (e.g. metres) and has
direction; it is a vector quantity. It is denoted by the symbol‘s’. Distance, however, is a
scalar quantity referring to the length of the path travelled; no direction is implied. For
example, if a car travels from Town A to Town B as shown in Fig. 4.2, displacement from A
to B = 10 km (magnitude) in an easterly direction (), while distance travelled from A to B =
15 km.
A
C
B
Angular
motion
Linear motion
Fig. 4.2 Displacement and distance
15 km
10 km
A B

Kinematics 4 - 3
4.2.2 Linear Speed and Velocity
(a) Speed is a scalar quantity indicating the distance travelled by a body per unit time. It is
measured in metres per second (m/s) or kilometres per hour (km/h). If a car has an
average speed of 60 km/h, this means that it can cover a distance of 60 km in an hour.
(b) Velocity is the rate of change of displacement with respect to time. It has both magnitude
and direction. The units commonly used are: m/s or km/h.
4.2.3 Velocity versus Time Graph
Velocity versus Time (v-t) Graphs are used to chart the motion of a body by plotting the
velocity on the vertical axis against time on the horizontal axis. Uniform velocity, which is
motion along a straight line at constant speed, can be shown on a velocity vs time graph as a
horizontal line (see Fig. 4.3).
Fig. 4.3 Velocity vs time graph for uniform velocity
The area under the v-t graph up to time, t, represents the displacement (in metres) of the body
after moving at uniform velocity v m/s for t seconds.
Linear displacement, s = v x t = area under v-t graph
Example 4.1
A car travels with a uniform velocity of 80 km/h for 30 seconds. Sketch the velocity vs time
graph for the motion. Hence, determine the displacement of the car during the 30 s.
t = 30 s
s = ?
t = 0 s

4 - 4 Kinematics
4.2.4 Acceleration
Acceleration is the rate of change of velocity with respect to time. It is a vector quantity
denoted by the symbol 'a' and has SI units of m/s2
. A body which changes its speed or its
direction is accelerating.
Motion (a) Uniform acceleration
(b) Uniform retardation or
deceleration
Definition
velocity increasing at a constant
rate from initial velocity v0 to final
velocity v1 in time t
velocity decreasing at a constant rate
from v0 to v1 in time t
Velocity-timegraph
straight line with a positive
gradient (Fig. 4.4a).
straight line with a negative
gradient (Fig. 4.4b).
Gradient
represents acceleration or retardation, a =
t
vv 01 
 v1 = v0 + a t ……... .(1)
Area
represents the displacement. From Fig. 4.4a,
s = area A + area B = v0 t + ½( v1 - vo ) t
From (1), (v1 - vo) = a t  s = v0 t + ½ a t 2
.…......(2)
From Fig. 4.4b, taking average velocity va = ½ (v1 + vo)
then s = va x t  s = ½ (v1 + vo) t ……. (3)
We can eliminate time, t, to obtain another useful equation:
From (1) t =
a
vv 01 
; substituting into (3) we get: s = ½ (v1 + vo)
a
vv 01 
Therefore, (v1 + vo ) (v1 – vo ) = 2 a s  v1
2
– vo
2
= 2 a
Hence v1
2
= vo
2
+ 2as .. .….. (4)
Fig. 4.4a
v0
v1
0
time
velocity
t
area A
area B
v1 – v0
velocity
vo
v1
0 time
t
Fig. 4.4b
va

Kinematics 4 - 5
Note: It is important to realize that the above four equations are applied to a single stage of
motion with constant acceleration. If the motion goes through several stages where
the acceleration is variable or changes, the problem is usually solved using the v-t
graph (see Example 4.2).
Example 4.2 (One vehicle linear kinematics)
A train travelling between two stations 5 km apart completes the journey in 8 minutes. During
the first 50 seconds the train is moving with a constant acceleration whilst a uniform
retardation brings the train to rest in the last 40 seconds. For the remaining portion of the
journey, the train is moving with uniform velocity.
(a) Sketch a velocity vs time graph for the journey,
Determine the:
(b) uniform velocity,
(c) acceleration,
(d) retardation,
(e) distances travelled in the first minute and the last minute of the journey.
(a)
t (s)
v (m/s)

4 - 6 Kinematics
Example 4.3 (Two vehicles linear kinematics)
When the traffic light turned green at a traffic junction, a car A moved off from rest with a
uniform acceleration of 0.4 m/s2
. Two seconds later, a motorbike B starts off from rest from the
same junction with a uniform acceleration of 0.8 m/s2
. Assume that A is travelling in a straight
path with B following along a parallel path.
(a) How much time is required for the motorbike rider to catch up with the car from the
instant B began moving?
(b) Sketch a v-t graph for the motion of A and also for the motion of B.
t (s)
v (m/s)
2 10 12

