A fan acquired a rotation rate of 420 cycles per minute with uniform angular acceleration in 11 seconds starting from rest. Find: 1. Angular acceleration 2. Angular displacement du... A fan acquired a rotation rate of 420 cycles per minute with uniform angular acceleration in 11 seconds starting from rest. Find: 1. Angular acceleration 2. Angular displacement during this time interval 3. Time taken to acquire half speed of its maximum. Read more

Steps to Solve

1. Calculate Angular Acceleration

To find the angular acceleration ($\alpha$), we can use the formula:

$$ \alpha = \frac{\Delta \omega}{\Delta t} $$

Where:

• $\Delta \omega$ is the change in angular velocity
• $\Delta t$ is the time taken

Given that the fan starts from rest (initial angular velocity $\omega_0 = 0$) and achieves a maximum speed (final angular velocity $\omega_f$), you need the values for $\omega_f$ and the time it takes to reach that speed.

2. Determine Angular Displacement

To find angular displacement ($\theta$), we can use the formula:

$$ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 $$

Where:

• $\omega_0$ is the initial angular velocity
• $t$ is the time taken to reach that angular displacement

Since the fan starts from rest, $\omega_0 = 0$ simplifies this to:

$$ \theta = \frac{1}{2} \alpha t^2 $$

3. Find Time to Achieve Half Maximum Speed

To achieve half the maximum speed, we need to find the time ($t_{1/2}$) at which the angular velocity is:

$$ \omega_{1/2} = \frac{1}{2} \omega_f $$

Using the angular velocity formula again:

$$ \omega_{1/2} = \alpha t_{1/2} $$

We can solve for $t_{1/2}$:

$$ t_{1/2} = \frac{\omega_{1/2}}{\alpha} $$

4. Substituting Values

Now, substitute $\omega_{1/2}$ into the equation:

$$ t_{1/2} = \frac{\frac{1}{2} \omega_f}{\alpha} $$

Where:

• $\alpha$ was previously calculated.
• $\omega_f$ is the maximum speed of the fan.