Question

The moment of inertia of a meter scale of mass \(0.6 \mathrm{~kg}\) about an axis perpendicular to the scale and passing through \(30 \mathrm{~cm}\) position on the scale is given by (Breath of scale is negligible). \(\\{\mathrm{A}\\} 0.104 \mathrm{kgm}^{2}\) \\{B \(\\} 0.208 \mathrm{kgm}^{2}\) \(\\{\mathrm{C}\\} 0.074 \mathrm{kgm}^{2}\) \(\\{\mathrm{D}\\} 0.148 \mathrm{kgm}^{2}\)

Short answer

The moment of inertia of the meter scale about the given axis is approximately \(0.148\,\text{kgm}^2\).

Step-by-step solution

Understand the given information

We are given: Mass of the meter scale (m) = 0.6 kg Position of the axis from one end (x) = 30 cm = 0.3 meters (since 1 meter = 100 cm) Total length of the scale (L) = 1 meter

Moment of inertia formula for the axis at one end

The moment of inertia of a rod of mass(m) and length (L) about an axis perpendicular to the rod and passing through one end is given by the formula: \(I_{end}=\frac{1}{3}mL^2\)

Calculate the moment of inertia at one end

Now, let's calculate the moment of inertia at one end using the given values: \(I_{end} = \frac{1}{3}(0.6\,\text{kg})(1\,\text{m})^2 =\frac{1}{3}(0.6)= 0.2\,\text{kgm}^2\)

Moment of inertia formula for a perpendicular axis

Now, we need to find the moment of inertia of the rod about the given axis at 30 cm position, For this, we will use the perpendicular axis theorem: \(I_x = I_{end} - mx^2\) Where, \(I_x\) = Moment of inertia about the given axis \(I_{end}\) = Moment of inertia about the axis at one end m = mass of the rod x = distance of the given axis from one end

Calculate the moment of inertia at the given axis

Now, let's find the moment of inertia at the given axis using the perpendicular axis theorem: \(I_x = 0.2\,\text{kgm}^2 - (0.6\,\text{kg})(0.3\,\text{m})^2 = 0.2 - (0.6)(0.09) = 0.2 - 0.054 = 0.146\,\text{kgm}^2\) Comparing the calculated value with the options given, we see that the answer is approximately equal to option D: \(\{\text{D}\} 0.148 \,\text{kgm}^2\) Therefore, the moment of inertia of the meter scale about the given axis is approximately \(0.148\,\text{kgm}^2\).