Kinematics 4 - 7
B B
C
A
(h –30 ) m
30 m
60 m/s
h
4.3 Free Falling Bodies (Acceleration Due to Gravity)
When a body is allowed to fall from a height without any forces resisting it, then the body is
said to be falling freely. When the body falls freely, it is subjected to a downward
acceleration. This downward acceleration is due to gravity.
We assume that acceleration due to gravity is uniform and is equal to 9.81 m/s2
. Motion of a
freely falling body can thus be solved using equations (1) to (4) as discussed above.
Note: a = + 9.81 m/s
2
for downward motion and a = – 9.81 m/s
2
for upward motion.
Example 4.4
A ball is projected vertically upwards from point A with an initial velocity of 60 m/s.
(a) To what height will it rise?
(b) How much time will it take from the instant of projection for the ball to reach a height of
30 m above A on its return journey, and
(c) What will be its velocity then?

4 - 8 Kinematics
TUTORIAL (Linear Kinematics)
4.1 An automobile maintains a constant speed of 60 km/h. Find the speed in metres per
second. (16.7 m/s)
4.2 An air-plane flying at a constant speed of 150 m/s passes over a distance of 240
km. Find the time taken. (26 min 40 s or 1600 s)
4.3 A car accelerates uniformly from rest to 80 km/h in 15 seconds. The car then
travels steadily at 80 km/h on the road. Determine:
(a) the distance travelled during the acceleration. (166.7 m)
(b) the time needed to travel the first 1 km. (52.5 s)
4.4 Figure Q4.4 represents graphically the velocity of a bus moving along a straight road
over a period of time.
(a) Find the kinematics quantities between 0 and A?
(s = 400 m; v0 = 0 m/s; v1 = 40 m/s; t = 20 s; a = 2 m/s2
)
(b) Find the kinematics quantities between B and C ?
(800 m; 20 m/s; 0 m/s2
)
(c) What is the retardation of the bus between C and D ? (1 m/s
2
)
(d) What is the total distance travelled by the bus in 100 s ? (2000 m)
0
10
20
30
40
A
B C
D
20 40 60 80 100
Velocity
(m/s)
time (s)
Fig Q4.4

Kinematics 4 - 9
4.5 A man MA starts to cycle from rest with a uniform acceleration of 1.2 m/s2
for 10
seconds and then maintains at that speed for another 10 seconds. Another man MB
starts to cycle from rest 5 seconds later with a uniform acceleration of 1.6 m/s2
for 15
seconds.
(a) Draw the v-t graph for both MA and MB together using the following scales: 10
mm to 2 m/s and 10 mm to 2 seconds
(b) From the graph, estimate how many seconds after MB starts, when both MA and
MB have the same velocity.
(7.5 s)

4 - 10 Kinematics
4.6 A train runs from starts to stop between 2 stations in 4 minutes. It is uniformly
accelerated for 30 seconds, during which time it travels 450 m. It then runs at constant
velocity and finally retards uniformly for 15 seconds. Draw a v-t graph to show the
complete motion and find the distance between the 2 stations.
(6525 m)

Kinematics 4 - 11
4.7 A car A starts from rest with a uniform acceleration of 0.6 m/s2
. A second car B starts
from the same point 4 seconds later and follows the same path with an acceleration of
0.9 m/s2
. At what time will B overtake A? How far will both cars have travelled when
B passes A? Draw a v-t graph for the motion of both cars.
(21.8 s ; 142.6 m)

4 - 12 Kinematics
4.8 A man standing at the top edge of a building 100 m tall throws a stone weighing 0.1 kg
vertically up into the air with a speed of 40 m/s. The stone rises vertically and later falls
freely all the way down to the street below.
(a) determine the time taken for the stone to reach the street level from the start of the
throw.
(b) construct the v-t graph for the upward and downward motion of the stone
using a velocity scale of 1 cm : 10 m/s and a time scale of 1 cm : 1 second.
(c) state from the v-t graph, the velocity of the stone when it hits the street.
(10.16 s ; 60 m/s )

Kinematics 4 - 13
4.9 A skydiver jumps from a plane at a height of 1.8 km above the ground. He falls freely
for 6 seconds and his parachute opens giving him a uniform deceleration for 3 seconds
to a velocity 5.56 m/s which he maintains until he lands.
(a) Sketch a v-t graph for the motion of the skydiver.
(b) Determine:
(i) the maximum velocity, and
(ii) total time elapsed during his motion from the plane to the ground. (Neglect
the air resistance on his body during free fall)
(58.86 m/s; 283.6 s)

