Problem 44 - University Physics with Modern Physics

Question

You need to design an industrial turntable that is 60.0 \(\mathrm{cm}\) in diameter and has a kinetic energy of 0.250 \(\mathrm{J}\) when turning at 45.0 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) . (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

Step-by-Step Solution

Answer

(a) The moment of inertia is 0.0225 kg⋅m². (b) The mass is 0.5 kg.

Step 1: Convert diameter to radius

The diameter of the turntable is given as 60.0 cm. To find the radius, divide the diameter by 2.\[ r = \frac{d}{2} = \frac{60.0 \text{ cm}}{2} = 30.0 \text{ cm} = 0.30 \text{ m} \]

Step 2: Calculate angular velocity

Convert the rotational speed from revolutions per minute to radians per second. Use the conversion factor \(2\pi \text{ radians} = 1 \text{ revolution}\) and \(1 \text{ min} = 60 \text{ s}\).\[ \omega = 45.0 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{1 \text{rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{45.0 \times 2\pi}{60} \text{ rad/s} = 4.712 \text{ rad/s} \]

Step 3: Relate kinetic energy to moment of inertia

The kinetic energy \(K\) of a rotating object is given by the equation:\[ K = \frac{1}{2} I \omega^2 \]where \(I\) is the moment of inertia, and \(\omega\) is the angular velocity. We need to solve for \(I\).\[ 0.250 = \frac{1}{2} I (4.712)^2 \]

Step 4: Solve for moment of inertia

Rearrange the equation from Step 3 to solve for \(I\):\[ I = \frac{2K}{\omega^2} = \frac{2 \times 0.250}{(4.712)^2} \approx \frac{0.5}{22.206} \approx 0.0225 \text{ kg} \cdot \text{m}^2 \]

Step 5: Use disk formula for moment of inertia

The moment of inertia \(I\) for a solid disk is given by:\[ I = \frac{1}{2} m r^2 \]where \(m\) is the mass and \(r\) is the radius. Substitute \(I = 0.0225 \text{ kg} \cdot \text{m}^2\) and \(r = 0.30 \text{ m}\) to solve for \(m\).

Step 6: Solve for mass of the disk

Using the formula from Step 5, rearrange to find mass \(m\):\[ m = \frac{2I}{r^2} = \frac{2 \times 0.0225}{(0.30)^2} = \frac{0.045}{0.09} = 0.5 \text{ kg} \]

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the context of rotation, this energy depends on the object's moment of inertia and its angular velocity. The equation for rotational kinetic energy is given by \[ K = \frac{1}{2} I \omega^2 \] where \(K\) is the kinetic energy, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity expressed in radians per second. To calculate this energy, you first need the correct measure of \(\omega\) by converting from revolutions per minute to radians per second using the conversion \[ \text{1 revolution} = 2\pi \text{ radians}. \]Determining kinetic energy accurately involves both precise measurements and calculations, as even slight variations in angular velocity or mass distribution can lead to significant changes in energy.

Rotational Motion

Rotational motion is the movement of an object in a circular path around a central axis. Every point of an object in rotational motion moves in a circle around the axis, with each point maintaining a constant radial distance from it. A solid disk, such as a turntable, illustrates this concept neatly. The speed of the rotating object is measured in angular units, like radians per second, because this provides a direct way to express the rotational speed. The angular velocity, denoted by \(\omega\), ties to how fast the object spins and is calculated by \[ \omega = \frac{\text{angle rotated (in radians)}}{\text{time (in seconds)}}. \]Understanding rotational motion is crucial as it lays the foundation for analyzing objects' stability, energy levels, and forces involved in a rotating body, all of which are essential for engineering and physics problems.

Solid Disk

A solid disk is a common geometric object in physics and engineering, characterized by its uniform mass distribution. When discussing the moment of inertia, a critical factor is how an object's mass is spread relative to the axis of rotation. For a solid disk, the moment of inertia \(I\) is determined by the formula:\[ I = \frac{1}{2} m r^2 \]where \(m\) is the disk's mass and \(r\) is its radius. This equation highlights that the moment of inertia depends on both how heavy the disk is and how far its mass is distributed from the center.By understanding this concept, one can calculate the amount of torque needed for a specific angular acceleration or, conversely, the resulting angular velocity when subjected to a given torque. The computation of a solid disk's moment of inertia is essential for applications ranging from industrial machinery to simple turntables.