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Chapter 9: Problem 59

Two disks are rotating about the same axis. Disk A has a moment of inertia of \(3.4 \mathrm{kg} \cdot \mathrm{m}^{2}\) and an angular velocity of \(+7.2 \mathrm{rad} / \mathrm{s} .\) Disk \(\mathrm{B}\) is rotating with an angular velocity of \(-9.8 \mathrm{rad} / \mathrm{s} .\) The two disks are then linked together without the aid of any external torques, so that they rotate as a single unit with an angular velocity of -2.4 rad/s. The axis of rotation for this unit is the same as that for the separate disks. What is the moment of inertia of disk B?

Short Answer

Expert verified
Moment of inertia of disk B is approximately 4.41 kg鈰卪虏.

Step by step solution

01

Understand the Principle of Conservation of Angular Momentum

When two disks are linked together and no external torques are acting on them, the total angular momentum before linking is equal to the total angular momentum after linking. This is based on the conservation of angular momentum principle.
02

Define Angular Momentum

The angular momentum of a rotating object is given by the product of its moment of inertia and its angular velocity: \( L = I \cdot \omega \), where \( L \) is the angular momentum, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.
03

Write the Equation for Total Angular Momentum Before Linking

Before the disks are linked, the total angular momentum is the sum of the angular momenta of disks A and B. So, \(L_{ ext{total before}} = I_A \cdot \omega_A + I_B \cdot \omega_B\), where \(I_A = 3.4\, \text{kg} \cdot \text{m}^2\), \(\omega_A = 7.2\, \text{rad/s}\), and \(\omega_B = -9.8\, \text{rad/s}\).
04

Write the Equation for Total Angular Momentum After Linking

After the disks are linked, the total angular momentum becomes \(L_{ ext{total after}} = (I_A + I_B) \cdot \omega_{ ext{after}}\), where \(\omega_{ ext{after}} = -2.4\, \text{rad/s}\).
05

Set Total Angular Momentum Before Equal to After

According to the conservation of angular momentum, \(L_{ ext{total before}} = L_{ ext{total after}}\). Set up the equation: \[ I_A \cdot \omega_A + I_B \cdot \omega_B = (I_A + I_B) \cdot \omega_{ ext{after}} \]
06

Substitute Known Values

Substitute the known values into the equation: \[ 3.4 \cdot 7.2 + I_B \cdot (-9.8) = (3.4 + I_B) \cdot (-2.4) \]
07

Solve for \(I_B\)

Rearrange the equation to solve for \(I_B\). Simplify and isolate \(I_B\):\[ 24.48 - 9.8 I_B = -8.16 - 2.4 I_B \]Combine like terms:\[ 24.48 + 8.16 = 9.8 I_B - 2.4 I_B \]\[ 32.64 = 7.4 I_B \]Divide through by 7.4 to isolate \(I_B\):\[ I_B = \frac{32.64}{7.4} \approx 4.41\, \text{kg} \cdot \text{m}^2 \]
08

Finalize the Answer

The moment of inertia of disk B is found to be approximately \(4.41\, \text{kg} \cdot \text{m}^2\).

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Key Concepts

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

Moment of Inertia
Moment of inertia is essentially the rotational equivalent of mass in linear motion. It tells us how resistant an object is to changes in its rotational state about a specific axis. Imagine trying to spin a lightweight frisbee compared to a heavy wheel; the wheel is much harder to spin due to its higher moment of inertia.
  • For a simple object like a solid disk or sphere, the moment of inertia depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. Mathematically, it is represented as: \( I = \sum m \, r^2 \), where \( m \) is the mass and \( r \) is the distance from the axis.
  • In our example, disk A has a moment of inertia of \( 3.4 \, \text{kg} \cdot \text{m}^2 \). This value was key in determining the angular behavior when the disks were combined.

