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Answer :
The number of revolutions completed by the wheel in the first 10 seconds is 25 revolutions (Option 4).
First, we need to find the angular acceleration of the wheel. We know that angular acceleration [tex](\( \alpha \))[/tex] is the change in angular velocity per unit time.
Given that the wheel starts from rest and makes 900 revolutions in the first minute, we can find the angular velocity at the end of the first minute. Since one revolution is [tex]\( 2\pi \)[/tex] radians, 900 revolutions equal [tex]\( 900 \times 2\pi \)[/tex] radians. Therefore, the angular velocity [tex](\( \omega \))[/tex] at the end of the first minute is [tex]\( \frac{{900 \times 2\pi}}{{60}} \) rad/s.[/tex]
Next, we use the formula[tex]\( \alpha = \frac{{\omega - \omega_0}}{{t}} \), where \( \omega_0 \)[/tex] is the initial angular velocity, which is 0 since the wheel starts from rest, and [tex]\( t \)[/tex] is the time. Since the first minute equals 60 seconds, we have [tex]\( \alpha = \frac{{\omega}}{{t}} = \frac{{900 \times 2\pi}}{{60}} \) rad/s².[/tex]
Now, to find the number of revolutions in the first 10 seconds, we use the equation [tex]\( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).[/tex]
Since [tex]\( \omega_0 = 0 \), this simplifies to \( \theta = \frac{1}{2} \alpha t^2 \).[/tex]Plugging in the values, we get [tex]\( \theta = \frac{1}{2} \times \frac{{900 \times 2\pi}}{{60}} \times (10)^2 \)[/tex]. Solving this gives [tex]\( \theta = 25 \)[/tex]revolutions.
Therefore, the correct answer is 25 revolutions (Option 4).
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