Find the linear speed v for each of the following.


the tip of the minute hand of a clock, if the hand is 7 cm long

Verified step by step guidance

1. Identify the radius of the circular path traced by the tip of the minute hand. Since the minute hand is 7 cm long, the radius \(r\) is 7 cm.
2. Determine the angular speed \(\omega\) of the minute hand. The minute hand completes one full revolution (360 degrees or \(2\pi\) radians) in 60 minutes, so calculate \(\omega\) in radians per minute using the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the period (60 minutes).
3. Recall the relationship between linear speed \(v\) and angular speed \(\omega\): \(v = r \times \omega\).
4. Substitute the values of \(r\) and \(\omega\) into the formula \(v = r \times \omega\) to express the linear speed of the tip of the minute hand.
5. Interpret the result as the linear speed in centimeters per minute, which represents how fast the tip of the minute hand moves along its circular path.

Key Concepts

Angular Velocity

Angular velocity measures how fast an object rotates or revolves relative to a fixed point, expressed in radians per second. For a clock's minute hand, it completes one full rotation (2π radians) every 3600 seconds, which helps determine the rate of rotation needed to find linear speed.

Relationship Between Linear and Angular Velocity

Linear speed (v) at a point on a rotating object is related to angular velocity (ω) by the formula v = ωr, where r is the radius or distance from the rotation axis. This relationship allows conversion from rotational speed to the actual speed along the circular path.

Radius of Rotation

The radius of rotation is the distance from the center of rotation to the point of interest. In this problem, the length of the minute hand (7 cm) serves as the radius, which is essential for calculating the linear speed of the tip of the hand.