4 - 14 Kinematics
* 4.10 A dart player stands 3 m from the wall on which the board hangs and throws a dart which
leaves his hand with a horizontal velocity at a point 1.8 m above the ground (see Fig.
Q4.10). The dart strikes the board at a point 1.5 m from the ground. Assuming air
resistance to be negligible, calculate the:
(a) time of the flight of the dart
(b) initial speed, vo, of the dart
(c) velocity of the dart when it hits the board.
Hint: the dart moves with two components of motion: an x-component with uniform
speed vo and a y-component with uniform acceleration of 9.81 m/s2
.
( 0.2473 s, 12.13 m/s, 12.37 m/s 11.3° )
3 m
1.5m
1.8m
vo
Fig. Q4.10

Kinematics 4 - 15
4.4 Angular Kinematics
4.4.1 Angular Displacement (  )
The movement of the lever AB about A is measured in terms of the angle through which it
turns. It may be in degrees or revolutions. The change in its angular position is known as
angular displacement, .
The basic SI unit for measuring angular displacement is the radian (rad). In Fig. 4.5, AB turns
through an angle of 1 radian about A when point B moves a distance on the circumference
equal to radius r.
when AB = arc BB’ = r,  = 1 rad
since one circumference length = 2  r metres
 angular displacement of 1 revolution = 2 radians
= 360o
…….
Fig. 4.5
4.4.2 Angular Velocity (  )
Angular velocity is the rate of change of angular displacement. It is commonly measured in
rad/s or rev/min (rpm).
Note: Quantities in rev/min are usually converted to rad/s before problem solving.
4.4.3 Uniform Angular Velocity ()
This can be indicated on an angular velocity vs time graph (see Fig. 4.6).
Note:
Angular velocity  is in rad/s
Angular displacement  is in rad
sradrev/min /
60
2
1


B’
B
r
r
A
 = 1 rad
Angular velocity (rad/s)
Area .t
Time (s)
 t

Fig. -t graph

4 - 16 Kinematics
4.4.4 Angular acceleration (  )
If angular velocity is not constant, but is increasing at a uniform rate (see Fig. 4.7), the motion is
described as rotating with a uniform angular acceleration. The rate at which angular velocity
is changing with time is called angular acceleration, . The unit used is rad/s2
.
t
ωω
α o1 
 = gradient of  vs t graph
where
 = angular acceleration (rad/s2
)
o = initial angular velocity (rad/s)
1 = final angular velocity (rad/s)
t = time (s)
Area under the angular velocity vs time graph is equal to the angular displacement, .
Example 4.5
Starting from rest, a wheel accelerates uniformly to 300 rev/min in 5 seconds. After rotating at
300 rev/min for some time, it is retarded uniformly to rest in 20 seconds. The total number of
revolutions made by the wheel is 500.
(a) Sketch the angular velocity vs time graph [-t graph] for the complete motion.
(b) Determine the:
(i) time taken when the wheel is rotating uniformly at 300 rev/min;
(ii) angular retardation.
(a)
t (s)
ω (rad/s)
t
Fig. 4.7 -t graph showing angular acceleration
o
1
0
time
Angular velocity

Kinematics 4 - 17
4.5 Relationship between linear and angular motion
There are many situations in which linear and angular motions are combined, eg. belts on
pulleys, wheels & axle of the car, etc.
A hoist drum with a cable wound around it can rotate and lower a weight (Fig. 4.8). Distance
lowered, s, depends on the radius and amount of rotation measured in radians i.e.
s = r  ............................. ( i ) where  is expressed in radians.
Dividing equation (i) by time t on both sides,
v = r  ......................... ( ii )
Equation (ii) relates the angular velocity  to the linear velocity v.
Dividing equation (ii) by time t on both sides,
We get the relationship:
a = r  .......................... ( iii )
s
t
 r

t
t
ω
r
t
v


s
hoist drum
(radius r)
v
cable


Fig. 4.8 Relationship between linear and angular motion

4 - 18 Kinematics
Example 4.6
A cyclist is pedalling at a rate such that the bicycle wheels rotate at an angular velocity of
30 rpm. If the diameter of the wheels is 0.6 m, determine the linear velocity of the cyclist if
the bicycle moves on the road without slipping.
Example 4.7 (Common peripheral velocity)
Pulley A and pulley B are connected by a belt as shown in the figure below. Pulley B and C
are both located along the same shaft. If pulley A rotates from rest to an angular velocity of
160 rad/s in 3 seconds, determine:
(a) angular acceleration of pulley A
(b) angular velocity and acceleration of pulley B
(c) angular velocity of pulley C
(d) velocity of a point on the rim of pulley C.
v
ωA ωB
A
B
C
diameter of pulley A = 150 mm
diameter of pulley B = 250 mm
diameter of pulley C = 100 mm

