Find the linear speed v for each of the following.


a point on the edge of a flywheel of radius 2 m, rotating 42 times per min

Verified step by step guidance

1. Start by identifying the given values: the radius of the flywheel \( r = 2 \) meters and the rotation rate is 42 times per minute.
2. Recall the formula for linear speed \( v \) in terms of angular speed \( \omega \): \( v = r \cdot \omega \).
3. Calculate the angular speed \( \omega \) in radians per minute. Since one full rotation is \( 2\pi \) radians, multiply the number of rotations by \( 2\pi \) to convert to radians: \( \omega = 42 \times 2\pi \) radians per minute.
4. Substitute the values of \( r \) and \( \omega \) into the linear speed formula: \( v = 2 \cdot (42 \times 2\pi) \).
5. Simplify the expression to find the linear speed \( v \).

Key Concepts

Linear Speed

Linear speed refers to the distance traveled by a point on a rotating object per unit of time. It is calculated by multiplying the angular speed (in radians per second) by the radius of the rotation. In this context, linear speed helps determine how fast a point on the edge of the flywheel moves in a straight line.

Angular Speed

Angular speed is the rate at which an object rotates around a central point, typically measured in radians per second. It can be derived from the number of rotations per minute (RPM) by converting it into radians, as one complete rotation equals 2π radians. Understanding angular speed is crucial for calculating linear speed in rotating systems.

Conversion of Units

Conversion of units is essential in physics and engineering to ensure that measurements are compatible. In this problem, converting the flywheel's rotation from times per minute to radians per second is necessary for accurate calculations. This process often involves using conversion factors, such as knowing that 1 minute equals 60 seconds and that one rotation equals 2π radians.