Problem 37 - College Physics

Question

You need to design an industrial turntable that is 60.0 cm in diameter and has a kinetic energy of 0.250 J when turning at 45.0 rpm v (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 approximately 0.0225 kg·m². (b) The mass of the turntable must be approximately 0.50 kg.

Step 1: Convert Diameter to Radius

The diameter of the turntable is given as 60.0 cm. The radius is half the diameter, so:\[ r = \frac{60.0 \text{ cm}}{2} = 30.0 \text{ cm} = 0.30 \text{ m} \](Note: Converted to meters for consistency with SI units.)

Step 2: Convert Rotational Speed to Radians per Second

The speed of the turntable is given in rpm (revolutions per minute), and we need to convert it to radians per second:\[ 45.0 \text{ rpm} = 45.0 \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 4.71 \text{ rad/s} \]

Step 3: Use Kinetic Energy Formula for Rotational Motion

The kinetic energy \( KE \) of a rotating object is given by:\[ KE = \frac{1}{2} I \omega^2 \]where \( I \) is the moment of inertia and \( \omega \) is the angular velocity in radians per second. We know \( KE = 0.250 \text{ J} \) and \( \omega = 4.71 \text{ rad/s} \), so:\[ 0.250 = \frac{1}{2} I (4.71)^2 \]

Step 4: Solve for Moment of Inertia

Rearrange the kinetic energy equation to solve for \( I \):\[ I = \frac{2 \times 0.250}{(4.71)^2} \approx 0.0225 \text{ kg} \cdot \text{m}^2 \]

Step 5: Use Disk Moment of Inertia Formula to Find Mass

For a uniform solid disk, the moment of inertia \( I \) is given by:\[ I = \frac{1}{2} m r^2 \]We have \( I \approx 0.0225 \text{ kg} \cdot \text{m}^2 \) and \( r = 0.30 \text{ m} \). Solve for \( m \):\[ 0.0225 = \frac{1}{2} m (0.30)^2 \]\[ m = \frac{0.0225 \times 2}{0.09} \approx 0.50 \text{ kg} \]

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. When we deal with objects that rotate, like a turntable, this type of energy is specifically termed rotational kinetic energy. It depends on two main factors: the moment of inertia and the angular velocity of the object.
In our industrial turntable scenario, the kinetic energy is given as 0.250 Joules. To find the moment of inertia, we can use the formula for rotational kinetic energy:

• KE = \( \frac{1}{2} I \omega^2 \)

Here, \( I \) is the moment of inertia and \( \omega \) is the angular velocity. By rearranging the formula, you can easily deduce the moment of inertia if the other two values are known.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or spins around a particular axis. It is commonly measured in radians per second (rad/s). In this exercise, the turntable spins at a speed of 45.0 revolutions per minute (rpm). To use the rotational equations, we must convert this speed into radians per second.
We achieve this conversion by linking the number of revolutions to radians. Remember, one revolution equals \( 2\pi \) radians. Thus, the conversion can be done using the relation:

• \( 45.0 \text{ rpm} = 45.0 \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 4.71 \text{ rad/s} \)

With the angular velocity in rad/s, you have the necessary parameter to compute other rotational dynamics such as kinetic energy or moment of inertia.

Uniform Solid Disk

A uniform solid disk is a perfect model for objects like industrial turntables, where mass is distributed evenly. The moment of inertia for a disk is determined using its mass focused at a certain distance from the axis of rotation, which affects how it spins.
For a solid disk of mass \( m \) and radius \( r \), the formula for moment of inertia is:

• \( I = \frac{1}{2} m r^2 \)

This equation tells us that both the mass and the square of the radius directly influence the moment of inertia. Thus, if the mass or the radius increases, the disk’s resistance to rotational acceleration increases. Solving this equation with known values allows us to determine the mass of the disk required to achieve a decided kinetic energy.

Rotation Axis

The rotation axis is the straight line around which the object spins. For a turntable, this would be a vertical line passing through its center. The distribution of mass around this axis determines the object's inertia, dictating how easily or difficultly it can start or stop spinning.
When manufacturers design rotational devices like the industrial turntable in this exercise, they must consider the axis of rotation to predict how the disk will behave under different applied forces. This understanding is crucial since it fundamentally influences the moment of inertia:

• Ensures balance by evenly distributing the mass.
• Influences the stability of the rotation.

Controlling these factors provides smoother and safer operation for practical applications such as industrial machinery, where precision and reliability are paramount.