Question

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

The rotational kinetic energy is approximately 3.26 x 10^-4 J.

Step-by-step solution

Understanding the Problem

We need to find the rotational kinetic energy of a spinning disk of pizza dough. It is given an angular speed (\(\omega\)) of 7.2 rad/s, and its moment of inertia (\(I\)) is \(6.3 \times 10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2}\). We will use the formula for rotational kinetic energy \(K = \frac{1}{2} I \omega^2\).

Recall the Rotational Kinetic Energy Formula

The formula for the rotational kinetic energy (\(K\)) of a spinning object is given by:\[ K = \frac{1}{2} I \omega^2 \]where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

Substitute Known Values into the Formula

Substitute the given values into the formula:\[ K = \frac{1}{2} \times 6.3 \times 10^{-6} \times (7.2)^2 \]

Calculate \(\omega^2\)

First, calculate \(\omega^2\):\(\omega^2 = (7.2)^2 = 51.84\).

Calculate the Rotational Kinetic Energy

Now substitute \(\omega^2 = 51.84\) into the formula:\[ K = \frac{1}{2} \times 6.3 \times 10^{-6} \times 51.84 \]\[ K = 3.25872 \times 10^{-4} \mathrm{~J}\]

Simplify the Result

The rotational kinetic energy of the spinning disk of pizza dough simplifies to approximately \(3.26 \times 10^{-4} \mathrm{~J}\).

Key concepts

Angular Speed

Angular speed is a crucial concept in understanding rotational motion. It refers to the rate at which an object rotates or spins around a specific axis. In simpler terms, it's how quickly something is spinning. Angular speed is expressed in radians per second (rad/s), which might sound complicated but it's just a measure of how much of a circle something travels in a second.
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

Moment of inertia is akin to mass in linear motion but for rotational motion. It measures an object's resistance to change in its rotational motion. Think of it as how difficult it is to start or stop spinning an object. This depends on how the object's mass is distributed relative to the axis of rotation.
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

Solving physics problems might seem daunting, but breaking them down into smaller steps can help. Here's a basic approach to tackling a problem like this:

• **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.

This structured approach not only simplifies solving the problem but also enhances a deeper understanding of the underlying concepts like angular speed and moment of inertia.

Kinetic Energy Formula

The kinetic energy formula is essential in physics to calculate the energy an object has due to its motion. For rotational motion, we specifically use the rotational 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.