A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.

Verified step by step guidance

1. Identify the given quantities: the linear speed of the thread being pulled off is 59.4 cm/sec, and the spool makes 152 revolutions per minute (rpm).
2. Convert the revolutions per minute to revolutions per second by dividing by 60, since there are 60 seconds in a minute. This gives the angular speed in revolutions per second.
3. Recall the relationship between linear speed (v), radius (r), and angular speed (ω) in revolutions per second: \(v = 2 \pi r \times \text{(revolutions per second)}\).
4. Rearrange the formula to solve for the radius \(r\): \(r = \frac{v}{2 \pi \times \text{(revolutions per second)}}\).
5. Substitute the known values for \(v\) and revolutions per second into the formula to express the radius \(r\) in terms of the given quantities.

Key Concepts

Relationship Between Linear and Angular Velocity

This concept connects the linear speed at which the thread is pulled to the angular speed of the spool's rotation. The linear velocity (v) equals the product of the radius (r) and the angular velocity (ω), expressed as v = rω. Understanding this relationship allows conversion between rotational and linear measurements.

Conversion of Units for Angular Velocity

Angular velocity is often given in revolutions per minute (rpm), but calculations require radians per second. To convert, multiply rpm by 2π (radians per revolution) and divide by 60 (seconds per minute). This ensures consistent units when applying formulas involving angular velocity.

Circumference of a Circle

The circumference of the spool is the distance the thread moves in one full revolution, calculated as 2πr. This concept helps relate the number of revolutions to the total length of thread pulled, linking rotational motion to linear displacement.