Question

While discussing the period of a pendulam, one of the student argued that period depends on the mass of the bob.

1. What is your opinion?

2. How will you prove your argument dimensionally?

Answer 1

1. Period is independent of mass of the bob

2. The principle of homogeneity of dimensions also helps to derive a relationship between the different physical quantities involved; This method is known as dimensional analysis.

The period of the simple pendulum may possibly depend upon:

• The mass of the bob, m
• The length of the pendulum, I
• Acceleration due to gravity, g
• The angle of swing, q

Let us write the equation for the time period as t = k m^a l^b g^c q^d

where, k is a constant having no dimensions; a, b, care to be found out.

The dimensions of, t = T¹

Dimensions of. m = M¹

Dimensions of, l = L¹

Dimensions of, g = L¹T^-2

Angle q has no dimensions (since, q = arc/radius = L/L)

Equating the dimensions of both sides of the equation, we get,

T¹ = M^aL^b (L¹T^-2)^c

ie. T¹ = M^aL^b+c+ T^-2c.

The dimensions of the terms on both sides must be the same. Equating the powers of M, L and T.

a = 0; b + c = 0; -2c = 1

∴ c = \(- \frac1{2}\), b = ^– c = \(- \frac1{2}\)

Hence, the equation becomes,

t = kl^1/2g^-1/2

ie, t = k √(1/g)

Experimentally, the value of k is found to be 2p.