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Chapter 9: Problem 19

Emilie's potter's wheel rotates with a constant 2.25 \(\mathrm{rad} / \mathrm{s}^{2}\) angular acceleration. After 4.00 \(\mathrm{s}\) , the wheel has rotated through an angle of 60.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval?

Short Answer

Expert verified
The initial angular velocity was 10.5 rad/s.

Step by step solution

01

Identify Known Variables

We need to solve for Emilie's initial angular velocity \( \omega_0 \). The known values are angular acceleration \( \alpha = 2.25 \, \text{rad/s}^2 \), time \( t = 4.00 \, \text{s} \), and final angular displacement \( \theta = 60.0 \, \text{rad} \).
02

Use the Angular Motion Formula

The equation for angular displacement when starting with an initial angular velocity and constant angular acceleration is: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \] Substitute the known values: \[ 60.0 = \omega_0 \times 4.00 + \frac{1}{2} \times 2.25 \times (4.00)^2 \]
03

Simplify the Equation

Calculate the term that involves the angular acceleration: \[ \frac{1}{2} \times 2.25 \times (4.00)^2 = 18 \] So, substitute back: \[ 60.0 = \omega_0 \times 4.00 + 18 \]
04

Solve for Initial Angular Velocity

Rearrange the equation to solve for \( \omega_0 \): \[ \omega_0 \times 4 = 60.0 - 18 \] \[ \omega_0 = \frac{42}{4} = 10.5 \, \text{rad/s} \]

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Key Concepts

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

Angular Velocity
Angular velocity refers to how fast an object rotates around a central point or axis. Unlike linear velocity, which measures how fast something moves in a straight direction, angular velocity measures rotation. It's a vector quantity, which means it has both a magnitude and a direction.
When dealing with rotating objects, like a potter's wheel, angular velocity is expressed in radians per second (rad/s). Radians are a unit of angular measure used in many areas of mathematics.
  • Initial Angular Velocity (\(\omega_0\)): This is the speed of rotation at the beginning of a time interval.
  • Final Angular Velocity (\(\omega\)): This is the speed at the end of the time interval, after the object has undergone some angular acceleration.
Angular velocity connects closely with angular displacement and angular acceleration, as it helps describe how the rotation of an object changes over time.
Angular Acceleration
Angular acceleration describes the rate at which angular velocity changes. It's similar to linear acceleration but applies to rotating objects.
Angular acceleration is expressed in radians per second squared (rad/s虏). When an object experiences angular acceleration, it means the speed of its rotation is either increasing or decreasing.
In the case of Emilie's potter's wheel:
  • The constant angular acceleration of 2.25 rad/s虏 means the wheel's angular velocity increases steadily over time.
  • This incremental increase results in the wheel spinning faster and covering more angle as time progresses.
  • Angular acceleration, like angular velocity, is a vector quantity.
Understanding this concept is essential when solving motion problems involving rotating objects, as it helps determine how the motion evolves.
Angular Displacement
Angular displacement measures how much an object has rotated from its starting point. It is the angle in radians that the object sweeps through during a given time frame.
Unlike distance in linear motion, which is a scalar quantity, angular displacement provides information about both direction and magnitude in the context of rotation.
  • In Emilie's exercise, the wheel rotates through an angular displacement of 60 radians over 4 seconds.
  • This displacement helps us understand the total extent of rotation regardless of any changes in velocity or acceleration.
  • Utilizing equations of motion for rotation can aid in determining other rotational parameters, like initial angular velocity, given angular displacement, time, and acceleration.
Understanding angular displacement is fundamental to grasping how rotations are quantified and analyzing the dynamics of rotational systems.

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Most popular questions from this chapter

An electric fan is turned off, and its angular velocity decreases uniformly from 500.0 rev/min to 200.0 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s' and the number of revolutions made by the motor in the 4.00 s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

\(\bullet\) Electric drill. 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 \(\mathrm{rev} / \mathrm{min}\) , find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

Storing energy in flywheels. It has been suggested that we should use our power plants to generate energy in the off-hours (such as late at night) and store it for use during the day. One idea put forward is to store the energy in large fly wheels. Suppose we want to build such a flywheel in the shape of a hollow cylinder of inner radius 0.500 \(\mathrm{m}\) and outer radius \(1.50 \mathrm{m},\) using concrete of density \(2.20 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .\) (a) If, for stability, such a heavy flywheel is limited to 1.75 second for each revolution and has negligible friction at its axle, what must be its length to store 2.5 \(\mathrm{MJ}\) of energy in its rotational motion? (b) Suppose that by strengthening the frame you could safely double the flywheel's rate of spin. What length of flywheel would you need in that case? (Solve this part without reworking the entire problem!)

\(\bullet\) A size-5 soccer ball of diameter 22.6 \(\mathrm{cm}\) and mass 426 \(\mathrm{g}\) rolls up a hill without slipping, reaching a maximum height of 5.00 \(\mathrm{m}\) above the base of the hill. We can model this ball as a thin-walled hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy did it then have?

\(\bullet\) (a) Calculate the angular velocity (in rad/s) of the second, minute, and hour hands on a wall clock. (b) What is the period of each of these hands?
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