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Chapter 9: Problem 60

Gear \(A\), with a mass of \(1.00 \mathrm{~kg}\) and a radius of \(55.0 \mathrm{~cm}\) is in contact with gear \(\mathrm{B}\), with a mass of \(0.500 \mathrm{~kg}\) and a radius of \(30.0 \mathrm{~cm} .\) The gears do not slip with respect to each other as they rotate. Gear A rotates at 120. rpm and slows to 60.0 rpm in \(3.00 \mathrm{~s}\). How many rotations does gear B undergo during this time interval?

Short Answer

Expert verified
Answer: Gear B undergoes approximately 7.87 rotations during the 3-second time interval.

Step by step solution

01

Convert angular speeds to rad/s

To convert rpm to rad/s, we can use the conversion factor: 1 rpm = \((2\pi \mathrm{~rad})/(\mathrm{60~s})\). For gear A, \(\omega_\mathrm{Ai} = 120\, \mathrm{rpm} \times \frac{2\pi \mathrm{~rad}}{\mathrm{60~s}} = 12.566\,\mathrm{rad/s}\) (initial angular speed) \(\omega_\mathrm{Af} = 60\, \mathrm{rpm} \times \frac{2\pi \mathrm{~rad}}{\mathrm{60~s}} = 6.283\, \mathrm{rad/s}\) (final angular speed)
02

Calculate the angular acceleration of gear A

We can calculate the angular acceleration (\(\alpha_\mathrm{A}\)) of gear A using the equation: \(\alpha_\mathrm{A} = \frac{\omega_\mathrm{Af} - \omega_\mathrm{Ai}}{t}\), where \(t = 3.00\, \mathrm{s}\). \(\alpha_\mathrm{A} = \frac{6.283 - 12.566}{3.00} = -2.094\, \mathrm{rad/s^2}\)
03

Find the angular acceleration of gear B

Given the radii of gears A and B, since they are in contact without slipping, their accelerations are related by: \(\alpha_\mathrm{B} = \frac{r_\mathrm{A}}{r_\mathrm{B}} \times \alpha_\mathrm{A}\) \(\alpha_\mathrm{B} = \frac{55.0}{30.0} \times (-2.094) = -3.824\, \mathrm{rad/s^2}\)
04

Calculate the final angular speed of gear B

We know the initial angular speed of gear B can be found from the gears' initial speeds: \(\omega_\mathrm{Bi} = \frac{r_\mathrm{A}}{r_\mathrm{B}} \times \omega_\mathrm{Ai}\) \(\omega_\mathrm{Bi} = \frac{55.0}{30.0} \times 12.566 = 23.045\, \mathrm{rad/s}\) Now we can calculate the final angular speed (\(\omega_\mathrm{Bf}\)) of gear B: \(\omega_\mathrm{Bf} = \omega_\mathrm{Bi} + \alpha_\mathrm{B} \times t\) \(\omega_\mathrm{Bf} = 23.045 - 3.824 \times 3.00 = 11.573\, \mathrm{rad/s}\)
05

Find the angular displacement of gear B

We can find the angular displacement (\(\theta_\mathrm{B}\)) covered by gear B during the time interval using the equation: \(\theta_\mathrm{B} = \omega_\mathrm{Bi} \times t + 0.5 \times \alpha_\mathrm{B} \times t^2\) \(\theta_\mathrm{B} = 23.045 \times 3.00 + 0.5 \times (-3.824) \times 3.00^2 = 49.455\, \mathrm{rad}\)
06

Convert angular displacement to number of rotations

To find the number of rotations that gear B underwent, divide the angular displacement by \(2\pi\). Number of rotations of gear B = \(\frac{\theta_\mathrm{B}}{2\pi}\) Number of rotations of gear B = \(\frac{49.455}{6.283} \approx 7.87\) Therefore, gear B undergoes approximately 7.87 rotations during the time interval.

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Key Concepts

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

Angular Acceleration
Angular acceleration is a measure of how quickly an object's rotational speed changes. It is similar to linear acceleration, but instead of linear speed, it refers to the rate of change of angular speed. Mathematically, angular acceleration is expressed as \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular speed, and \( \Delta t \) is the change in time.

