Question

. Calculate 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 of the ball is \( 1.312 \times 10^{-7} \text{kg} \cdot \text{m}^2/\text{s} \).

Step-by-step solution

Identify Given Values

We are given that the moment of inertia \( I \) of the ball is \( 1.6 \times 10^{-8} \, \text{kg} \cdot \text{m}^2 \) and the angular speed \( \omega \) is \( 8.2 \, \text{rad/s} \).

Understand the Angular Momentum Formula

The angular momentum \( L \) of a rotating object can be calculated using the formula \( L = I \cdot \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

Substitute Given Values into the Formula

Substitute the values of \( I \) and \( \omega \) into the formula: \[ L = (1.6 \times 10^{-8} \, \text{kg} \cdot \text{m}^2) \times (8.2 \, \text{rad/s}) \]

Calculate the Angular Momentum

Perform the multiplication to find \( L \): \[ L = 1.312 \times 10^{-7} \, \text{kg} \cdot \text{m}^2/\text{s} \]

Key concepts

Moment of Inertia

The moment of inertia is an intrinsic property of rotating objects that depends on the mass distribution around the axis of rotation. It plays a similar role to mass in linear motion, essentially acting as the "rotational mass" that resists changes to the object's state of rotational motion.
- It is often denoted by the symbol \( I \). - The formula for calculating the moment of inertia varies depending on the shape and distribution of mass within an object. For instance, a solid sphere, hollow cylinder, or disc all have different formulas.In the given example, the ball has a moment of inertia value of \( 1.6 \times 10^{-8} \, \text{kg} \cdot \text{m}^2 \). This small value indicates that the ball will easily start to rotate, as the mass is concentrated closer to the axis of rotation. Understanding the moment of inertia allows us to predict how different parts of an object will react to applied forces, making it crucial in the study of dynamics and design of rotating systems.

Angular Velocity

Angular velocity refers to how fast an object rotates or revolves around an axis. It indicates the rate of change of angular position of the object and is typically measured in radians per second (rad/s). Unlike linear velocity, which addresses straight-line motion, angular velocity concerns circular or rotational motion.
- The symbol for angular velocity is usually \( \omega \), pronounced as "omega."- One full revolution corresponds to \( 2\pi \) radians, so an angular velocity of \( \omega \) radians per second means the object covers \( \omega \) radians in each second.In the problem presented, the ball possesses an angular velocity of \( 8.2 \, \text{rad/s} \). This means the ball is spinning at a rate such that every second, its angular position changes by \( 8.2 \) radians. Knowing the angular velocity is essential to calculating other related quantities, such as angular momentum.

Rotational Motion

Rotational motion is the movement of an object around a central axis or point. This can be seen in various real-world examples, such as Earth's rotation around its axis or a wheel spinning on a car. The principles governing this type of motion are essential in fields like engineering, physics, and robotics.
- The primary quantities involved in rotational motion include angular displacement, angular velocity, angular acceleration, and angular momentum. - A fundamental relationship in rotational motion is Newton's second law for rotation, which states that the net external torque acting on a body is equal to the rate of change of its angular momentum. In the exercise, we combine two aspects of rotational motion—moment of inertia and angular velocity—to find the ball's angular momentum. This concept illustrates how different physical properties interrelate under rotational dynamic conditions, providing insight into both theoretical and applied mechanics.