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Chapter 8: Problem 21

(II) A wheel 31 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?

Short Answer

Expert verified
The point travels approximately 3757.22 cm.

Step by step solution

01

Convert RPM to Radians per Second

First, we need to convert the wheel's speeds from revolutions per minute (RPM) to radians per second (rad/s). We know that 1 revolution is \(2\pi\) radians. Therefore: - Initial angular velocity \( \omega_i = 240 \text{ rpm} \times \frac{2\pi}{60} = 8\pi \text{ rad/s} \)- Final angular velocity \( \omega_f = 360 \text{ rpm} \times \frac{2\pi}{60} = 12\pi \text{ rad/s} \)
02

Calculate Angular Acceleration

Angular acceleration (\( \alpha \)) is calculated using the formula \( \alpha = \frac{\omega_f - \omega_i}{t} \). Substituting the values we have:\[ \alpha = \frac{12\pi - 8\pi}{6.8} = \frac{4\pi}{6.8} = \frac{2\pi}{3.4} \text{ rad/s}^2\]
03

Find Angular Displacement

The angular displacement (\(\theta\)) can be calculated using the formula \(\theta = \omega_i t + \frac{1}{2} \alpha t^2\). Substituting the values, we get:\[ \theta = 8\pi \times 6.8 + \frac{1}{2} \times \frac{2\pi}{3.4} \times (6.8)^2 \]After solving, \(\theta = 54.4\pi + 22.72\pi = 77.12\pi \text{ radians} \).
04

Calculate Distance Traveled by a Point on the Edge

The distance (\(s\)) traveled by a point on the edge is given by the formula \(s = r\theta\), where \(r\) is the radius. The wheel has a diameter of 31 cm, so the radius \(r = \frac{31}{2} = 15.5\) cm.Substituting the values, \(s = 15.5 \times 77.12\pi \). Solving this gives:\[ s \approx 3757.22 \text{ cm} \]

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Key Concepts

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

Understanding Angular Velocity
Angular velocity describes how quickly an object rotates around a specific point or axis. Imagine a merry-go-round, spinning around its center.
  • It shows how fast a point on the spinning object is moving.
  • Angular velocity is measured in radians per second (rad/s).
In the provided exercise, we start by converting the wheel's speed from revolutions per minute (RPM) to radians per second. This is important because standard formulas for rotational motion use radians rather than full revolutions.To calculate the initial and final angular velocities, we use the formula:\[\omega = \text{RPM} \times \frac{2\pi}{60}\]This helps us understand how the wheel's speed increases from 240 to 360 RPM.
Grasping Angular Acceleration
Angular acceleration is about changes in angular velocity. Essentially, it's how quickly an object's spinning speed changes, like pressing harder on a car's gas pedal to speed up.
  • It is crucial in understanding how fast an object's spinning changes over time.
  • Measured in "radians per second squared" (rad/s虏), it tells us how many rad/s the velocity changes every second.
Using the exercise:\[\alpha = \frac{\omega_f - \omega_i}{t}\]This formula shows us the change in speed between the initial and final states of motion, over a specific time period. We calculate that the angular acceleration of the wheel is \(\frac{2\pi}{3.4} \text{ rad/s}^2\). Knowing this helps us assess how effectively the wheel gains speed during those 6.8 seconds.
Exploring Angular Displacement
Angular displacement introduces the concept of how much an object has turned during its motion. It is analogous to distance in linear motion.
  • It's measured in radians and indicates the total change in the angular position of an object.
  • Knowing angular displacement helps us figure out how far an object has turned.
With the exercise, the formula used:\[\theta = \omega_i t + \frac{1}{2} \alpha t^2\]allows us to calculate that the wheel turned through \(77.12\pi\) radians in total. Recognizing angular displacement gives us a sense of the total spin or turn, making it a vital metric in rotational dynamics.
Determining Distance Traveled
When dealing with rotations, "distance traveled" refers to the path a point on the edge of the rotating object covers. This is more complex than simple straight-line distances since it deals with circular paths.
  • We calculate distance as a product of the radius and the angular displacement.
  • This provides the route length a specific spot on the rotating body takes.
Using the formula:\[s = r\theta\]where \(r\) stands for the radius of the wheel, and \(\theta\) is the angular displacement, we find that a point on the edge traveled approximately 3757.22 cm. Thus, this concept helps us understand and quantify movement on the circumference, especially critical for engineering and physics problems involving wheels or gears.

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Most popular questions from this chapter

A spherical asteroid with radius \(r = 123 m\) and mass \(M = 2.25 \times 10^{10} kg\) rotates about an axis at four revolutions per day. A "tug" spaceship attaches itself to the asteroid's south pole (as defined by the axis of rotation) and fires its engine, applying a force \(F\) tangentially to the asteroid's surface as shown in Fig. 8-65. If \(F = 285 N\), how long will it take the tug to rotate the asteroid's axis of rotation through an angle of 5.0\(^{\circ}\) by this method?

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius \(R_1 = 2.5 cm\) and winds its way out to radius \(R_2 = 5.8 cm\). To read the digital information, a CD player rotates the CD so that the player's readout laser scans along the spiral's sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency \(f\) of the CD as the laser moves outward. Determine the values for \(f\) (in units of rpm) when the laser is located at \(R_1\) and when it is at \(R_2\).

(II) A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from 3.8 rev/s to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s.

(III) Suppose a 65-kg person stands at the edge of a 5.5-m diameter merry-go- round turntable that is mounted on frictionless bearings and has a moment of inertia of \(1850 kg\cdot m^2\). The turntable is at rest initially, but when the person begins running at a speed of 4.0 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

(I) A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. \((a)\) What is the maximum torque she exerts? \((b)\) How could she exert more torque?
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