Question

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

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

Step-by-step solution

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)

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 \(t = 3.00\, \mathrm{s}\). \(\alpha_\mathrm{A} = \frac{6.283 - 12.566}{3.00} = -2.094\, \mathrm{rad/s^2}\)

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: \(\alpha_\mathrm{B} = \frac{r_\mathrm{A}}{r_\mathrm{B}} \times \alpha_\mathrm{A}\) \(\alpha_\mathrm{B} = \frac{55.0}{30.0} \times (-2.094) = -3.824\, \mathrm{rad/s^2}\)

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: \(\omega_\mathrm{Bf} = \omega_\mathrm{Bi} + \alpha_\mathrm{B} \times t\) \(\omega_\mathrm{Bf} = 23.045 - 3.824 \times 3.00 = 11.573\, \mathrm{rad/s}\)

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: \(\theta_\mathrm{B} = \omega_\mathrm{Bi} \times t + 0.5 \times \alpha_\mathrm{B} \times t^2\) \(\theta_\mathrm{B} = 23.045 \times 3.00 + 0.5 \times (-3.824) \times 3.00^2 = 49.455\, \mathrm{rad}\)

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.

Key concepts

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.