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Chapter 11: Problem 46

The rotational inertia of a collapsing spinning star drops to \(\frac{1}{3}\) its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy?

Short Answer

Expert verified
The ratio of new to initial kinetic energy is 3:1.

Step by step solution

01

Understanding Rotational Inertia

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotation. Denoted as \( I \), it is given that the star's rotational inertia drops to \( \frac{1}{3} \) of its initial value.
02

Analyze Conservation of Angular Momentum

The law of conservation of angular momentum states that if no external torque acts on a system, its angular momentum \( L \) remains constant. Angular momentum is given by \( L = I \cdot \omega \), where \( \omega \) is the angular velocity. If \( I \) changes, \( \omega \) must change to keep \( L \) constant. If \( I \) becomes \( \frac{1}{3}I_0 \), then to conserve angular momentum, \( \omega \) must change such that \( \frac{1}{3}I_0 \cdot \omega = I_0 \cdot \omega_0 \). Thus, \( \omega = 3\omega_0 \).
03

Finding Rotational Kinetic Energy

The kinetic energy due to rotation is given by \( KE_{rot} = \frac{1}{2} I \omega^2 \). Initially, this is \( KE_{rot, initial} = \frac{1}{2} I_0 \omega_0^2 \). After the rotational inertia changes, it becomes \( KE_{rot, new} = \frac{1}{2} \left(\frac{1}{3} I_0\right) (3\omega_0)^2 \). Simplifying this gives \( KE_{rot, new} = \frac{1}{2} \times \frac{1}{3} I_0 \times 9 \omega_0^2 = \frac{3}{2} I_0 \omega_0^2 \).
04

Calculating the Ratio of Kinetic Energies

To find the ratio of the new rotational kinetic energy to the initial kinetic energy, divide \( KE_{rot, new} \) by \( KE_{rot, initial} \): \( \frac{KE_{rot, new}}{KE_{rot, initial}} = \frac{\frac{3}{2} I_0 \omega_0^2}{\frac{1}{2} I_0 \omega_0^2} = 3 \).

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Key Concepts

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

Rotational Inertia
Rotational inertia is akin to the concept of mass in linear motion. Imagine how a solid object spins. Rotational inertia determines how much an object resists any change in this spinning motion. For instance, a figure skater pulling their arms in can spin faster because of their reduced rotational inertia. In this exercise, when the star's rotational inertia decreases to one-third of its initial value, it directly impacts how the star's spinning motion reacts.
  • Rotational Inertia is denoted by the letter \( I \).
  • It is a crucial factor in determining the spin rate of celestial objects like stars.
  • A lower rotational inertia means the object can spin faster.
Understanding and calculating rotational inertia is vital when examining astronomical phenomena.
Conservation of Angular Momentum
Angular momentum is preserved when there is no external torque, meaning the spinning remains efficient. It's much like when a dancer spins, keeping their balance and speed steady by controlling their arms' position. For our collapsing star, the conservation law plays a crucial role in explaining how angular velocity compensates for the decrease in rotational inertia. This concept tells us that as the star's inertia drops to one-third, its angular velocity must triple to maintain angular momentum.
  • Angular momentum \( L \) is given by the formula \( L = I \cdot \omega \).
  • If \( I \) decreases, \( \omega \) must increase if no external forces are involved.
  • Without conservation, the star would dramatically change its motion, which doesn鈥檛 typically happen in the universe.
Hence, understanding this principle helps us predict changes in rotation without measuring spin directly.
Angular Velocity
Angular velocity refers to how fast an object rotates or spins around a point or axis. In simpler terms, think of it as the number of spins an ice skater makes per second. When our star's rotational inertia decreases, to conserve angular momentum, the angular velocity has to increase. As explained earlier, it triples in this scenario.
  • Angular velocity \( \omega \) reflects the speed of rotation.
  • It is directly influenced by changes in rotational inertia.
  • In most physics problems, it鈥檚 crucial to calculate \( \omega \) to determine the kinetic energy.
Knowing how angular velocity changes can assist in understanding larger cosmic events affecting rotating celestial bodies.
Moment of Inertia
Sometimes used interchangeably with rotational inertia, the moment of inertia ( I ) greatly impacts rotational kinetic energy. It describes how mass is distributed relative to the axis of rotation. A narrower distribution, such as a core, allows for faster spins compared to a broad one like that of a flat disk. In our star's puzzle, a change in this key value demonstrates why its spinning behavior shifts noticeably.
  • The closer the mass is to the axis, the smaller the moment of inertia.
  • Moment of inertia impacts stability and speed during rotation.
  • It's also pivotal when calculating energy transformations in closed systems.
Understanding the moment of inertia's impact helps when delving deep into physics, especially in mechanics and rotational dynamics.

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Most popular questions from this chapter

A particle is acted on by two torques about the origin: \(\vec{\tau}_{1}\) has a magnitude of \(2.0 \mathrm{~N} \cdot \mathrm{m}\) and is directed in the positive direction of the \(x\) axis, and \(\vec{\tau}_{2}\) has a magnitude of \(4.0 \mathrm{~N} \cdot \mathrm{m}\) and is directed in the negative direction of the \(y\) axis. In unit-vector notation, find \(d \vec{\ell} / d t\), where \(\vec{\ell}\) is the angular momentum of the particle about the origin.

A body of radius \(R\) and mass \(m\) is rolling smoothly with speed \(v\) on a horizontal surface. It then rolls up a hill to a maximum height \(h\). (a) If \(h=3 v^{2} / 4 g\), what is the body's rotational inertia about the rotational axis through its center of mass? (b) What might the body be?

A uniform disk of mass \(10 \mathrm{~m}\) and radius \(3.0 r\) can rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of mass \(m\) and radius \(r\) lies on top of the larger disk, concentric with it. Initially the two disks rotate together with an angular velocity of \(20 \mathrm{rad} / \mathrm{s}\). Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their angular velocity about the center of the larger disk? (b) What is the ratio \(K / K_{0}\) of the new kinetic energy of the two-disk system to the system's initial kinetic energy?

A uniform solid ball rolls smoothly along a floor, then up a ramp inclined at \(15.0^{\circ} .\) It momentarily stops when it has rolled \(1.50 \mathrm{~m}\) along the ramp. What was its initial speed?

Force \(\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}-(3.0 \mathrm{~N}) \hat{\mathrm{k}}\) acts on a pebble with position vector \(\vec{r}=(0.50 \mathrm{~m}) \hat{\mathrm{j}}-(2.0 \mathrm{~m}) \hat{\mathrm{k}}\) relative to the origin. In unit-vector notation, what is the resulting torque on the pebble about (a) the origin and (b) the point \((2.0 \mathrm{~m}, 0,-3.0 \mathrm{~m})\) ?
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