A tire 0.500 m in radius rotates at a constant rate of 200...

A tire $0.500 \mathrm{~m}$ in radius rotates at a constant rate of 200 rev $/ \mathrm{min}$. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge). (Hint: In one revolution, the stone travels a distance equal to the circumference of its path, $2 \pi r .$ )

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the inward acceleration required to keep an object moving in a circular path. It is determined using the formula a = v^2 / r, where v is the tangential speed and r is the radius of the circle. This concept is fundamental in understanding the forces acting on an object in circular motion.

Tangential Speed

Tangential speed is the linear speed experienced by a point on the edge of a rotating object. It is directly related to angular speed and is calculated by multiplying the angular speed by the radius of the circle, linking rotational motion with linear displacement.

Circumference of a Circle

The circumference is the total distance around the edge of a circle and is calculated by the formula 2?r, where r is the radius. This concept is key in determining the distance an object travels in one complete revolution.

Uniform Circular Motion

Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed. Even though the speed remains constant, the object experiences acceleration toward the center of the circle, known as centripetal acceleration, due to the continuously changing direction of velocity.

Angular Speed and Conversion

Angular speed is the rate at which an object rotates and is typically measured in radians per second. When angular speed is provided in other units, such as revolutions per minute, it must be converted to radians per second to be used properly in calculations involving linear speed and acceleration.

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