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Chapter 9: Problem 9

A bicycle wheel has an initial angular velocity of 1.50 \(\mathrm{rad} / \mathrm{s}\) . (a) If its angular acceleration is constant and equal to 0.300 \(\mathrm{rad} / \mathrm{s}^{2}\) , what is its angular velocity at \(t=2.50 \mathrm{s} ?\) (b) Through what angle has the wheel turned between \(t=0\) and \(t=2.50 \mathrm{s} ?\)

Short Answer

Expert verified
(a) 2.25 rad/s; (b) 4.6875 radians.

Step by step solution

01

Identify Known Variables

First, note the given variables. The initial angular velocity \( \omega_0 \) is 1.50 rad/s and the constant angular acceleration \( \alpha \) is 0.300 rad/s虏.
02

Use Angular Velocity Formula

To find the angular velocity \( \omega \) at \( t = 2.50 \) s, use the equation \( \omega = \omega_0 + \alpha t \). Substituting the known values gives \( \omega = 1.50 + (0.300)(2.50) \).
03

Calculate Angular Velocity

Perform the calculation: \( \omega = 1.50 + 0.300 \times 2.50 = 1.50 + 0.750 = 2.25 \) rad/s. Thus, the angular velocity at \( t = 2.50 \) s is 2.25 rad/s.
04

Use Angular Displacement Formula

To find the angle turned \( \theta \), use the formula \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). Substituting the known values gives \( \theta = (1.50)(2.50) + \frac{1}{2}(0.300)(2.50)^2 \).
05

Calculate Angular Displacement

Perform the calculations: \( \theta = 3.75 + \frac{1}{2}(0.300)(6.25) = 3.75 + 0.9375 = 4.6875 \) rad. Thus, the wheel turns through an angle of 4.6875 radians.

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Key Concepts

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

Angular Velocity
Angular velocity is a key concept when dealing with rotational motion. It measures how fast an object rotates around a central point or axis. Unlike linear velocity, which measures how fast an object moves in a straight line, angular velocity is concerned with the rate of rotation.
  • Initial Angular Velocity (\( \omega_0 \)): This is the starting point of the rotation. In the exercise, the bicycle wheel begins with an angular velocity of 1.50 rad/s. This means that initially, the wheel rotates 1.50 radians every second.
  • Calculation: To find the angular velocity at a later time, we use the formula \( \omega = \omega_0 + \alpha t \), where \( \alpha \) is the angular acceleration, and \( t \) is time. This is the rate at which the wheel's angular speed increases or decreases. For instance, at \( t = 2.50 \) seconds and an angular acceleration of 0.300 rad/s虏, the wheel's angular velocity becomes 2.25 rad/s.
Angular velocity is vital for understanding how quickly an object turns. It helps in predicting future positions and times by knowing the initial speed of rotation and how that speed changes over time.
Understanding angular velocity offers insight into the wheel's dynamic behavior as it spins.
Angular Acceleration
Angular acceleration describes how quickly the angular velocity of an object changes with time. It is an essential factor in determining how a rotating object speeds up or slows down.
  • Constant Angular Acceleration (\( \alpha \)): In the given exercise, the wheel's angular acceleration is 0.300 rad/s虏. This means that every second, the angular velocity of the wheel increases by 0.300 radians per second.
  • Effect on Angular Velocity: Angular acceleration is used in the formula \( \omega = \omega_0 + \alpha t \) to calculate the final angular velocity. The constant angular acceleration ensures a predictable change in velocity over time.
Angular acceleration is crucial because it reflects how rapidly an object can change its state of rotation. It helps in determining the time it would take for a spinning object to reach a particular speed, or conversely, how long it might take for it to stop.
Angular Displacement
Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It is the angle between the initial and final positions of a rotating body.
  • Formula for Angular Displacement (\( \theta \)): In the scenario, we use \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \) to find out how much the bicycle wheel has turned. This formula combines the initial angular velocity and the effect of angular acceleration over time to calculate total displacement.
  • Calculated Example: With an initial angular velocity of 1.50 rad/s and angular acceleration of 0.300 rad/s虏, after 2.50 seconds, the wheel has turned through 4.6875 radians. This reveals the cumulative effect of both its starting speed and steady acceleration on position.
Understanding angular displacement helps visualize the entire extent an object has rotated over time. It's a fundamental part of rotational dynamics as it connects with both angular velocity and angular acceleration to describe a complete picture of rotational movement.

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Most popular questions from this chapter

The motor of a table saw is rotating at 3450 rev/min. A pulley attached to the motor shaft drives a second pulley of half the diameter by means of a V-belt. A circular saw blade of diameter 0.208 \(\mathrm{m}\) is mounted on the same rotating shaft as the second pulley. (a) The operator is careless and the blade catches and throws back a small piece of wood. This piece of wood moves with linear speed equal to the tangential speed of the rim of the blade. What is this speed? (b) Calculate the radial acceleration of points on the outer edge of the blade to see why sawdust doesn't stick to its teeth.

Centrifuge. An advertisement claims that a centrifuge takes up only 0.127 \(\mathrm{m}\) of bench space but can produce a radial acceleration of 3000 \(\mathrm{g}\) at 5000 rev/min. Calculate the required radius of the centrifuge. Is the claim realistic?

An airplane propeller is 2.08 \(\mathrm{m}\) in length (from tip to tip) with mass 117 \(\mathrm{kg}\) and is rotating at 2400 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0\(\%\) of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

A thin, flat, uniform disk has mass \(M\) and radius \(R\) . A circular hole of radius \(R / 4,\) centered at a point \(R / 2\) from the disk's center, is then punched in the disk. (a) Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (Hint: Find the moment of inertia of the piece punched from the disk. (b) Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

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?
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