__Linear and Angular Motion__

__Linear and Angular Motion__

__The Radian__

__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**

**Linea****r 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:

**Angula****r 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,

**, indicate a vector quantity and the order of the letters indicate that the time of action is from a to b.**

*ab***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

**the velocity of B.**

*ob*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**.

**I****n 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

**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**

*ab***. By Pythagoras’ theorem:**

*ob*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