Moments of inertia can vary significantly between different shapes and mass distributions, making it a crucial quantity in rotational dynamics. In solving for the moment of inertia of disk B, we needed to understand how its own rotational characteristics contributed to the combined system with disk A.
Angular Velocity
Angular velocity represents how fast an object is rotating. It shows the rate at which the angular position of an object changes with time. Its direction indicates whether the rotation is clockwise or counterclockwise.
  • Measured in radians per second (\( \text{rad/s} \)), angular velocity is symbolized by \( \omega \). In the context of this exercise, disk A was spinning at \( +7.2 \, \text{rad/s} \) while disk B was rotating at \( -9.8 \, \text{rad/s} \).
  • The positive or negative sign indicates the direction of rotation. Positive is typically taken as counterclockwise, and negative as clockwise.
In our exercise, after the disks were joined, the combined system had an angular velocity of \( -2.4 \, \text{rad/s} \). Understanding angular velocity helps us in comprehending how fast and in which direction an object or combined system will rotate.
Rotational Dynamics
Rotational dynamics is akin to linear dynamics but applies to rotating objects. It deals with the motion of rotating bodies and the forces and torques that act upon them. This field helps us understand the behavior of objects from a spinning top to planetary systems.
  • When analyzing rotational systems, key concepts include torque (the rotational counterpart of force), angular acceleration, and angular momentum.
  • The principle of conservation of angular momentum states that when no external torques are present, the total angular momentum of a closed system remains constant. This is exactly what we applied in solving the given problem, as the two disks rotated together without external influence.

In this exercise, understanding rotational dynamics allowed us to set up an equation equating the total angular momentum before and after the disks were connected. This straightforward yet powerful principle helped us solve for the moment of inertia of disk B, revealing the beauty and simplicity behind complex rotational systems.

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Most popular questions from this chapter

Two spheres are each rotating at an angular speed of \(24 \mathrm{rad} / \mathrm{s}\) about axes that pass through their centers. Each has a radius of \(0.20 \mathrm{m}\) and a mass of 1.5 kg. However, as the figure shows, one is solid and the other is a thin-walled spherical shell. Suddenly, a net external torque due to friction (magnitude \(=0.12 \mathrm{N} \cdot \mathrm{m}\) ) begins to act on each sphere and slows the motion down. Concepts: (i) Which sphere has the greater moment of inertia and why? (ii) Which sphere has the angular acceleration (a deceleration) with the smaller magnitude? (iii) Which sphere takes a longer time to come to a halt? Calculations: How long does it take each sphere to come to a halt?

A 9.75-m ladder with a mass of \(23.2 \mathrm{kg}\) lies flat on the ground. A painter grabs the top end of the ladder and pulls straight upward with a force of \(245 \mathrm{N}\). At the instant the top of the ladder leaves the ground, the ladder experiences an angular acceleration of \(1.80 \mathrm{rad} / \mathrm{s}^{2}\) about an axis passing through the bottom end of the ladder. The ladder's center of gravity lies halfway between the top and bottom ends. (a) What is the net torque acting on the ladder? (b) What is the ladder's moment of inertia?

A Ride Inside a Tractor Tire. You and your friends plan to roll down a hill on the inside of 600 -pound tractor tire (diameter \(D=1.80 \mathrm{m}\) ). The hill is inclined at an angle of \(25.0^{\circ}\) and you initially plan to start from a distance \(L=100 \mathrm{m}\) up the hill, but decide to first check whether it will be safe. (a) Assuming the masses of the tire and your 105 -pound body are concentrated at the outer rim of a thin-walled cylinder/hoop, what is the effective acceleration your body experiences at the bottom of the hill where your angular speed is greatest, i.e., how many "g's" will you experience? Assuming the human body can withstand a g-force of \(8.00 \mathrm{g}\) 's \(\left(1 \mathrm{g}=9.80 \mathrm{m} / \mathrm{s}^{2}\right),\) is it safe to make the ride from \(100 \mathrm{m}\) up the hill? (b) What is the maximum starting distance \(\left(L_{\text {max }}\right)\) up the hill that is safe?

A 15.0 -m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is \(0.44 \mathrm{kg} \cdot \mathrm{m}^{2},\) and its radius is \(0.160 \mathrm{m} .\) When the reel is turning, friction at the axle exerts a torque of magnitude \(3.40 \mathrm{N} \cdot \mathrm{m}\) on the reel. If the hose is pulled so that the tension in it remains a constant \(25.0 \mathrm{N},\) how long does it take to completely unwind the hose from the reel? Neglect the mass and thickness of the hose on the reel, and assume that the hose unwinds without slipping.

The drawing shows a rectangular piece of wood. The forces applied to corners \(\mathrm{B}\) and \(\mathrm{D}\) have the same magnitude of \(12 \mathrm{N}\) and are directed parallel to the long and short sides of the rectangle. The long side of the rectangle is twice as long as the short side. An axis of rotation is shown perpendicular to the plane of the rectangle at its center. A third force (not shown in the drawing) is applied to corner A, directed along the short side of the rectangle (either toward \(\mathrm{B}\) or away from \(\mathrm{B}\) ), such that the piece of wood is at equilibrium. Find the magnitude and direction of the force applied to corner A.
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