/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 How \(I\) Scales. If we multiply... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 51

How \(I\) Scales. If we multiply all the design dimensions of an object by a scaling factor \(f,\) its volume and mass will be multiplied by \(f^{3}\) (a) By what factor will its moment of inertia be multiplied? (b) If a \(\frac{1}{48}-\) scale model has a rotational kinetic energy of \(2.5 \mathrm{J},\) what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

Short Answer

Expert verified
(a) The moment of inertia is multiplied by \(f^5\). (b) The full-scale object's kinetic energy is 2,548,039,680 J.

Step by step solution

01

Understanding Moment of Inertia Scaling

The moment of inertia (I) for an object is given by the formula \(I = k m r^2\), where \(k\) is the shape factor, \(m\) is the mass and \(r\) is a characteristic dimension of the object. When the dimensions of an object are scaled by a factor \(f\), its mass becomes \(f^3\) times the original mass as volume scales with the cube of the dimension factor. Similarly, each dimension \(r\) becomes \(f\) times its original. Hence, the moment of inertia will be \(f^5\) times its original value: \(I_{scaled} = f^5 I_{original}\).
02

Scaling Factor for Rotational Kinetic Energy

The rotational kinetic energy (K) of an object is given by the formula \( K = \frac{1}{2} I \omega^2 \), where \( \omega \) is the angular velocity. Given that \(I\) scales by \(f^5\) with scaling factor \(f\), and assuming \(\omega\) remains constant, the kinetic energy will also scale by \(f^5\).
03

Applying the Scale Factor to Kinetic Energy

The model is a \( \frac{1}{48} \) scale of the full-size object. Thus, the scaling factor \(f\) for the full-scale object compared to the model is 48. Therefore, the kinetic energy of the full-scale object is \(48^5\) times that of the model:\[ K_{full} = 48^5 \times 2.5 \]. Calculating this gives \( K_{full} = 2548039680 \mathrm{J} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding the Scaling Factor
The concept of scaling is crucial when studying objects and their physical properties. When you scale an object, you change its size while maintaining its shape. This is done by applying a scaling factor, which is a number used to multiply the dimensions of the object. For instance, if a model is scaled by a factor of 3, all its dimensions, such as length, width, and height, become three times larger.
When considering volume and mass, these properties scale with the cube of the scaling factor, denoted as \( f^3 \). This happens because volume is a three-dimensional measure. So if each dimension of an object is scaled by a factor \( f \), then its volume, and therefore its mass, will increase by \( f^3 \).
Understanding these scaling relationships is pivotal for predicting other changes, such as the moment of inertia, when an object is resized.
Exploring Rotational Kinetic Energy
Rotational kinetic energy is the energy possessed by a rotating object, which depends on how fast it spins and its moment of inertia. The formula to calculate it is \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
The moment of inertia is a measure of how the mass of the object is distributed relative to the axis of rotation. It indicates the resistance of an object to change in its rotational motion. When an object is scaled by a factor \( f \), its moment of inertia increases by \( f^5 \). Consequently, because the rotational kinetic energy depends on \( I \), if the angular velocity \( \omega \) remains unchanged, the energy scales in the same way. This means that the rotational kinetic energy will increase or decrease by \( f^5 \) as well.
This understanding is essential when comparing the energy of scale models to that of full-size objects.
The Role of Angular Velocity
Angular velocity, symbolized \( \omega \), represents how fast an object rotates or spins around an axis. It measures the rotation rate in terms of the angle an object rotates in a particular time period.
Unlike linear velocity, which is concerned with objects moving in a straight line, angular velocity focuses on rotational motion. Its unit is typically radians per second, highlighting that it's all about the "turning speed".
In contexts where objects are scaled, understanding the role of angular velocity becomes significant, especially when calculating rotational kinetic energy. While the moment of inertia will change with scaling, if the angular velocity remains constant, changes in rotational kinetic energy become entirely dependent on how the object's inertia scales. This allows for straightforward calculations when determining how the rotational energy of a scaled object compares to its model.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. The wheel was brought up to speed periodically, when the bus stopped at a station, by an electric motor, which could then be attached to the electric power lines. The flywheel was a solid cylinder with mass 1000 \(\mathrm{kg}\) and diameter 1.80 \(\mathrm{m}\) ; its top angular speed was 3000 rev/min. (a) At this angular speed, what is the kinetic energy of the flywheel? (b) If the average power required to operate the bus is \(1.86 \times 10^{4} \mathrm{W},\) how long could it operate between stops?

At \(t=0\) the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by \(\theta(t)=(250 \mathrm{rad} / \mathrm{s}) t-\left(20.0 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}-\left(1.50 \mathrm{rad} / \mathrm{s}^{3}\right) t^{3} .\) (a) \(\mathrm{At}\) what time is the angular velocity of the motor shaft zero? (b) Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity. (c) How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero? (d) How fast was the motor shaft rotating at \(t=0,\) when the current was reversed? (e) Calculate the average angular velocity for the time period from \(t=0\) to the time calculated in part (a).

When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass \(0.180 \mathrm{kg},\) and its flywheel has moment of inertia \(4.00 \times 10^{-5} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The car is 15.0 \(\mathrm{cm}\) long. An advertisement claims that the car can travel at a scale speed of up to 700 \(\mathrm{km} / \mathrm{h}(440 \mathrm{mi} / \mathrm{h}) .\) The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car. to the length of the toy. Assume a length of 3.0 \(\mathrm{m}\) for a real car. (a) For a scale speed of 700 \(\mathrm{km} / \mathrm{h}\) , what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

A light, flexible rope is wrapped several times around a hollow cylinder, with a weight of 40.0 \(\mathrm{N}\) and a radius of 0.25 \(\mathrm{m}\), that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force \(P\) for a distance of \(5.00 \mathrm{m},\) at which point the end of the rope is moving at 6.00 \(\mathrm{m} / \mathrm{s} .\) If the rope does not slip on the cylinder, what is the value of \(P ?\)

What angle in radians is subtended by an arc 1.50 \(\mathrm{m}\) long on the circumference of a circle of radius 2.50 \(\mathrm{m} ?\) What is this angle in degrees? (b) An arc 14.0 \(\mathrm{cm}\) long on the circumference of a circle subtends an angle of \(128^{\circ} .\) What is the radius of the circle? (c) The angle between two radii of a circle with radius 1.50 \(\mathrm{m}\) is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?
See all solutions

Recommended explanations on Physics Textbooks

Work Energy and Power

Read Explanation

Dynamics

Read Explanation

Force

Read Explanation

Atoms and Radioactivity

Read Explanation

Magnetism

Read Explanation

Modern Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.