Answer :
Sure! Let's work through this step by step to find the angular displacement.
1. Identify the initial and final rotational speeds:
- Initial rotational speed, [tex]\( f_{\text{initial}} \)[/tex] = 1.2 Hz
- Final rotational speed, [tex]\( f_{\text{final}} \)[/tex] = 1.8 Hz
2. Time duration:
- Time, [tex]\( t \)[/tex] = 5 seconds
3. Convert rotational speeds to angular velocities:
- Angular velocity is given by [tex]\( \omega = 2\pi f \)[/tex] where [tex]\( f \)[/tex] is the rotational speed in Hz.
- Initial angular velocity, [tex]\( \omega_{\text{initial}} \)[/tex]:
[tex]\[ \omega_{\text{initial}} = 2\pi \cdot f_{\text{initial}} \][/tex]
[tex]\[ \omega_{\text{initial}} \approx 2\pi \cdot 1.2 = 7.5398223686155035 \text{ rad/s} \][/tex]
- Final angular velocity, [tex]\( \omega_{\text{final}} \)[/tex]:
[tex]\[ \omega_{\text{final}} = 2\pi \cdot f_{\text{final}} \][/tex]
[tex]\[ \omega_{\text{final}} \approx 2\pi \cdot 1.8 = 11.309733552923255 \text{ rad/s} \][/tex]
4. Calculate the angular displacement (θ) for uniformly accelerated rotational motion:
- The formula for angular displacement with uniformly accelerated rotational motion is:
[tex]\[ \theta = \frac{1}{2} (\omega_{\text{initial}} + \omega_{\text{final}}) \cdot t \][/tex]
[tex]\[ \theta = \frac{1}{2} (7.5398223686155035 + 11.309733552923255) \cdot 5 = 47.12388980384689 \text{ radians} \][/tex]
5. Express the angular displacement in terms of π:
- To match the possible answers, we need to express θ in terms of π.
[tex]\[ \theta \approx 47.12388980384689 \text{ radians} \][/tex]
Since 47.12388980384689 radians is very close to [tex]\( 15\pi \)[/tex] (because [tex]\( 15\pi \approx 47.12388980384689 \)[/tex]), we can express it as:
[tex]\[ \theta \approx 15\pi \text{ radians} \][/tex]
Therefore, the angular displacement covered within 5 seconds is [tex]\(\boxed{15\pi \text{ rad}}\)[/tex].
1. Identify the initial and final rotational speeds:
- Initial rotational speed, [tex]\( f_{\text{initial}} \)[/tex] = 1.2 Hz
- Final rotational speed, [tex]\( f_{\text{final}} \)[/tex] = 1.8 Hz
2. Time duration:
- Time, [tex]\( t \)[/tex] = 5 seconds
3. Convert rotational speeds to angular velocities:
- Angular velocity is given by [tex]\( \omega = 2\pi f \)[/tex] where [tex]\( f \)[/tex] is the rotational speed in Hz.
- Initial angular velocity, [tex]\( \omega_{\text{initial}} \)[/tex]:
[tex]\[ \omega_{\text{initial}} = 2\pi \cdot f_{\text{initial}} \][/tex]
[tex]\[ \omega_{\text{initial}} \approx 2\pi \cdot 1.2 = 7.5398223686155035 \text{ rad/s} \][/tex]
- Final angular velocity, [tex]\( \omega_{\text{final}} \)[/tex]:
[tex]\[ \omega_{\text{final}} = 2\pi \cdot f_{\text{final}} \][/tex]
[tex]\[ \omega_{\text{final}} \approx 2\pi \cdot 1.8 = 11.309733552923255 \text{ rad/s} \][/tex]
4. Calculate the angular displacement (θ) for uniformly accelerated rotational motion:
- The formula for angular displacement with uniformly accelerated rotational motion is:
[tex]\[ \theta = \frac{1}{2} (\omega_{\text{initial}} + \omega_{\text{final}}) \cdot t \][/tex]
[tex]\[ \theta = \frac{1}{2} (7.5398223686155035 + 11.309733552923255) \cdot 5 = 47.12388980384689 \text{ radians} \][/tex]
5. Express the angular displacement in terms of π:
- To match the possible answers, we need to express θ in terms of π.
[tex]\[ \theta \approx 47.12388980384689 \text{ radians} \][/tex]
Since 47.12388980384689 radians is very close to [tex]\( 15\pi \)[/tex] (because [tex]\( 15\pi \approx 47.12388980384689 \)[/tex]), we can express it as:
[tex]\[ \theta \approx 15\pi \text{ radians} \][/tex]
Therefore, the angular displacement covered within 5 seconds is [tex]\(\boxed{15\pi \text{ rad}}\)[/tex].
