Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s\(^2\), what is its angular velocity at \(t =\) 2.50 s? (b) Through what angle has the wheel turned between \(t =\) 0 and \(t =\) 2.50 s?

Short Answer

Expert verified
(a) 2.00 rad/s, (b) 4.375 rad

Step by step solution

01

Identify Given Values

From the problem, we have:- Initial angular velocity, \( \omega_0 = 1.50 \, \text{rad/s} \)- Angular acceleration, \( \alpha = 0.200 \, \text{rad/s}^2 \)- Time, \( t = 2.50 \, \text{s} \)
02

Find Final Angular Velocity

We use the equation for angular velocity with constant angular acceleration:\[ \omega = \omega_0 + \alpha t \]Substitute the given values:\[ \omega = 1.50 \, \text{rad/s} + (0.200 \, \text{rad/s}^2)(2.50 \, \text{s}) \]\[ \omega = 1.50 \, \text{rad/s} + 0.50 \, \text{rad/s} \]\[ \omega = 2.00 \, \text{rad/s} \]Thus, the final angular velocity at \( t = 2.50 \, \text{s} \) is 2.00 rad/s.
03

Calculate Total Angular Displacement

Use the angular displacement formula:\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]Substitute the known values:\[ \theta = (1.50 \, \text{rad/s})(2.50 \, \text{s}) + \frac{1}{2}(0.200 \, \text{rad/s}^2)(2.50 \, \text{s})^2 \]\[ \theta = 3.75 \, \text{rad} + \frac{1}{2}(0.200)(6.25) \, \text{rad} \]\[ \theta = 3.75 \, \text{rad} + 0.625 \, \text{rad} \]\[ \theta = 4.375 \, \text{rad} \]Thus, the total angle through which the wheel has turned is 4.375 radians.

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 Vaia!

Key Concepts

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

Angular Velocity
Angular velocity is a measure of how quickly an object rotates or spins around a central point or axis. In this context of the bicycle wheel, it describes how fast the wheel is spinning. It is a vector quantity, meaning it has both a magnitude and a direction, but in most simple scenarios such as a bicycle wheel, we focus primarily on the magnitude.
Angular velocity is often represented by the Greek letter omega (\( \omega \)), and it is measured in radians per second (rad/s). Imagine it like the speed of a car, but for a rotating object instead. When a wheel spins faster, its angular velocity increases.

To calculate angular velocity when angular acceleration is constant, you use this equation:
\[ \omega = \omega_0 + \alpha t \]
Where:
  • \( \omega_0 \) is the initial angular velocity (how fast it was spinning to begin with)
  • \( \alpha \) is the angular acceleration
  • \( t \) is the time elapsed

In the original problem, when the bicycle wheel started with an angular velocity of 1.50 rad/s and an angular acceleration of 0.200 rad/s² was applied for 2.5 seconds, its final angular velocity increased to 2.00 rad/s as calculated using the formula. This indicates the wheel spins faster over time due to the constant acceleration.
Angular Acceleration
Angular acceleration is the rate at which angular velocity changes with time. It tells us how quickly the angular speed of a spinning object, like a wheel, is increasing or decreasing. It is represented by the Greek letter alpha (\( \alpha \)) and measured in radians per second squared (rad/s²).

Think of it as how hard you're pushing or pulling on a spinning wheel to change its speed. If a wheel is accelerating, it means the angular velocity of the wheel is either increasing or decreasing.

In our exercise, the bicycle wheel has a constant angular acceleration of 0.200 rad/s². This means every second, the speed at which the wheel spins increases by 0.200 rad/s. This constant acceleration is crucial when using the formula for finding the change in angular velocity over time, as it simplifies the calculation by allowing us to assume a linear increase. Without constant acceleration, the equations would be more complex, involving calculus to account for variable changes over time.
Angular Displacement
Angular displacement is the measure of how much an object has rotated or turned, generally represented by the Greek letter theta (\( \theta \)). It’s akin to distance in linear motion but deals with angles in rotational motion.
Measured in radians, it gives us the total change in the angle of the rotating object from start to finish.

To calculate angular displacement, especially when dealing with constant angular acceleration, you can use the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Where:
  • \( \omega_0 \) is the initial angular velocity
  • \( \alpha \) is the angular acceleration
  • \( t \) is time

Using this formula, you start with the initial rotation and add the effect of acceleration over a specified time.
In this problem, with the given values, the total amount that the wheel rotates in 2.5 seconds comes out to be 4.375 radians. This tells you exactly how far the wheel has turned from its initial position.

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 compact disc (CD) stores music in a coded pattern of tiny pits 10\(^-\)7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant \(linear\) speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

A light, flexible rope is wrapped several times around a \(hollow\) cylinder, with a weight of 40.0 N and a radius of 0.25 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 m, at which point the end of the rope is moving at 6.00 m/s. If the rope does not slip on the cylinder, what is \(P\)?

A sphere with radius \(R =\) 0.200 m has density \(\rho\) that decreases with distance \(r\) from the center of the sphere according to \(r =\) 3.00 \(\times\) 103 kg/m\(^3 -\) (9.00 \(\times\) 103 kg/m\(^4\))\(r\). (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

According to the shop manual, when drilling a 12.7-mm-diameter hole in wood, plastic, or aluminum, a drill should have a speed of 1250 rev/min. For a 12.7-mm-diameter drill bit turning at a constant 1250 rev/min, find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

The eel is observed to spin at 14 spins per second clockwise, and 10 seconds later it is observed to spin at 8 spins per second counterclockwise. What is the magnitude of the eel's average angular acceleration during this time? (a) 6/10 rad/s\(^2\); (b) 6\(\pi\)/10 rad/s\(^2\); (c) 12\(\pi\)/10 rad/s\(^2\); (d) 44\(\pi\)/10 rad/s\(^2\).

See all solutions

Recommended explanations on Physics Textbooks

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.

Sign-up for free