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Chapter 12: Problem 55

What is the moment of inertia of a \(2.0 \mathrm{kg}, 20-\mathrm{cm}\) -diameter disk for rotation about an axis (a) through the center, and (b) through the edge of the disk?

Short Answer

Expert verified
The moment of inertia of the disk for rotation about an axis (a) through the center is \(0.01 \mathrm{kg\ m^2}\), and (b) through the edge of the disk is \(0.02 \mathrm{kg\ m^2}\).

Step by step solution

01

Determine the radius of the disc

The diameter of the disc is given as \(20 \mathrm{cm}\). The radius is half of the diameter. Thus, \(r = \frac{20}{2} = 10 \mathrm{cm}\). For convenience, convert to meters by dividing by 100. Therefore, \(r = 0.1 \mathrm{m}\).
02

Calculate as per part (a) of the exercise

The formula of moment of inertia of a disc being rotated about an axis through its center is given by \(I = 0.5mr^2\). Substituting given \(m = 2.0 \mathrm{kg}\) and \(r = 0.1 \mathrm{m}\), we get \(I = 0.5 * 2.0 * (0.1)^2 = 0.01 \mathrm{kg\ m^2}\).
03

Calculate as per part (b) of the exercise

The formula of moment of inertia of a disc being rotated about an axis through its edge is given by \(I = mr^2\). Substituting given \(m = 2.0 \mathrm{kg}\) and \(r = 0.1 \mathrm{m}\), we get \(I = 2.0 * (0.1)^2 = 0.02 \mathrm{kg\ m^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotation Axis
The rotation axis is essentially the line about which the object spins. It plays a key role in determining how the moment of inertia is calculated. When considering a disk, if the axis goes through the center (part a), then the moment of inertia becomes less compared to when the axis goes through the edge (part b). This is because the distribution of mass in relation to the axis affects inertia. The further the mass is from this axis, the larger the inertia.
  • Axis through the center: Inertia is minimized as mass is closest to the axis.
  • Axis through the edge: Inertia increases as mass is further from the axis, increasing resistance to rotation.
Understanding the rotation axis helps in visualizing how difficult it is to get an object spinning or to stop it.
Mass Distribution
How mass is spread out or distributed across an object influences its moment of inertia. For a disk, since mass is uniformly distributed, this impacts the inertia calculations significantly. Mass lying further from the rotation axis contributes more to the moment of inertia. This is because the force needed to rotate the mass increases with the distance from the axis. Hence, if the axis is at the center, masses have less impact compared to when the axis is at the edge.
  • Uniform distribution leads to predictable calculations for a standard geometry like a disk.
  • Placement of mass relative to axis significantly alters rotational resistance.
Grasp understanding of mass distribution to predict how objects behave under rotational forces effectively.
Disk Geometry
Understanding the geometry of the object is crucial for calculating the moment of inertia. Here, the geometry in question is that of a disk.Disks have a circular shape, which means their mass is equally spread out around the center point. This consistent shape leads to specific inertia formulas:
  • Central axis: \(I = \frac{1}{2}mr^2\)
  • Edge axis: \(I = mr^2\)
These formulas reflect how the circular geometry and mass distribution collectively affect the object鈥檚 rotational characteristics. Being aware of disk geometry aids in applying the right formula to predict rotational behavior accurately.
Radius Conversion
Calculating the moment of inertia requires an accurate measure of the radius. Often given as a diameter, the radius is simply half of the diameter. For practical calculations, converting this into meters is usually essential because standard physics formulas require it. In this exercise, you see:
  • Original diameter = 20 cm
  • Radius conversion = Diameter / 2 = 10 cm
  • Conversion to meters = Radius / 100 = 0.1 m
These conversions are necessary to match the units in which physical constants are defined, ensuring that computations are consistent and accurate. Always remember to convert measurements appropriately for seamless calculations and results.

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