Linear and Angular Motion
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 15.1
The relationship between angle in radians (8), arc length (s) and radius of a circle (r) is:
Since the arc length of a complete circle is 2nr and the angle subtended at the centre is 360° , then from equation (1), for a complete circle,
Linear and Angular Velocity
Linear velocity
Linear velocity v is defined as the rate of change of linear displacement s with respect to time t, and for motion in a straight line:
The unit of linear velocity is metres per second (m/s)
Angular velocity
The speed of revolution of a wheel or a shaft is usually measured in revolutions per minute or revolutions per second but these units do not form part of a coherent system of units. The basis used in SI units is the angle turned through in one second.
Angular velocity ω is defined as the rate of change of angular displacement 8, with respect to time t, and for an object rotating about a fixed axis at a constant speed:
The unit of angular velocity is radians per second (rad/s)
An object rotating at a constant speed of n revolutions per second subtends an angle of 2nn radians in one second, that is, its angular velocity,
Equation (6) gives the relationship between linear velocity, v, and angular velocity, ω.
For example, if a wheel of diameter 540 mm is rotating at (1500/n) rev/min,
the angular velocity of the wheel and the linear velocity of a point on the rim of the wheel is calculated as follows:
From equation (5), angular velocity ω D 2nn, where n is the speed of revolution in revolutions per second, i.e.
Linear and Angular Acceleration
Linear acceleration, a, is defined as the rate of change of linear velocity with respect to time (as introduced in Chapter 8). For an object whose linear velocity is increasing uniformly:
The unit of linear acceleration is metres per second squared (m/s2). Rewriting equation (7) with v2 as the subject of the formula gives:
Angular acceleration, a, is defined as the rate of change of angular velocity with respect to time. For an object whose angular velocity is increasing uniformly:
The unit of angular acceleration is radians per second squared (rad/s2). Rewriting equation (9) with ω2 as the subject of the formula gives:
For example, if the speed of a shaft increases uniformly from 300 revolutions per minute to 800 revolutions per minute in 10s, the angular acceleration is determined as follows:
Initial angular velocity,
Further Equations of Motion
From equation (3), s D vt, and if the linear velocity is changing uniformly from v1 to v2 , then s D mean linear velocity x time
From equation (4), 8 = ωt, and if the angular velocity is changing uniformly from ω1 to ω2, then 8 D mean angular velocity ð time
Two further equations of linear motion may be derived from equations (8) and (12):
Two further equations of angular motion may be derived from equations (10) and (13):
Table 15.1 summarises the principal equations of linear and angular motion for uniform changes in velocities and constant accelerations and also gives the relationships between linear and angular quantities.
For example, the shaft of an electric motor, initially at rest, accelerates uniformly for 0.4 s at 15 rad/s2 . To determine the angle (in radians) turned through by the shaft in this time:
Relative Velocity
A vector quantity is represented by a straight line lying along the line of action of the quantity and having a length that is proportional to the size of the quantity, as shown in chapter 3. Thus ab in Figure 15.2 represents a velocity of 20 m/s, whose line of action is due west. The bold letters, ab, indicate a vector quantity and the order of the letters indicate that the time of action is from a to b.
For example, consider two aircraft A and B flying at a constant altitude, A travelling due north at 200 m/s and B travelling 30° east of north, written N 30°E, at 300 m/s, as shown in Figure 15.3.
Relative to a fixed point o, oa represents the velocity of A and ob the velocity of B.
The velocity of B relative to A, that is, the velocity at which B seems
to be travelling to an observer on A, is given by ab, and by measurement is 160 m/s in a direction E 22° N.
The velocity of A relative to B, that is, the velocity at which A seems to be travelling to an observer on B, is given by ba and by measurement is 160 m/s in a direction W 22° S.
In another example, a crane is moving in a straight line with a constant horizontal velocity of 2 m/s. At the same time it is lifting a load at a vertical velocity of 5 m/s. The velocity of the load relative to a fixed point on the earth’s surface is calculated as follows:
A vector diagram depicting the motion of the crane and load is shown in Figure 15.4. oa represents the velocity of the crane relative to a fixed point on the earth’s surface and ab represents the velocity of the load relative to the crane. The velocity of the load relative to the fixed point on the earth’s surface is ob. By Pythagoras’ theorem:
i.e. the velocity of the load relative to a fixed point on the earth’s surface is 5.385 m/s in a direction 68.20° to the motion of the crane