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160 PHYSICS

length. This point is called the fulcrum. A see-
saw on the children’s playground is a typical
example of a lever. Two forces F and F , parallel
1 2
to each other and usually perpendicular to the
lever, as shown here, act on the lever at
distances d and d respectively from the
1 2
fulcrum as shown in Fig. 7.23.

Fig. 7.21(b) The Earth’s magnetic field exerts equal
and opposite forces on the poles of a Fig. 7.23
compass needle. These two forces form
a couple. The lever is a system in mechanical
equilibrium. Let R be the reaction of the support
u Example 7.7 Show that moment of a at the fulcrum; R is directed opposite to the
couple does not depend on the point about forces F and F . For translational equilibrium,
which you take the moments. 1 2
R – F – F = 0 (i)
Answer 1 2
For considering rotational equilibrium we
take the moments about the fulcrum; the sum
of moments must be zero,
d F – d F = 0 (ii)
1 1 2 2
Normally the anticlockwise (clockwise)
moments are taken to be positive (negative). Note
R acts at the fulcrum itself and has zero moment
about the fulcrum.
Fig. 7.22
In the case of the lever force F is usually
1
Consider a couple as shown in Fig. 7.22 some weight to be lifted. It is called the load
acting on a rigid body. The forces F and -F act and its distance from the fulcrum d is called
1
respectively at points B and A. These points have the load arm. Force F is the effort applied to lift
2
position vectors r and r with respect to origin the load; distance d of the effort from the
1 2 2
O. Let us take the moments of the forces about fulcrum is the effort arm.
the origin. Eq. (ii) can be written as
The moment of the couple = sum of the d F =d F (7.32a)
1 1 2 2
moments of the two forces making the couple or load arm × load = effort arm× effort

= r × (–F) + r × F The above equation expresses the principle
1 2
= r × F – r × F of moments for a lever. Incidentally the ratio
2 1
= (r –r ) × F F /F is called the Mechanical Advantage (M.A.);
2 1 1 2
But r + AB = r , and hence AB = r – r .
1 2 2 1 F 1 d 2
The moment of the couple, therefore, is M.A. = = (7.32b)
AB× F. F 2 d 1

Clearly this is independent of the origin, the If the effort arm d is larger than the load
2
point about which we took the moments of the arm, the mechanical advantage is greater than
forces. t one. Mechanical advantage greater than one
means that a small effort can be used to lift a
7.8.1 Principle of moments
large load. There are several examples of a lever
An ideal lever is essentially a light (i.e. of around you besides the see-saw. The beam of a
negligible mass) rod pivoted at a point along its balance is a lever. Try to find more such

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