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Linear velocity refers to how fast a point is moving along a straight path. Imagine a point on the edge of a CD; as the CD spins, this point traces a circular path. The linear velocity is the speed of this point along the circular path. This can be calculated using the formula:

• \[ v = r \times \text{Angular Velocity} \]

Where \( r \) is the radius of the circle and the Angular Velocity is given in radians per second.
For example, if the diameter of a CD is 12 cm, the radius \( r \) is half of that, which is 6 cm. To find the linear velocity when the angular velocity is known, simply plug in the values to the formula.
This calculation helps us understand how fast a point on the edge of a spinning object is moving in terms of distance per unit of time.
In our case, the linear velocity of a point on the CD's edge turned out to be approximately 141.36 cm/second.
This means that if the point were to move along a straight path at this speed, it would cover 141.36 centimeters in one second.

Radians per second is a measure of angular velocity, which tells us how quickly something is rotating. A radian is a unit that measures angles based on the radius of a circle. When we say something is moving at a certain number of radians per second, we are indicating how many radian-lengths the object covers in one second.
To find the angular velocity in radians per second from revolutions per minute, follow these steps:

• Convert revolutions per minute to revolutions per second by dividing by 60.
• Multiply this by \( 2\pi \) (since there are \( 2\pi \) radians in one full revolution).

In the case of the CD which spins at 225 revolutions per minute, converting this gives us 3.75 revolutions per second. When translated to radians per second, using the formula\[ \text{Angular Velocity} = 2 \times \pi \times 3.75 \],
this gives us approximately 23.56 radians/second. This indicates how swiftly the CD is spinning.

Revolutions per minute (RPM) is a way to express the number of turns a rotating object makes around an axis in one minute. It’s a common unit used to represent the speed of rotation.
In practice, you might see RPMs when dealing with CDs, fans, engines, and other mechanical devices.
To relate RPM to other measurements, you might need to convert it. For example:

• To obtain revolutions per second, divide the RPM by 60 (since there are 60 seconds in a minute).
• To convert to radians per second, further multiply by \( 2\pi \).

In our CD example, 225 RPM is converted to 3.75 revolutions per second.
Analyzing RPMs helps in understanding how fast a rotating device or component is functioning relative to its maximum speed or other moving parts.
Understanding RPMs allows one to more easily predict and calculate related motions, like angular or linear speeds.