Problem 35 - University Physics with Modern Physics

Question

Determine the moment of inertia for three children weighing \(60.0 \mathrm{lb}, 45.0 \mathrm{lb}\) and \(80.0 \mathrm{lb}\) sitting at different points on the edge of a rotating merry-go-round, which has a radius of \(12.0 \mathrm{ft}\).

Step-by-Step Solution

Answer

Answer: The moment of inertia for the three children sitting on the edge of the merry-go-round is approximately \(1121.993 \mathrm{kg \cdot m^2}\).

Step 1: Convert weights to mass

First, we need to convert the weights of the children from pounds (lb) to mass in kilograms (kg). To do this, we can use the following conversion factor: 1 lb = 0.453592 kg. Child 1: \(60.0 \mathrm{lb} \times 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} = 27.216 \mathrm{kg}\) Child 2: \(45.0 \mathrm{lb} \times 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} = 20.412 \mathrm{kg}\) Child 3: \(80.0 \mathrm{lb} \times 0.453592 \frac{\mathrm{kg}}{\mathrm{lb}} = 36.288 \mathrm{kg}\)

Step 2: Convert the radius to meters

Next, we need to convert the radius of the merry-go-round from feet (ft) to meters (m). To do this, we can use the following conversion factor: 1 ft = 0.3048 m. Radius: \(12.0 \mathrm{ft} \times 0.3048 \frac{\mathrm{m}}{\mathrm{ft}} = 3.6576 \mathrm{m}\)

Step 3: Calculate the moment of inertia for each child

Now, we can calculate the moment of inertia for each child using the formula \(I = m \times r^2\). Child 1: \(I_1 = 27.216 \mathrm{kg} \times (3.6576 \mathrm{m})^2 = 364.172 \mathrm{kg\cdot m^2}\) Child 2: \(I_2 = 20.412 \mathrm{kg} \times (3.6576 \mathrm{m})^2 = 273.018 \mathrm{kg\cdot m^2}\) Child 3: \(I_3 = 36.288 \mathrm{kg} \times (3.6576 \mathrm{m})^2 = 484.803 \mathrm{kg\cdot m^2}\)

Step 4: Calculate the total moment of inertia

Finally, we can find the total moment of inertia by adding the moments of inertia for each child. Total moment of inertia: \(I_{total} = I_1 + I_2 + I_3 = 364.172 + 273.018 + 484.803 = 1121.993 \mathrm{kg\cdot m^2}\) So, the moment of inertia for the three children sitting on the edge of the merry-go-round is approximately \(1121.993 \mathrm{kg \cdot m^2}\).

Key Concepts

Moment of Inertia

The moment of inertia is a property of any object that describes how difficult it is to change its rotational motion about an axis. Think of it as the rotational equivalent of mass in linear motion. For a point mass, the moment of inertia is calculated by the product of the mass and the square of the distance to the rotation axis, represented by the formula:
\( I = m \times r^2 \)
In the context of our merry-go-round problem, each child represents a point mass located at a distance from the axis of rotation (the center). The further the child is from the center, the higher the moment of inertia. It's essential to understand that moment of inertia is additive for multiple point masses, which means you can simply add up the moment of inertia of each child to get the total moment of inertia of the system.

Physics Problem Solving

Problem solving in physics often follows a systematic approach that involves identifying the problem, gathering information, forming a plan, applying physics concepts and equations, and finally, solving the problem. In our example, we first identified the aim, which is to calculate the moment of inertia for children on a merry-go-round. We gathered information on the weights of the children and the radius of the merry-go-round. Then we formed a plan by determining the steps needed, including unit conversions and the application of the moment of inertia formula. Understanding the underlying concepts like moment of inertia and applying it alongside mathematical operations is key to solving physics problems effectively.

Unit Conversion

Unit conversion is crucial in physics to ensure all quantities are expressed in compatible units before performing calculations. In our example, we converted weights from pounds to kilograms and the radius from feet to meters, using the conversion factors \(1 lb = 0.453592 kg\) and \(1 ft = 0.3048 m\) respectively. Mastering unit conversions is often the first step in solving a physics problem since the correctness of the final result heavily depends on using the correct units throughout the calculation. This requires familiarity with the metric system, which is the standard system of measurement in physics.