Question

A square sheet of mass m and side l is rotating about an axis passing through one side as shown moment of inertia of sheet about the axis is

1)Ml²/2

2)Ml²/3

3)2Ml²/3

4)Ml²/6

pls dont spam ans asap pls

Answer 1

The correct option to the above question is (4), which is Ml²/6, and the explanation is given below -

• Moment of inertia is a quantity expressing a body's tendency to resist angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation

• To determine the moment of inertia of a square plate we have to consider a few things.

• First, we will assume that the plate has mass (M) and sides of length (L). The surface area A = L × L = L²

Now we will define the mass per unit area as;

Surface density, ρ = mass/area = M/L²

We will use integration,

The Inertia of plate = ∫dl = ∫(dIcom + dIparallel axis)

                                = x=-L/2∫x=L/2 (1/12) ρ L³dx + ρ Lx²dx

                                = ρ (L³ / 12) [x |-L/2L/2 + ρ L [ ⅓ x³ |-L/2L/2

                                =  ρ (L³ / 12) [ L / 2 – (-L / 2)] + ρ L [(⅓ L³ /8) – (- ⅓ L³ / 8)]

                                =  ρ (L³ / 12) (L) + ρ L (⅔ L³ / 8)

                                = (ρ / 12) L₄ + (ρ / 12) L⁴

                                 = (1 / 6) ρ L⁴

                                 = (1 / 6) (M / L²) L⁴

                                 = (1 / 6) M L²

Therefore, the moment of inertia of the sheet about the axis is ML²/6, and the correct option is (4).