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Chapter 10: Problem 79

(a) Show that the rotational inertia of a solid cylinder of mass \(M\) and radius \(R\) about its central axis is equal to the rotational inertia of a thin hoop of mass \(M\) and radius \(R / \sqrt{2}\) about its central axis. (b) Show that the rotational inertia \(I\) of any given body of mass \(M\) about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass \(M\) and a radius \(k\) given by $$ k=\sqrt{\frac{I}{M}} . $$ The radius \(k\) of the equivalent hoop is called the radius of gyration of the given body.

Short Answer

Expert verified
The cylinder and the hoop both have the same rotational inertia of \( \frac{1}{2}MR^2 \). The radius of gyration is \( k = \sqrt{\frac{I}{M}} \).

Step by step solution

01

Rotational Inertia of a Solid Cylinder

The rotational inertia (or moment of inertia) of a solid cylinder about its central axis is given by the formula \( I = \frac{1}{2}MR^2 \), where \( M \) is the mass of the cylinder and \( R \) is its radius.
02

Rotational Inertia of a Thin Hoop

The rotational inertia of a thin hoop of mass \( M \) and radius \( r \) about its central axis is given by \( I = Mr^2 \). In this case, since the radius of the hoop is specified as \( R/\sqrt{2} \), substitute this value into the formula to get \( I = M\left(\frac{R}{\sqrt{2}}\right)^2 = \frac{1}{2}MR^2 \).
03

Comparing Inertias of the Cylinder and Hoop

Both the solid cylinder and the hoop have the same moment of inertia, \( \frac{1}{2}MR^2 \), showing they have the same rotational inertia when the hoop鈥檚 radius is \( R/\sqrt{2} \).
04

Understanding the Radius of Gyration

The radius of gyration \( k \) for any given body is defined such that its rotational inertia \( I \) about any axis is equivalent to that of a hoop of the same mass \( M \) and radius \( k \).
05

Deriving the Radius of Gyration Formula

Given that \( I = Mk^2 \), solving for \( k \) gives \( k = \sqrt{\frac{I}{M}} \). This equation confirms the relationship and defines \( k \) as the radius of gyration.

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Key Concepts

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

Solid Cylinder
A solid cylinder is a common geometric shape encountered in physics, particularly when studying rotational dynamics. It features two circular bases connected by a curved surface. Understanding its rotational inertia helps us describe how it will behave when subjected to rotational forces.
The rotational inertia, specifically, is a measure of how much torque is needed to rotate the cylinder around its axis. For a solid cylinder with mass \( M \) and radius \( R \), the formula to calculate its rotational inertia is:
  • \( I = \frac{1}{2}MR^2 \)
This tells us the inertia depends on both the mass and the square of the radius, revealing how distribution of mass around the axis affects its resistance to rotation.
Thin Hoop
A thin hoop is another common geometric shape used in physics to understand rotational dynamics. Imagine a thin wire bent into a perfect circle; this is the thin hoop.
Its rotational inertia is crucial when it's rotating around its central axis. For a hoop with mass \( M \) and a given radius \( r \), the formula for rotational inertia is:
  • \( I = Mr^2 \)
In the given exercise, a thin hoop with a radius of \( \frac{R}{\sqrt{2}} \) is compared to a solid cylinder. The resulting rotational inertia for this specific hoop computation becomes identical to that of a solid cylinder:
  • \( I = M\left(\frac{R}{\sqrt{2}}\right)^2 = \frac{1}{2}MR^2 \)
Thus, when a hoop is designed with these dimensions, it demonstrates the same rotational resistance as a solid cylinder.
Radius of Gyration
The radius of gyration is a theoretical construct aimed at simplifying the understanding of an object's inertia. This concept gives a distance from the rotational axis where the entire mass of the object could be considered to be concentrated, while maintaining the same moment of inertia.
If an object of mass \( M \) has a rotational inertia \( I \) about a given axis, the radius of gyration, denoted by \( k \), is defined by the formula:
  • \( k = \sqrt{\frac{I}{M}} \)
This provides a valuable tool in physics for predicting the rotational behavior of bodies, combining mass and shape aspects into one easily understandable parameter.
Moment of Inertia
The moment of inertia is a fundamental concept in physics, denoting an object's resistance against rotational motion. It's directly analogous to mass in linear motion but focuses on rotation around a specific axis.
Each shape and distribution of mass will have a unique moment of inertia, calculated based on both its geometry and mass distribution.
For most objects, as the mass spreads away from the axis, the moment of inertia increases, requiring greater torque for rotation. In this way, it serves as a critical bridge between linear and rotational dynamics, making it an essential component of Newton鈥檚 laws applied to rotational systems.
Physics Education
Physics education plays a vital role in shaping our understanding of the natural world, including concepts like rotational dynamics. By exploring concepts such as moment of inertia and radius of gyration, learners gain insight into the rotational characteristics of objects.
Practical applications of these concepts include engineering, where they help in designing stable structures and machinery.
  • By introducing real-world problems, learners engage deeply with theory.
  • Understanding inertia aids in developing solutions for practical challenges.
Ultimately, physics education not only helps explain the physical phenomena but also equips students with skills to apply these ideas creatively in various fields.

