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Angular speed is a measure of how fast something is rotating. It's often given in revolutions per minute (rev/min). But sometimes we need it in a different unit, like radians per second (rad/s), for calculation purposes.
To convert from rev/min to rad/s, we use the fact that one revolution equals \(2\pi\) radians.
This is simply because \(2\pi \) is the measure of a full circle in radians.
Knowing that there are 60 seconds in a minute helps us factor out the time and get the correct unit.

• Start by multiplying the rev/min by \(2\pi\) to convert to radians.
• Then, divide by 60 to adjust for seconds.

For instance, to convert 200 rev/min to rad/s, calculate: \( \omega = 200 \times \frac{2\pi \ ext{radians}}{60} \approx 20.94\, \text{rad/s}\).
And there you have it, the angular speed in radians per second!

Linear speed is the speed at which a point on the rim of a rotating object is moving along its path.
It's different from angular speed, which measures the rotation itself.
To find linear speed, we use the formula: \( v = r \times \omega \), where \( v \) is the linear speed, \( r \) is the radius of the circular path, and \( \omega \) is the angular speed.

• First, determine the radius of the flywheel. For a diameter of 1.20 meters, the radius is \(0.60\, \text{meters}\).
• Next, use the angular speed in rad/s for calculation.

So using \( \omega \approx 20.94 \text{ rad/s} \) previously calculated, the linear speed is: \( v = 0.60 \times 20.94 \approx 12.56 \text{ m/s} \).
This means a point on the rim moves at 12.56 meters every second!

Angular acceleration indicates how quickly the angular velocity of a rotating object changes.
It's important for understanding situations where the speed of rotation is increasing or decreasing at a constant rate.
The formula for angular acceleration \( \alpha \) is: \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular speed and \( \Delta t \) is the time over which this change occurs.

• First, determine the change in angular speed. If the final speed is 1000 rev/min and initial is 200 rev/min, the change is 800 rev/min.
• Convert this change into rev/s by dividing by 60, resulting in roughly 13.33 rev/s.

Given a time period \( \Delta t = 60\, \text{s} \), calculate \( \alpha \) as: \( \alpha = \frac{13.33}{60} \approx 0.222 \text{ rev/s}^2 \).
This means the flywheel's speed increases by 0.222 revolutions per second squared.

Once we know the angular acceleration, we can find how many revolutions the flywheel makes while it speeds up.
This is described by the equation: \( \theta = \omega_i \times t + \frac{1}{2} \times \alpha \times t^2 \), where \( \theta \) is the total number of revolutions, \( \omega_i \) is the initial angular speed, and \( \alpha \) is the angular acceleration.

• The initial speed \( \omega_i \) needs to be converted from 200 rev/min to rev/s, giving \(3.33\, \text{rev/s}\).
• Substitute this and \( \alpha = 0.222 \text{ rev/s}^2 \)into the equation.

This becomes: \( \theta = 3.33 \times 60 + \frac{1}{2} \times 0.222 \times 60^2 \), which simplifies to about 799.6 revolutions.
During the 60 seconds of accelerating, the flywheel makes approximately 799.6 complete turns!