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 \(0.148\,\text{kgm}^2\).

Key concepts

Perpendicular Axis Theorem

The Perpendicular Axis Theorem is a useful tool in physics when dealing with moments of inertia for objects in two-dimensional planes. It states that the moment of inertia of a flat body about an axis perpendicular to its plane can be found by the sum of the moments of inertia about two mutually perpendicular axes in the plane of the body. For a uniform rod like a meter scale, this theorem simplifies calculations when the axis does not pass through typical pivots, such as ends or centers.
In our problem, the axis was at the 30 cm position on the meter scale, requiring consideration beyond standard symmetrical placements.
The theorem exemplifies how principles in physics offer simplifications and solutions to complex real-world situations.

Rod

A rod in physics is a model used to describe an object that has one significant dimension (length) but negligible width and height. This concept is crucial for simplifying the problem of calculating moments of inertia. When modeling something like a meter scale as a rod, its mass and length are the primary considerations.

• Mass: The mass of the rod affects how it rotates around an axis. In our example, the rod had a mass of 0.6 kg.
• Length: The meter scale is 1 meter long, which is standard for such problems.

When you think of the scale as a rod, it makes applying theories like the perpendicular axis theorem more intuitive and manageable.

Meter Scale

The meter scale is a simple, versatile tool used in both teaching measurements and applying fundamental physics concepts. In exercises dealing with physical principles, a meter scale’s regular dimensions (1 meter in length) allow for straightforward calculations.
In this problem, sizing of the meter scale gives us clear values for calculating moments of inertia. With width being negligible, it resembles an ideal rod, further simplifying theoretical analysis. Understanding its properties helps grasp essential concepts in physics, like balance and dynamics.
Makeshift aids like the meter scale can transition from ordinary use to illustrating key physical principles, revealing the elegance and simplicity underlying much of physics.

Physics Problem Solving

Solving physics problems like this involves a series of logical steps taking us from what's given to the final answer. Here, our main task was finding the moment of inertia using known formulas and applying fundamental physics concepts:

• Start with understanding all given information, such as mass and dimensions, and translate them into consistent units.
• Use appropriate formulas; in this case, the moment of inertia of a rod at one end.
• Incorporate advanced concepts like the perpendicular axis theorem for more complex scenarios.
• Perform calculations step-by-step, verifying each stage ensures accuracy.

These practices help navigate through even abstract or theoretical problems, leading to a greater comprehension of underlying physics and developing problem-solving skills.