The Chemistry Maths Book, Second Edition

464 Chapter 16Vectors

EXAMPLE 16.16Moment of force (torque)

In mechanics the magnitude of the moment of force F,

or torque T, about a point O is defined as the product

T 1 = 1 |F|d, where dis the perpendicular distance from O to

the line of action of the force (Figure 16.26).

If ris the vector from O to a point A on the line, then

d 1 = 1 |r| 1 sin 1 θand

T 1 = 1 |r||F|sin 1 θ 1 = 1 |r 1 z 1 F|

The corresponding vector

T 1 = 1 r 1 z 1 F (16.54)

is perpendicular to rand F, and its direction is that of the axis through O about which

the force tends to produce rotation.

0 Exercise 41

EXAMPLE 16.17An electric dipole in an electric field

An electric dipoleμ 1 = 1 qr(see Example 16.6)

in an electric field Eexperiences a torque

T 1 = 1 μ 1 z 1 E (16.55)

that tends to align the dipole along the

direction of the field. Thus, from Figure 16.27,

the total torque about O is

T 1 = 1 r

1

1 z 1 F

1

1 + 1 r

2

1 z 1 F

2

= 1 q(r

1

1 − 1 r

2

) 1 z 1 E 1 = 1 qr 1 z 1 E 1 = 1

μ 1 z 1 E

0 Exercises 42, 43

EXAMPLE 16.18Angular velocity and angular momentum

Angular velocity

A particle moving in a circle of radius rwith speed vhas angular velocity y(angular

speed ω) about an axis through the centre with magnitude given by

v 1 = 1 ωr (16.56)

and direction at right angles to the plane of motion (out of the page for the motion

shown in Figure 16.28).

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Figure 16.26

o

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Figure 16.27