Chapter 8: Problem 30
Calculate 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
Step by step solution
Understanding the Problem
Recall the Rotational Kinetic Energy Formula
Substitute Known Values into the Formula
Calculate \(\omega^2\)
Calculate the Rotational Kinetic Energy
Simplify the Result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Speed
In the problem, the pizza dough disk spins at an angular speed of 7.2 rad/s. This helps us grasp how swift the spinning action is. The faster the object spins, the larger its angular speed. Understanding angular speed is essential as it links together with other factors like moment of inertia to determine the rotational kinetic energy.
Moment of Inertia
In the exercise, the moment of inertia for the pizza dough is given as \(6.3 \times 10^{-6}\, \mathrm{kg}\cdot \mathrm{m}^2\). This small value indicates that the dough is quite easy to spin due to its uniform shape and light mass. Knowing the moment of inertia is vital because it directly affects the object's rotational kinetic energy, allowing us to solve physics problems involving spinning objects more effectively.
Physics Problem Solving
- **Understand the Problem:** Identify what's asked, and note given values like angular speed and moment of inertia.
- **Choose the Right Formulas:** Use formulas suitable for the problem, such as the rotational kinetic energy formula.
- **Substitute and Solve:** Plug the known values into the formula and perform calculations step-by-step.
- **Interpret the Results:** Understand what the result means in the context of the problem.
Kinetic Energy Formula
\[ K = \frac{1}{2} I \omega^2 \]
where \(K\) is rotational kinetic energy, \(I\) is the moment of inertia, and \(\omega\) is the angular speed.
This formula is simple yet powerful. It combines the moment of inertia and angular speed to reveal how much energy is stored in a spinning object. Using this formula, we calculated the rotational kinetic energy of the pizza dough. By substituting the given values and following through with calculations, we found it to be roughly \(3.26 \times 10^{-4}\, \mathrm{J}\), demonstrating how energy concepts apply to everyday scenarios.