Gear B has an angular acceleration, a = ¹²+t+rad/s² (t is in seconds). The radius of Gear A is 2 inches and the radius of Gear B is 4 inches. Determine the angular velocity and the angular displacement of gear A at t = 3 seconds. A B Angular Velocity, WA = Angular Displacement, A = Include units.
Gear B has an angular acceleration, a = ¹²+t+rad/s² (t is in seconds). The radius of Gear A is 2 inches and the radius of Gear B is 4 inches. Determine the angular velocity and the angular displacement of gear A at t = 3 seconds. A B Angular Velocity, WA = Angular Displacement, A = Include units.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Gear Mechanics Problem Solving**
**Problem Statement:**
Gear B has an angular acceleration, \( \alpha_B = \frac{1}{4}t^2 + \frac{2}{5}t + \frac{1}{6} \) rad/\(s^2\) (t is in seconds).
The radius of Gear A is 2 inches and the radius of Gear B is 4 inches. Determine the angular velocity and the angular displacement of Gear A at \( t = 3 \) seconds.
**Visual Aid Description:**
The provided image displays two interlocking gears labeled as Gear A and Gear B. Gear A is the smaller gear, positioned to the left, and Gear B is the larger gear, positioned to the right. The image aids in visualizing the relative sizes and positions of the gears, which are important for understanding the mechanical relationship and calculations.
**Formulas Needed:**
1. Angular Velocity: \( \omega_A \)
2. Angular Displacement: \( \theta_A \)
**Given Data:**
- Angular acceleration of Gear B, \( \alpha_B = \frac{1}{4}t^2 + \frac{2}{5}t + \frac{1}{6} \) rad/\( s^2 \)
- Radius of Gear A, \( r_A = 2 \) inches
- Radius of Gear B, \( r_B = 4 \) inches
**Tasks:**
Determine the following at \( t = 3 \) seconds:
1. Angular Velocity, \( \omega_A \)
2. Angular Displacement, \( \theta_A \)
**Solution Steps:**
To find the angular velocity \( \omega_A \) and angular displacement \( \theta_A \) of Gear A, follow these steps:
1. Integrate the angular acceleration of Gear B (\( \alpha_B \)) with respect to time to find the angular velocity (\( \omega_B \)).
2. Integrate \( \omega_B \) to determine the angular displacement (\( \theta_B \)).
3. Relate the angular motion of Gear B to Gear A using the gear ratio (\( \frac{r_A}{r_B} \)).
**Calculations:**
(Include detailed calculations)
**Results:**
- **Angular Velocity, \( \omega_A \) =** [Enter value with appropriate units]
- **Angular Displacement, \( \theta_A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42a8b670-f4b9-41aa-8a38-e9f581824926%2F1e9f5560-077e-4458-921b-59f91d85346b%2F3pskc1_processed.png&w=3840&q=75)
Transcribed Image Text:**Gear Mechanics Problem Solving**
**Problem Statement:**
Gear B has an angular acceleration, \( \alpha_B = \frac{1}{4}t^2 + \frac{2}{5}t + \frac{1}{6} \) rad/\(s^2\) (t is in seconds).
The radius of Gear A is 2 inches and the radius of Gear B is 4 inches. Determine the angular velocity and the angular displacement of Gear A at \( t = 3 \) seconds.
**Visual Aid Description:**
The provided image displays two interlocking gears labeled as Gear A and Gear B. Gear A is the smaller gear, positioned to the left, and Gear B is the larger gear, positioned to the right. The image aids in visualizing the relative sizes and positions of the gears, which are important for understanding the mechanical relationship and calculations.
**Formulas Needed:**
1. Angular Velocity: \( \omega_A \)
2. Angular Displacement: \( \theta_A \)
**Given Data:**
- Angular acceleration of Gear B, \( \alpha_B = \frac{1}{4}t^2 + \frac{2}{5}t + \frac{1}{6} \) rad/\( s^2 \)
- Radius of Gear A, \( r_A = 2 \) inches
- Radius of Gear B, \( r_B = 4 \) inches
**Tasks:**
Determine the following at \( t = 3 \) seconds:
1. Angular Velocity, \( \omega_A \)
2. Angular Displacement, \( \theta_A \)
**Solution Steps:**
To find the angular velocity \( \omega_A \) and angular displacement \( \theta_A \) of Gear A, follow these steps:
1. Integrate the angular acceleration of Gear B (\( \alpha_B \)) with respect to time to find the angular velocity (\( \omega_B \)).
2. Integrate \( \omega_B \) to determine the angular displacement (\( \theta_B \)).
3. Relate the angular motion of Gear B to Gear A using the gear ratio (\( \frac{r_A}{r_B} \)).
**Calculations:**
(Include detailed calculations)
**Results:**
- **Angular Velocity, \( \omega_A \) =** [Enter value with appropriate units]
- **Angular Displacement, \( \theta_A
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