Answer :
The question is to find time taken to complete the first revolution
Answer:
1.81 seconds
Explanation:
The constant angular equation for first revolution
[tex]\theta = {\theta _0} + {\omega _0} + \frac{1}{2}\alpha {t_1^2}[/tex]
Where, [tex] \theta[/tex] is angular displacement at time [tex]t_1[/tex], [tex]{\theta _0}[/tex] is the angular displacement at time t = 0, [tex]{\omega _0}[/tex] is the initial angular velocity, [tex]\alpha[/tex] is angular acceleration and [tex]t_1[/tex] is the time takes to complete first revolution.
Angular displacement at time [tex]{t_1}[/tex] to complete the first revolution is,
[tex]\theta = 2\pi[/tex]
Substitute [tex]2\pi[/tex] for [tex]\theta[/tex], 0 for [tex]{\theta _0}[/tex] , 0 for [tex]{\omega _0}[/tex] then we have
[tex]\theta = {\theta _0} + {\omega _0} + \frac{1}{2}\alpha {t_1}^2[/tex]
[tex]\begin{array}{c}\\2\pi = 0 + 0 + \frac{1}{2}\alpha {t_1}^2\\\\2\pi = \frac{1}{2}\alpha {t_1}^2\\\end{array}[/tex]
Also [tex]\alpha[/tex]
[tex]\alpha = \frac{{4\pi }}{{{t_1}^2}}[/tex]
The time takes to complete second revolution is,
[tex]{t_2} = 0.75s[/tex]
The time takes to complete first two revolutions is,
[tex]\[{t^'} = {t_1} + {t_2}\][/tex]
The constant angular equation for first two revolution is,
[tex]\[\theta = {\theta _0} + {\omega _0} + \frac{1}{2}\alpha {t^'}^2\][/tex]
Where, [tex]\theta[/tex] is angular displacement at time [tex]\[{t^'}\][/tex], [tex]{\theta _0}[/tex] is the angular displacement at time [tex]\[t = {t^'}\][/tex], [tex]{\omega _0}[/tex] is the initial angular velocity, [tex]\alpha[/tex] is angular acceleration and [tex]\[{t^'}\][/tex] is the time takes to complete the first two revolutions.
The angular displacement at time t to complete the first two revolutions is,
[tex]\theta = 4\pi[/tex]
Substitute [tex]4\pi[/tex] for [tex]\theta[/tex] 0 for [tex]{\theta _0}[/tex] , 0 for [tex]{\omega _0}[/tex] and [tex]{t_1} + {t_2}[/tex] for [tex]\[{t^'}\][/tex] in equation [tex]\[\theta = {\theta _0} + {\omega _0} + \frac{1}{2}\alpha {t^'}^2\][/tex]
[tex]\begin{array}{c}\\4\pi = 0 + 0 + \frac{1}{2}\alpha {\left( {{t_1} + {t_2}} \right)^2}\\\\4\pi = \frac{1}{2}\alpha {\left( {{t_1} + {t_2}} \right)^2}\\\end{array}[/tex]
Substitute [tex]\frac{{4\pi }}{{{t_1}^2}}[/tex] for [tex]\alpha[/tex] in the above equation.
[tex]\begin{array}{c}\\4\pi = \frac{1}{2}\left( {\frac{{4\pi }}{{{t_1}^2}}} \right){\left( {{t_1} + {t_2}} \right)^2}\\\\2{t_1}^2 = {\left( {{t_1} + {t_2}} \right)^2}\\\\\sqrt 2 {t_1} = \pm \left( {{t_1} + {t_2}} \right)\\\end{array}[/tex]
Since time is positive
[tex]\begin{array}{c}\\\sqrt 2 {t_1} = + \left( {{t_1} + {t_2}} \right)\\\\{t_1} = \frac{{{t_2}}}{{\sqrt 2 - 1}}\\\end{array}[/tex]
Substitute 0.75s for [tex]{t_2}[/tex]
[tex]\begin{array}{c}\\{t_1} = \frac{{0.75s}}{{\sqrt 2 - 1}}\\\\ = 1.81s\\\end{array}[/tex]
(a) The computer disk drive did take a time of approximately 0.530 seconds.
(b) The angular acceleration of the computer disk drive is approximately 44.680 radians per square second.
An example of rotational motion
(a) The change in angular displacement ([tex]\theta[/tex]) is determined by the following kinematic formula:
[tex]\theta = \omega_{o}\cdot t +\frac{1}{2}\cdot \alpha\cdot t^{2}[/tex] (1)
Where:
- [tex]\omega_{o}[/tex] - Initial angular speed, in radians per second.
- [tex]t[/tex] - Time, in seconds
- [tex]\alpha[/tex] - Angular acceleration, in radians per square second.
First, we find the angular acceleration: ([tex]\theta = 4\pi[/tex], [tex]\omega_{o} = 0\,\frac{rad}{s}[/tex], [tex]t = 0.750\,s[/tex])
[tex]\alpha = \frac{2\cdot (\theta-\omega_{o}\cdot t)}{t^{2}}[/tex]
[tex]\alpha = \frac{2\cdot \left[4\pi-\left(0\,\frac{rad}{s} \right)\cdot (0.750\,s)\right]}{(0.750\,s)^{2}}[/tex]
[tex]\alpha \approx 44.680\,\frac{rad}{s^{2}}[/tex]
And lastly we find the time required for the disk drive to make its first revolution: ([tex]\theta = 2\pi[/tex], [tex]\omega_{o} = 0\,\frac{rad}{s}[/tex], [tex]\alpha \approx 44.680\,\frac{rad}{s^{2}}[/tex])
[tex]\omega_{o}\cdot t + \frac{1}{2}\cdot \alpha\cdot t^{2} - \theta = 0[/tex] (2)
[tex]22.34\cdot t^{2}-2\pi = 0[/tex]
[tex]t \approx 0.530\,s[/tex]
The computer disk drive did take a time of approximately 0.530 seconds. [tex]\blacksquare[/tex]
(b) The angular acceleration of the computer disk drive is approximately 44.680 radians per square second. [tex]\blacksquare[/tex]
Remark
The statement is incomplete and poorly formatted. Correct form is presented below:
A computer disk drive is turned on starting from the rest and has constant angular acceleration. If it took 0.750 seconds for the drive to make its second revolution,
(a) How long did it take to make the first revolution?
(b) What is the angular acceleration, in radians per square second?
To learn more on rotational motion, we kindly invite to check this verified question: https://brainly.com/question/15120445