Kinematics 4 - 19
4.6 Summary
Physical Quantity Linear Motion Angular Motion
Displacement
Initial Velocity
Final Velocity
Acceleration
s
vo
v1
a

o
1

Motion with Uniform
Velocity (a = 0)
s = v t  =  t
Uniformly
Accelerated Motion
v1 = vo + a t
s = ½ (vo + v1) t
s = vo t + ½ a t2
v1
2
= vo
2
+ 2 a s
1 = o +  t
 = ½ (o + 1) t
 = o t + ½  t2
1
2
= o
2
+ 2  
Relationship between
Linear and Angular
quantities
s = r  v = r  a = r 
TUTORIAL (Angular kinematics)
4.11 How many radians are there in 5 revolutions? 90 revolutions? 360 revolutions?
(31.4 rads, 565.5 rads, 2262 rads)
4.12 Change the following from degrees to radians:
(a) 60o
(b) 90 o
(c) 135o
(d) 15o
[ (a) /3, (b) /2 , (c) 3/4, (d) /12 ]

4 - 20 Kinematics
4.13 A grinding wheel is rotating at 2500 rpm. When the power is switched off, it takes 3
minutes for the wheel to come to rest.
(a) Sketch an angular velocity vs time graph for the motion.
(b) Determine the angular retardation. (-1.45 rad/s2
)
(c) Determine the number of revolutions made by the wheel
in the 3-minute period. (3750 rev)
(d) What is the initial linear velocity of a point on the rim of
the wheel if the diameter of the wheel is 0.5 m? (65.45 m/s)
4.14 A hoist drum of diameter 1.5 m accelerates uniformly from rest for 2 seconds until it is
rotating at 90 rpm. At this speed, the drum makes 5 revolutions before retarding
uniformly to rest in 1.2 seconds.
(a) Sketch the -t graph for the motion.
(b) Determine:
(i) the angular acceleration and the angular retardation.
(ii) the total time the drum is rotating.
(4.71 rad/s2
, -7.85 rad/s2
, 6.53 s)
(c) What is the total linear displacement of a lift connected by a cable to the drum?
(34.87 m)

Kinematics 4 - 21
4.15 A flywheel is accelerated uniformly from rest to attain a speed of 220 rev/min while
turning through 80 complete revolutions. Find:
(a) time taken, (43.6 s)
(b) angular acceleration (0.53 rad/s2
)
(c) linear speed at a point on the flywheel rim when it
revolves at 220 rpm and the flywheel diameter is 1.4 m. (16.1 m/s)
4.16 Starting from rest, a wheel accelerates uniformly to reach a speed of 180 rpm in
60 seconds. It runs for some time at this speed before it is brought to rest with a uniform
retardation in 50 seconds. The total number of revolutions made by the wheel is 1000.
(a) Sketch the -t graph for the complete motion.
(b) Determine:
(i) the time taken when the wheel is rotating at the constant speed of 180 rpm.
(ii) the angular acceleration of the wheel.
( 278.3 s, 0.314 rad/s2
)

4 - 22 Kinematics
4.17 The combined pulley (B & D) shown in Fig. Q4.17 has two cables wound around it at
different diameters and fastened to blocks A and C respectively. Block A is released
from rest and reaches a velocity of 240 mm/s after 0.6 second.
(a) Determine the acceleration of block A.
(b) Determine the angular acceleration of pulley B.
(c) At the instant when block A reaches 240 mm/s, determine:
(i) the angular velocity of pulley B
(ii) the velocity of block C.
(iii) the displacement made by block C.
( 0.4 m/s2
, 5 rad/s2
, 3 rad/s, 150 mm/s, 45 mm )
A
B
C
Ø 160 mm
Ø 100 mm
240 mm/s
Fig. Q4.17
D

Kinematics 4 - 23
*4.18 A 80 mm diameter pulley is connected to a 40 mm diameter pulley as shown in Fig.
Q4.18. The bigger pulley rotates from rest to 150 revolutions per minute (rpm) in 3
seconds. It then maintains the speed for another 10 seconds before decelerating to rest
in the next 5 seconds.
(a) Sketch the angular velocity – time diagram for the 80 mm pulley for the
complete motion.
(b) For the 80 mm pulley, determine the:
(i) angular acceleration
(ii) total angular displacement in radian
(c) Determine the peripheral velocity of the pulleys.
(d) Determine the total number of revolutions made by the smaller pulley for the
complete motion.
( 5.236 rad/s2
, 219.9 rad, 0.6284 m/s, 70 revolutions )
************************************
Fig.Q4.18

Unit 4 mm9400 ver1.1 (2014)

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