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Chapter 9: Problem 11

The rotating blade of a blender turns with constant angular acceleration 1.50 rad/s\(^2\). (a) How much time does it take to reach an angular velocity of 36.0 rad/s, starting from rest? (b) Through how many revolutions does the blade turn in this time interval?

Short Answer

Expert verified
(a) 24.0 s; (b) approximately 68.8 revolutions.

Step by step solution

01

Identify Given Values and Required Outcome for Part (a)

We are given an angular acceleration \( \alpha = 1.50 \, \text{rad/s}^2 \), initial angular velocity \( \omega_0 = 0 \, \text{rad/s} \), and final angular velocity \( \omega = 36.0 \, \text{rad/s} \). We need to find the time \( t \) required to reach this angular velocity.
02

Use the Angular Kinematics Equation to Find Time

Using the kinematic equation \( \omega = \omega_0 + \alpha t \), where \( \omega \) is the final angular velocity, rearrange the formula to solve for time \( t \): \( t = \frac{\omega - \omega_0}{\alpha} \). Substitute in the given values: \( t = \frac{36.0 \, \text{rad/s} - 0}{1.50 \, \text{rad/s}^2} \).
03

Calculate the Time for Part (a)

Calculate: \( t = \frac{36.0}{1.50} \approx 24.0 \text{ s} \). So, the time taken to reach an angular velocity of 36.0 rad/s is 24.0 seconds.
04

Determine Given Values for Part (b)

We know the time \( t = 24.0 \, \text{s} \), initial angular velocity \( \omega_0 = 0 \, \text{rad/s} \), and angular acceleration \( \alpha = 1.50 \, \text{rad/s}^2 \). We need to find the number of revolutions the blade goes through in this time.
05

Use Angular Displacement Equation

The angular displacement \( \theta \) can be found using the equation \( \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \). Substitute the known values: \( \theta = 0 \times 24.0 + \frac{1}{2} \times 1.50 \times (24.0)^2 \).
06

Calculate Angular Displacement in Radians

Calculate \( \theta = \frac{1}{2} \times 1.50 \times 576 \approx 432 \, \text{rad} \). So, the angular displacement is 432 radians.
07

Convert Radians to Revolutions

One revolution is \( 2\pi \) radians. Convert the angular displacement from radians to revolutions using \( \text{revolutions} = \frac{\theta}{2\pi} \). Calculate: \( \text{revolutions} = \frac{432}{2\pi} \approx 68.8 \).
08

Provide Final Answer for Part (b)

The blade turns through approximately 68.8 revolutions.

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Key Concepts

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

Angular Acceleration
Angular acceleration refers to the rate at which an object's angular velocity changes over time. It is like the engine of rotational motion, giving a rotating object like a blender blade the ability to increase or decrease its speed. This is essential to understanding how objects begin spinning faster or slower. The unit for angular acceleration is radians per second squared (\(\text{rad/s}^2\)).
  • If an object is spinning and gaining speed, it has a positive angular acceleration.
  • If it is slowing down, the angular acceleration is negative.
In our exercise, the blade of a blender starts with an angular acceleration of 1.50 \(\text{rad/s}^2\), meaning it speeds up its rotation at this rate.
Angular Velocity
Angular velocity describes how fast an object is rotating. It is like telling you how quickly your blender blade spins around its center. The faster it spins, the greater its angular velocity. The standard unit for angular velocity is radians per second (\(\text{rad/s}\)).
  • It can be thought of as the rotational version of linear speed.
  • If the angular velocity increases, the object spins faster. If it decreases, the object slows down.
In our problem, the blender blade starts from rest, which means an initial angular velocity (\(\omega_0\)) of 0. It reaches a final angular velocity (\(\omega\)) of 36.0 \(\text{rad/s}\) after some time.
Angular Displacement
Angular displacement measures the change in the angle as an object rotates, in simple terms鈥攊t's how far something has spun. The unit is usually radians. You can think of it as measuring the spinning journey of an object, similar to how distance measures linear motion.There is a specific formula to find angular displacement: \[\theta = \omega_0 t + \frac{1}{2}\alpha t^2\]
  • \( \omega_0 \) is the initial angular velocity.
  • \( \alpha \) is the angular acceleration.
  • \( t \) is the time of rotation.
In the exercise, the blade's angular displacement is calculated to be 432 radians. This shows the extent to which the blade has rotated over a time period of 24 seconds.
Revolutions
Revolutions are a way of expressing how many times something has completed a full circle during its rotation. One revolution is equivalent to an angular displacement of \(2\pi\) radians.Calculating revolutions helps translate radians into a more tangible measure鈥攈ow many times around did it go?In the problem you encountered, to find out how many full rotations the blade made, convert the angular displacement from radians to revolutions using the formula:\[\text{revolutions} = \frac{\theta}{2\pi}\]For our 432 radians, this is approximately 68.8 revolutions. Therefore, during the time interval in question, the blender blade completed just under 69 full rotations.

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Most popular questions from this chapter

It has been argued that power plants should make use of off-peak hours (such as late at night) to generate mechanical energy and store it until it is needed during peak load times, such as the middle of the day. One suggestion has been to store the energy in large flywheels spinning on nearly frictionless ball bearings. Consider a flywheel made of iron (density 7800 kg/m\(^3\)) in the shape of a 10.0-cm-thick uniform disk. (a) What would the diameter of such a disk need to be if it is to store 10.0 megajoules of kinetic energy when spinning at 90.0 rpm about an axis perpendicular to the disk at its center? (b) What would be the centripetal acceleration of a point on its rim when spinning at this rate?

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A roller in a printing press turns through an angle \(\theta(t)\) given by \(\theta(t) = \gamma t^2 - \beta t^3\), where \(\gamma =\) 3.20 rad/s\(^2\) and \(\beta =\) 0.500 rad/s\(^3\). (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of t does it occur?

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