Question

Calculate the M.I. of a thin uniform rod of the mass $100g$ and length $60cm$ about an axis perpendicular to its length and passing through

1. Its centre and
2. one end

Answer 1

Moment of inertia or rotational inertia of a body is the resistance that the body offers to any change in its speed of rotation about a particular axis by the application of a torque. Naturally, the M.I. for a body is different for different axes of rotation. It is given by the sum of the products of the mass of all the particles and their respective distances from the particular axis of rotation.

Complete step by step solution:

We have been given the mass and length of the rod. Let us first convert these quantities into their SI units.

Mass of the rod $= 100g = \dfrac{{100}}{{1000}}kg = 0.1kg$

Length of the rod $= 60cm = \dfrac{{60}}{{100}} = 0.6m$

(a) Now, we know, the M.I. of a thin rod about an axis perpendicular to its length and passing through its centre of mass is given by $\dfrac{{M{L^2}}}{{12}}$ , where $M$ and $L$ denote the mass and length of the rod respectively.

Therefore, $M.{I_i} = \dfrac{{0.1 \times {{(0.6)}^2}}}{{12}} = \dfrac{{0.1 \times 0.36}}{{12}} = 0.003kg{m^2}$

(b) And, the M.I. of a thin rod about an axis perpendicular to its length and passing through one of its end is given by $\dfrac{{M{L^2}}}{3}$ , where $M$ and $L$ denote the mass and length of the rod respectively.

Therefore, $M.{I_{ii}} = \dfrac{{0.1 \times {{(0.6)}^2}}}{3} = \dfrac{{0.1 \times 0.36}}{3} = 0.012kg{m^2}$ .

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$\therefore$ a) M.I. of a thin uniform rod of the mass $100g$ and length $60cm$ about an axis perpendicular to its length and passing through its center is $0.003kg{m^2}$.

b) M.I. of a thin uniform rod of the mass $100g$ and length $60cm$ about an axis perpendicular to its length and passing through one end is $0.012kg{m^2}$.

Note:
- The moment of inertia of a body about different axes of rotation is always different, as the distance of all the particles in the body from the axis is different.
- Also, we need to keep in mind the units in which the quantities are given, and convert them into SI units, if they are not already.