Let's address the question step-by-step. We'll first convert the specified angle to radians and then tackle the questions related to the flywheel's angular velocity.
### 3.1 Convert to Radians
Given:
- Degrees: 112°
- Minutes: 24'
- Seconds: 6''
Step-by-Step Conversion:
1. Convert minutes to degrees: [tex]\( \text{Minutes to degrees} = \frac{24}{60} \)[/tex]
[tex]\[
= 0.4°
\][/tex]
2. Convert seconds to degrees: [tex]\( \text{Seconds to degrees} = \frac{6}{3600} \)[/tex]
[tex]\[
= 0.00166667°
\][/tex]
3. Total degrees: [tex]\( 112° + 0.4° + 0.00166667° \)[/tex]
[tex]\[
= 112.40166667°
\][/tex]
4. Convert degrees to radians: The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex]. So,
[tex]\[
\text{Radians} = 112.40166667° \times \frac{\pi}{180}
\][/tex]
[tex]\[
= 1.961779168062493 \text{ radians}
\][/tex]
Answer for 3.1: [tex]\( 1.961779168062493 \text{ radians} \)[/tex]
### 3.2 Calculations with Angular Velocity
Given:
- Angular velocity, [tex]\( \omega = 300 \text{ rad/s} \)[/tex]
(a) The Rotational Frequency in Revolutions per Second (r/s)
1. Angular velocity to frequency:
The relationship between angular velocity and rotational frequency is given by [tex]\( f = \frac{\omega}{2\pi} \)[/tex].
[tex]\[
f = \frac{300 \text{ rad/s}}{2\pi}
= 47.7464829275686 \text{ r/s}
\][/tex]
Answer for 3.2(a): [tex]\( 47.7464829275686 \text{ r/s} \)[/tex]
(b) The Angular Displacement in 1 Minute
1. Convert time to seconds: 1 minute = 60 seconds.
2. Angular displacement: The angular displacement [tex]\( \theta \)[/tex] can be found by multiplying angular velocity by time:
[tex]\[
\theta = \omega \times \text{time}
= 300 \text{ rad/s} \times 60 \text{ s}
= 18000 \text{ radians}
\][/tex]
Answer for 3.2(b): [tex]\( 18000 \text{ radians} \)[/tex]
(c) The Rotational Frequency in Revolutions per Minute (r/min)
1. Convert frequency from r/s to r/min:
To convert the rotational frequency from revolutions per second to revolutions per minute, multiply by 60.
[tex]\[
\text{Frequency (r/min)} = 47.7464829275686 \text{ r/s} \times 60
= 2864.788975654116 \text{ r/min}
\][/tex]
Answer for 3.2(c): [tex]\( 2864.788975654116 \text{ r/min} \)[/tex]
In summary:
- 3.1: [tex]\( 112°24'6" \)[/tex] in radians is [tex]\( 1.961779168062493 \)[/tex] radians.
- 3.2(a): The rotational frequency in r/s is [tex]\( 47.7464829275686 \)[/tex].
- 3.2(b): The angular displacement in 1 minute is [tex]\( 18000 \)[/tex] radians.
- 3.2(c): The rotational frequency in r/min is [tex]\( 2864.788975654116 \)[/tex].
