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. Understand that the linear speed \( v \) is the distance traveled by a point on the circumference of a circle per unit of time.
2. Recognize that the minute hand of a clock completes one full revolution in 60 minutes.
3. Calculate the circumference of the circle traced by the tip of the minute hand using the formula \( C = 2\pi r \), where \( r = 7 \) cm.
4. Determine the distance traveled by the tip of the minute hand in one full revolution, which is the circumference.
5. Calculate the linear speed \( v \) by dividing the circumference by the time taken for one full revolution (60 minutes).

Key Concepts

Linear Speed

Linear speed refers to the distance traveled per unit of time. In the context of circular motion, it can be calculated using the formula v = rω, where v is the linear speed, r is the radius of the circular path, and ω is the angular speed in radians per second. Understanding linear speed is crucial for determining how fast a point on a rotating object moves along its circular path.

Angular Speed

Angular speed is the rate at which an object rotates around a central point, measured in radians per second. For a clock's minute hand, it completes one full rotation (2π radians) in 60 minutes. Knowing the angular speed allows us to relate it to linear speed, as the minute hand's movement can be described in terms of both its rotational and linear characteristics.

Circumference of a Circle

The circumference of a circle is the total distance around it, calculated using the formula C = 2πr, where r is the radius. For the minute hand of a clock, the length of the hand serves as the radius. This concept is essential for determining the distance traveled by the tip of the minute hand in one complete rotation, which directly influences the calculation of linear speed.