4.6 Principle of Moments
To understand the principle of moments, let us perform an activity.
lightbulbActivity 4.2 Balance a metre rule on a wedge at its centre of gravity such that the metre rule stays horizontal. Suspend two weights w₁ and w₂ on one side of the metre rule at distances ℓ₁ and ℓ₂ from the centre, and a third weight w₃ on the other side at distance ℓ₃ until the rule is again balanced. The weights w₁ and w₂ tend to rotate the rod anticlockwise about the CG, and the weight w₃ tends to rotate it clockwise. The values of the moments of the weights are w₁ × ℓ₁, w₂ × ℓ₂, and w₃ × ℓ₃. When the metre rule is balanced:
Total anticlockwise moments = Total clockwise moments w₁ × ℓ₁ + w₂ × ℓ₂ = w₃ × ℓ₃ ............ (4.6)
This is known as the principle of moments, which is stated as:
lightbulbWhen a body is in a balanced position, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that point.
Example 4.4
A metre stick is pinned at one end O on a table so that it can rotate freely. One force of magnitude 18 N is applied perpendicular to the length of the stick at its free end. Another force of magnitude 60 N is acting at an angle of 30° with the stick as shown in Fig. 4.13(a). At what distance from the pinned end should the second force act such that the stick does not rotate?
Solution:
The weight of the stick does not affect the horizontal plane. Resolving force F of magnitude 60 N into rectangular components that act at distance d from point O:
F_x = 60 N × cos 30° = 60 N × 0.866 = 51.96 N
F_y = 60 N × sin 30° = 60 N × 0.5 = 30 N
As the component F_x passes through the axis of rotation, its torque is zero. Torque τ₁ of 30 N is positive and τ₂ of 18 N force is negative. The stick will not rotate when these two torques balance each other, i.e., τ₁ = τ₂.
F_y × d = 18 N × 1 m
30 N × d = 18 N m
d = 18 N m / 30 N = 0.6 m