Mechanical Engineering Principles

Part 2 Dynamics

11

Linear and angular motion

At the end of this chapter you should be

able to:

• appreciate that 2πradians corresponds to
  360 °


• define linear and angular velocity


• perform calculations on linear and angular
  velocity usingω= 2 πnandv=ωr


• define linear and angular acceleration


• perform calculations on linear and angular
  acceleration usingω 2 = ω 1 +αt and
  a=rα


• select appropriate equations of motion
  when performing simple calculations


• appreciate the difference between scalar
  and vector quantities


• use vectors to determine relative veloci-
  ties, by drawing and by calculation

11.1 The radian

The unit of angular displacement is the radian,
where one radian is the angle subtended at the centre
of a circle by an arc equal in length to the radius,
as shown in Figure 11.1.
The relationship between angle in radians (θ), arc
length (s) and radius of a circle (r)is:

s=rθ ( 11. 1 )

1 rad

r

r

r

Figure 11.1

Since the arc length of a complete circle is 2πrand

the angle subtended at the centre is 360°, then from

equation (11.1), for a complete circle,

2 πr=rθ or θ= 2 πradians

Thus,^2 πradians corresponds to 360° ( 11. 2 )

11.2 Linear and angular velocity

Linear velocityvis defined as the rate of change

of linear displacementswith respect to timet,and

for motion in a straight line:

Linear velocity=

change of displacement

change of time

i.e. v=

s

t (^11.^3 )

The unit of linear velocity is metres per second

(m/s).