Chapter 9: Problem 40
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 1rev/min2. (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?
Short Answer
Step by step solution
Understand the Problem
Convert Angular Velocity
Relate Kinetic Energy to Moment of Inertia
Use Moment of Inertia Formula for Disk
Validate the Solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rotational Kinetic Energy
- \( K = \frac{1}{2} I \omega^2 \)
To find the rotational kinetic energy, you need to know the object's moment of inertia and its rate of rotation. In the exercise given, the turntable has a kinetic energy of 0.250 J while rotating. This helps us relate the moment of inertia directly to the kinetic energy and the given angular velocity through the equation.
Understanding this relationship is crucial when designing rotating objects such as shafts, wheels, and turntables, as it helps determine how much energy is needed for desired rotational speeds.
Uniform Solid Disk
For a uniform solid disk, the calculation of its moment of inertia becomes simplified. This is because such a disk's mass is not concentrated in any one area but is spread evenly throughout the entire disk.
The formula for the moment of inertia of a solid disk about an axis perpendicular to its center is given by:
- \( I = \frac{1}{2} m r^2 \)
Using this formula allows engineers and physicists to easily compute how the mass of the disk affects its rotational characteristics, important for dynamic systems analysis.
Angular Velocity
In practice, angular velocity is often given in revolutions per minute (rpm), but for calculations, it is usually converted to radians per second (rad/s). This conversion is done using the conversion factors:
- \(1 \text{ rev} = 2\pi \text{ rad} \)
- \(1 \text{ min} = 60 \text{ s} \)
Understanding and calculating angular velocity is vital because it directly influences the rotational kinetic energy an object will have, and is necessary for determining the dynamics of rotating systems.
Mass Calculation
In the case of a uniform solid disk, once you know the moment of inertia, you can rearrange its formula to find the mass:
- \(m = \frac{2I}{r^2} \)
In the exercise, by knowing the turntable's moment of inertia and its radius (0.30 m), we can calculate its mass, finding it to be approximately 0.50 kg.
This calculation not only aids in understanding the object's dynamics but also ensures that the right material and stability are maintained in the physical world, which is essential for designing functional and safe equipment.