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

Vel

ocit

y-ti

me

grap

h

straight line with a positive gradient (Fig. 4.4a).

straight line with a negative gradient (Fig. 4.4b).

Gra

die

nt

represents acceleration or retardation, a = t

vv 01

v1 = v0 + a t ……... .(1)

Are

a

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 = vo2 + 2as .. .….. (4)

Fig. 4.4a

v0

v1

0time

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/s2)

(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

1.5

m

1.8

m

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

ωωα 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

ωr

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

12

= o2 + 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) 135 o (d) 15 o

[ (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

Engineering

Unit 4 mm9400 ver1.1 (2014)