In our gear example, angular acceleration is crucial to determine how gear A is slowing down over time. It experienced a decrease in speed from 12.566 rad/s to 6.283 rad/s over 3 seconds, resulting in an angular acceleration of \( -2.094 \mathrm{\, rad/s^2} \). This negative value signifies a deceleration. Such calculations are essential when analyzing the dynamics of any rotating system.
Rotational Kinematics
Rotational kinematics is the study of motion without considering the forces that cause it. It is essentially similar to linear kinematics but focuses on rotation. The primary variables are angular displacement, angular velocity, and angular acceleration.

In this scenario, we explored how the gears' rotations relate to each other without considering friction or other forces. Using rotational kinematics, we establish relationships between angular displacement (\( \theta \)), initial and final angular velocities (\( \omega_i \) and \( \omega_f \)), and angular acceleration (\( \alpha \)). These three variables, along with time, allow us to predict the gears' behavior over a given period. This is important for engineers and designers to ensure the smooth functioning of mechanical systems.
Gear Ratio
The gear ratio is a critical aspect of understanding how interconnected gears affect each other's motion. In essence, it describes the relationship between the sizes, speeds, or torques of two meshed gears. The gear ratio can be calculated with the formula \( \text{Gear Ratio} = \frac{r_A}{r_B} \), where \( r_A \) and \( r_B \) are the radii of gear A and gear B respectively.

In our example, the gear ratio between gear A and gear B is \( \frac{55.0}{30.0} \). This ratio is used to determine how the motion of one gear influences the other. By knowing this ratio, we can find out how fast gear B rotates compared to gear A. Understanding gear ratios is vital for transmitting power efficiently between mechanical components.
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 rotational analogue of linear displacement. In simpler terms, it tells us how much rotation has occurred.

For gear B, the angular displacement over 3 seconds was calculated as 49.455 rad. This was determined using the formula \( \theta = \omega_i \cdot t + 0.5 \cdot \alpha \cdot t^2 \). This displacement was then converted into the number of rotations by dividing by \( 2\pi \). Understanding angular displacement is key when analyzing the performance of rotating systems, allowing us to visualize just how much rotation has taken place over a given time period.

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Most popular questions from this chapter

In a department store toy display, a small disk (disk 1) of radius \(0.100 \mathrm{~m}\) is driven by a motor and turns a larger disk (disk 2) of radius \(0.500 \mathrm{~m}\). Disk 2 , in turn, drives disk 3 , whose radius is \(1.00 \mathrm{~m}\). The three disks are in contact and there is no slipping. Disk 3 is observed to sweep through one complete revolution every \(30.0 \mathrm{~s}\) a) What is the angular speed of disk \(3 ?\) b) What is the ratio of the tangential velocities of the rims of the three disks? c) What is the angular speed of disks 1 and \(2 ?\) d) If the motor malfunctions, resulting in an angular acceleration of \(0.100 \mathrm{rad} / \mathrm{s}^{2}\) for disk 1 , what are disks 2 and 3's angular accelerations?

A typical Major League fastball is thrown at approximately \(88 \mathrm{mph}\) and with a spin rate of \(110 \mathrm{rpm} .\) If the distance between the pitcher's point of release and the catcher's glove is exactly \(60.5 \mathrm{ft},\) how many full turns does the ball make between release and catch? Neglect any effect of gravity or air resistance on the ball's flight.

Mars orbits the Sun at a mean distance of 228 million \(\mathrm{km},\) in a period of 687 days. The Earth orbits at a mean distance of 149.6 million \(\mathrm{km},\) in a period of 365.26 days. a) Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line? b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth. c) Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

Life scientists use ultracentrifuges to separate biological components or to remove molecules from suspension. Samples in a symmetric array of containers are spun rapidly about a central axis. The centrifugal acceleration they experience in their moving reference frame acts as "artificial gravity" to effect a rapid separation. If the sample containers are \(10.0 \mathrm{~cm}\) from the rotation axis, what rotation frequency is required to produce an acceleration of \(1.00 \cdot 10^{5} g ?\)

A CD starts from rest and speeds up to the operating angular frequency of the CD player. Compare the angular velocity and acceleration of a point on the edge of the CD to those of a point halfway between the center and the edge of the CD. Do the same for the linear velocity and acceleration.
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