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Most popular questions from this chapter

Two uniform solid spheres have the same mass of \(1.65 \mathrm{~kg}\), but one has a radius of \(0.226 \mathrm{~m}\) and the other has a radius of \(0.854 \mathrm{~m}\). Each can rotate about an axis through its center. (a) What is the magnitude \(\tau\) of the torque required to bring the smaller sphere from rest to an angular speed of \(317 \mathrm{rad} / \mathrm{s}\) in \(15.5 \mathrm{~s} ?(\mathrm{~b})\) What is the magnitude \(F\) of the force that must be applied tangentially at the sphere's equator to give that torque? What are the corresponding values of (c) \(\tau\) and (d) \(F\) for the larger sphere?

A disk rotates at constant angular acceleration, from angular position \(\theta_{1}=10.0 \mathrm{rad}\) to angular position \(\theta_{2}=70.0 \mathrm{rad}\) in \(6.00 \mathrm{~s}\). Its angular velocity at \(\theta_{2}\) is \(15.0 \mathrm{rad} / \mathrm{s}\). (a) What was its angular velocity at \(\theta_{1} ?\) (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph \(\theta\) versus time \(t\) and angular speed \(\omega\) versus \(t\) for the disk, from the beginning of the motion \((\operatorname{let} t=0\) then \()\).

At \(t=0\), a flywheel has an angular velocity of \(4.7 \mathrm{rad} / \mathrm{s}, \mathrm{a}\) constant angular acceleration of \(-0.25 \mathrm{rad} / \mathrm{s}^{2}\), and a reference line at \(\theta_{0}=0 .\) (a) Through what maximum angle \(\theta_{\max }\) will the reference line turn in the positive direction? What are the (b) first and (c) second times the reference line will be at \(\theta=\frac{1}{2} \theta_{\max } ?\) At what (d) negative time and (e) positive time will the reference line be at \(\theta=10.5\) rad? (f) Graph \(\theta\) versus \(t\), and indicate the answers to (a) through (e) on the graph.

Four particles, each of mass, \(0.20 \mathrm{~kg}\), are placed at the vertices of a square with sides of length \(0.50 \mathrm{~m}\). The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis \(A\) that passes through one of the particles. The body is released from rest with rod \(A B\) horizontal (Fig. \(10-61\) ). (a) What is the rotational inertia of the body about axis \(A ?\) (b) What is the angular speed of the body about axis \(A\) when rod \(A B\) swings through the vertical position?

A wheel, starting from rest, rotates with a constant angular acceleration of \(2.00 \mathrm{rad} / \mathrm{s}^{2}\). During a certain \(3.00 \mathrm{~s}\) interval, it turns through \(90.0\) rad. (a) What is the angular velocity of the wheel at the start of the \(3.00 \mathrm{~s}\) interval? (b) How long has the wheel been turning before the start of the \(3.00\) s interval?
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