Question

A chef spins a disk of pizza dough over her head, giving it an angular speed of \(7.2 \mathrm{rad} / \mathrm{s}\). If the moment of inertia of the pizza dough is \(6.3 \times 10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2}\), what is its rotational kinetic energy? (Assume that the disk of dough is uniform.)

Short answer

The rotational kinetic energy is \( 1.63 \times 10^{-4} \mathrm{~J} \).

Step-by-step solution

Understand the formula for rotational kinetic energy

The formula for rotational kinetic energy is given by: \( K = \frac{1}{2} I \omega^2 \), where \( K \) is the rotational kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular speed.

Identify the given values

From the problem statement, we know that the angular speed \( \omega = 7.2 \mathrm{rad} / \mathrm{s} \) and the moment of inertia \( I = 6.3 \times 10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2} \).

Substitute the values into the formula

Plug the given values into the equation for rotational kinetic energy: \( K = \frac{1}{2} \times 6.3 \times 10^{-6} \times (7.2)^2 \).

Calculate the rotational kinetic energy

First, calculate \( (7.2)^2 = 51.84 \). Then multiply by the moment of inertia: \( 6.3 \times 10^{-6} \times 51.84 = 3.26792 \times 10^{-4} \). Finally, divide by 2: \( K = \frac{1}{2} \times 3.26792 \times 10^{-4} = 1.63396 \times 10^{-4} \mathrm{~J} \).

Present the final answer

The rotational kinetic energy of the pizza dough is \( 1.63 \times 10^{-4} \mathrm{~J} \).

Key concepts

Angular Speed

Angular speed is an important concept when examining objects that rotate or spin. It is the rate at which an object turns around an axis and is commonly measured in radians per second (rad/s). Angular speed is akin to linear speed but for rotating systems.
Imagine a chef spinning a disk of pizza dough. The dough's angular speed is how quickly it is rotating as it spins in the air. The faster the pizza spins, the higher its angular speed. In systematic terms, angular speed is denoted by the Greek letter omega (\( \omega \)).To relate it to physical systems, angular speed is affected by how much force you apply to rotate an object and can change over time due to interactions like friction or applied torques. Understanding angular speed helps in analyzing how fast something spins and is foundational for calculating rotational kinetic energy.

Moment of Inertia

The moment of inertia is a crucial measure in rotational dynamics. It describes how the mass is distributed in relation to the axis of rotation, affecting how easy or difficult it is to spin the object. Think of it as a rotational equivalent to mass for linear motion.
For the spinning pizza dough, the moment of inertia (\( I \)) determines how the mass of the dough is spread out from its center. When mass is further from the axis, the moment of inertia is greater, making it harder to initiate or alter its rotation. In this case, a uniform disk model simplifies the calculation.The moment of inertia depends on both the mass and the shape of an object. Many common geometric shapes have predetermined formulas to calculate their moment of inertia due to their symmetrical properties. Knowing this helps predict how rotational forces impact the motion of objects.

Physics Calculations

Physics calculations for rotational dynamics involve understanding and applying several foundational formulas. When dealing with rotating objects, key variables such as angular speed and moment of inertia become part of larger calculations.
To find rotational kinetic energy, we use the formula:\[ K = \frac{1}{2} I \omega^2 \]This relates three elements:

• \( K \) for rotational kinetic energy, indicating the energy due to rotation.
• \( I \) for moment of inertia, considering how mass is distributed.
• \( \omega \) for angular speed, indicating the rate of rotation.

Substitute the known values to perform the calculation, as seen in our exercise. Start with the provided angular speed and moment of inertia values, square the angular speed, multiply by the moment of inertia, and finally, adjust with the one-half factor to solve for \( K \).
Being precise with these calculations is important as they represent real-world energy transformations and can describe complex systems. It simplifies understanding how rotating suns, planets, or in our example, pizza dough behave in motion.