Question

The moment of inertia of a ball is \(1.6 \times 10^{-8} \mathrm{~kg} \cdot \mathrm{m}^{2}\). If the ball spins with an angular speed of \(8.2 \mathrm{rad} / \mathrm{s}\), what is its angular momentum?

Short answer

The angular momentum is \(1.312 \times 10^{-7} \mathrm{~kg} \cdot \mathrm{m}^{2}/s\).

Step-by-step solution

Identify Known Values

We are given the following values:- Moment of inertia, \( I = 1.6 \times 10^{-8} \mathrm{~kg} \cdot \mathrm{m}^{2} \)- Angular speed, \( \omega = 8.2 \mathrm{~rad/s} \).

Recall Angular Momentum Formula

The formula for angular momentum \( L \) of a rotating object is:\[ L = I \times \omega \]where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

Substitute Known Values into Formula

Insert the known values into the angular momentum formula:\[ L = (1.6 \times 10^{-8} \mathrm{~kg} \cdot \mathrm{m}^{2}) \times 8.2 \mathrm{~rad/s} \].

Calculate Angular Momentum

Perform the multiplication to calculate the angular momentum:\[ L = 1.6 \times 8.2 \times 10^{-8} \mathrm{~kg} \cdot \mathrm{m}^{2}/\mathrm{s} \].First, multiply 1.6 by 8.2 which gives 13.12. Then,\[ L = 13.12 \times 10^{-8} \mathrm{~kg} \cdot \mathrm{m}^{2}/s = 1.312 \times 10^{-7} \mathrm{~kg} \cdot \mathrm{m}^{2}/s \].

Key concepts

Understanding Moment of Inertia

Moment of inertia is like a measure of how hard it is to change the rotational motion of an object. Think about it as the rotational cousin of mass. Just like mass resists changes in linear motion, moment of inertia resists changes in rotational motion.
This concept helps us understand how different objects with the same mass can rotate differently. For example, a solid disk and a ring of the same mass and size will have different moments of inertia. The reason is that their mass is distributed differently in relation to their axis of rotation.

• A solid shaft has a smaller moment of inertia compared to a hollow one of the same mass and size.
• The formula typically depends upon the mass of an object and how that mass is spread out from the axis of rotation.

In mathematical terms, it's symbolized as \( I \), and usually calculated based on the object’s shape and rotational axis. For practical problems like the one in the exercise, it's often given as a value to use in other calculations.

Exploring Rotational Motion

Rotational motion occurs when an object spins or revolves around a central axis. It is different from linear motion, where objects move in a straight line. Imagine the Earth rotating on its axis, or a ballerina pirouetting; these are examples of rotational motion.
This type of motion is all around us and can be described using specific rotational quantities such as angular displacement, angular velocity, and angular acceleration, among others.

• Angular displacement refers to the angle through which an object rotates.
• Angular velocity, similar to speed in linear motion, indicates how fast the object is rotating.
• Angular acceleration describes how quickly the rotational speed changes.

In many physics problems, understanding rotational motion is essential because it helps us relate forces and movements in a way that can be analyzed, such as using the equations of rotational motion to solve for unknowns.

Grasping Angular Speed

Angular speed is a measure of how quickly an object is rotating. It tells us how many radians an object spins through per second, and it's an important part of understanding rotational dynamics. In essence, it's like the speedometer for spinning objects.
You can think of it as the equivalent of speed in the linear world, but it applies to how fast something spins rather than how fast it travels.

• It is usually denoted by the Greek letter \( \omega \) and commonly measured in radians per second (\( \mathrm{rad/s} \)).
• Calculating the angular speed could involve using the formula \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angular displacement and \( t \) is time.

In our original problem, we are given an angular speed of \( 8.2 \mathrm{rad/s} \), which means the ball completes 8.2 radians of rotation every second, which is key to solving for the angular momentum.