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Chapter 10: Problem 25

When a carpenter shuts off his circular saw, the 10.0 -inchdiameter blade slows from 4440 rpm to 0.00 rpm in 2.50 s. (a) What is the angular acceleration of the blade? (b) What is the distance traveled by a point on the rim of the blade during the deceleration? (c) What is the magnitude of the net displacement of a point on the rim of the blade during the deceleration?

Short Answer

Expert verified
(a) \(-185.98\) rad/s虏. (b) 73.8 m. (c) 0.254 m.

Step by step solution

01

Convert Angular Velocity

First, convert the initial and final angular velocities from revolutions per minute (rpm) to radians per second (rad/s). Use the conversion factor \( 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s} \).Initial angular velocity, \( \omega_i = 4440 \text{ rpm} = 4440 \times \frac{2\pi}{60} \text{ rad/s} = 464.96 \text{ rad/s} \).Final angular velocity, \( \omega_f = 0.00 \text{ rpm} = 0 \text{ rad/s} \).
02

Calculate Angular Acceleration

Angular acceleration (\( \alpha \)) is calculated using the formula \( \alpha = \frac{\omega_f - \omega_i}{t} \).Substitute the values: \( \alpha = \frac{0 - 464.96}{2.50} = -185.98 \text{ rad/s}^2 \).
03

Calculate the Distance Traveled by a Point on the Rim

The distance traveled by a point on the rim can be found by the formula \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \), where \( \theta \) is the angular distance in radians.Calculate \( \theta: \theta = 464.96 \times 2.5 + \frac{1}{2} \times (-185.98) \times (2.5)^2 = 581.2 \text{ rad} \).Convert the angular distance to linear distance using \( s = r \times \theta \), where \( r \) is the radius of the blade. The diameter is 10 inches, hence radius \( r = 5 \text{ inches} = 0.127 \text{ m} \).Thus, \( s = 0.127 \times 581.2 = 73.8 \text{ m} \).
04

Calculate the Magnitude of the Net Displacement

Net displacement for a point on the rim can be visualized as the straight line chord under the arc created by the initial and final position of the point on the rim.Using the formula for the chord length \( d = 2r \sin\left(\frac{\theta}{2}\right) \). Given that \( \theta = 581.2 \), for practicality in circular motion dynamics, this large angle results in a nearly complete revolution, making the net displacement equivalent to twice the radius.Thus, \( d \approx 0.254 \text{ m} \).
05

Conclusion

We found the angular acceleration to be \(-185.98 \text{ rad/s}^2\), the distance traveled by a point on the rim is \(73.8 \text{ m}\), and the magnitude of the net displacement for a point on the rim is approximately \(0.254 \text{ m}\).

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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 how quickly the rate of rotation of an object changes over time. It is represented by the symbol \( \alpha \) and is measured in radians per second squared (\( \text{rad/s}^2 \)). In our exercise, the carpenter鈥檚 saw blade slows down, indicating a negative angular acceleration.
Angular acceleration can be calculated using the formula:
  • \( \alpha = \frac{\omega_f - \omega_i}{t} \)
  • Where \( \omega_f \) is the final angular velocity, \( \omega_i \) is the initial angular velocity, and \( t \) is the time taken for the change.
Understanding angular acceleration is crucial in analyzing rotational dynamics as it informs how much torque is needed to change an object's rotational speed over a certain period.
Angular Velocity
Angular velocity is the measure of the rate of rotation of an object, or how fast something spins. It is denoted by the symbol \( \omega \) and measured in radians per second (\( \text{rad/s} \)).
To convert angular velocity from revolutions per minute (rpm) to radians per second, use the conversion:
  • \( 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s} \)
In the case of our saw blade, the initial angular velocity was 4440 rpm, converted to 464.96 \( \text{rad/s} \). The final angular velocity was 0, as the blade came to a stop. This concept is essential because understanding how fast something spins helps determine other attributes like angular acceleration and distance traveled.
Linear Displacement
Linear displacement refers to how far a point on a rotating object travels along the edge during its rotation. It's essential to convert angular measurements to linear ones to understand the actual path something follows in a real-world context.
For a circular object:
  • Linear displacement \( s \) can be calculated using \( s = r \times \theta \)
  • Where \( r \) is the radius and \( \theta \) is the angle in radians.
In our example, when the blade decelerated, a point on its rim traveled a linear distance of 73.8 meters. This conversion is useful for determining how much ground an object covers during its rotational motion.
Circular Motion
Circular motion involves an object moving in a path along a circle. In this scenario, understanding the dynamics of the motion, including angular velocity, acceleration, and displacement, is critical.
A point on a rotating object like our saw blade will move in circular motion throughout;
  • The actual path of travel is an arc, which makes calculation of distances and other dynamics slightly different from linear motion.
  • The net displacement usually forms a chord, the straight line connecting the starting and ending point of the motion across the circle. For small angles, this chord is a good approximation, but for larger angles like hundreds of radians, it visually becomes nearly a complete circle.
Understanding these principles helps us quantify and foresee the behavior of objects in rotational motion accurately.

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Most popular questions from this chapter

When a pitcher throws a curve ball, the ball is given a fairly rapid spin. If a \(0.15-\mathrm{kg}\) baseball with a radius of \(3.7 \mathrm{cm}\) is thrown with a linear speed of \(48 \mathrm{m} / \mathrm{s}\) and an angular speed of \(42 \mathrm{rad} / \mathrm{s},\) how much of its kinetic energy is translational and how much is rotational? Assume the ball is a uniform, solid sphere.

Pigeons are bred to display a number of interesting characteristics. One breed of pigeon, the "roller," is remarkable for the fact that it does a number of backward somersaults as it drops straight down toward the ground. Suppose a roller pigeon drops from rest and free falls downward for a distance of \(14 \mathrm{m}\). If the pigeon somersaults at the rate of \(12 \mathrm{rad} / \mathrm{s}\), how many revolutions has it completed by the end of its fall?

A pilot performing a horizontal turn will lose consciousness if she experiences a centripetal acceleration greater than 7.00 times the acceleration of gravity. What is the minimum radius turn she can make without losing consciousness if her plane is flying with a constant speed of \(245 \mathrm{m} / \mathrm{s} ?\)

Express the angular velocity of the second hand on a clock in the following units: (a) rev / hr, (b) \(\operatorname{deg} / \mathrm{min},\) and (c) \(\mathrm{rad} / \mathrm{s}\).

A person rides on a 12-m-diameter Ferris wheel that rotates at the constant rate of 8.1 rpm. Calculate the magnitude and direction of the force that the seat exerts on a \(65-\mathrm{kg}\) person when he is (a) at the top of the wheel, (b) at the bottom of the wheel, and (c) halfway up the wheel.
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