Centripetal Acceleration and Force in Uniform Circular Motion

This page introduces the concepts of centripetal acceleration and centripetal force in the context of uniform circular motion. It provides fundamental definitions and formulas essential for understanding circular motion in physics.

Definition: Uniform circular motion occurs when a body moves in a circular path at a constant speed.

Although the speed remains constant, the velocity changes direction continuously, resulting in acceleration. This acceleration, known as centripetal acceleration, is directed towards the center of the circle.

Vocabulary: Centripetal acceleration is the acceleration experienced by an object moving in a circular path, always pointing towards the center of the circle.

The formula for centripetal acceleration is given as:

Highlight: Centripetal acceleration = (velocity of body)² / radius of circular path

Centripetal force is the force required to keep an object moving in a circular path. Its magnitude is calculated using the formula:

Highlight: Centripetal force = mv² / r, where m is mass, v is velocity, and r is the radius of the circular path.

The page includes two examples demonstrating centripetal force calculation examples. In the first example, a ball whirled on a string illustrates the calculation of centripetal acceleration. The second example involves a car rounding a turn, showcasing how to determine the required centripetal force.

Example: A 1000-kg car rounds a turn of radius 30 m at a velocity of 9 m/s. The centripetal force required is calculated as F = mv²/r = (1000 kg)(9 m/s)² / 30 m = 2700 N.

These examples help solidify the understanding of uniform circular motion in physics and provide practical applications of the formulas introduced.

Applications of Centripetal Force in Various Scenarios

This page expands on the concept of centripetal force by presenting more complex examples and introducing the idea of motion in a vertical circle.

The page begins with two additional examples of centripetal force calculations:

1. A skater moving in a circular path, demonstrating the conversion of weight to mass in imperial units.
2. Determining the maximum velocity a car can achieve while rounding a turn, given the maximum force the road can exert on the tires.

Example: For a 160-lb skater moving in a 20 ft radius circle at 10 ft/s, the centripetal force is calculated as 25 lb.

The page then introduces the concept of motion in a vertical circle, which is crucial for understanding tension in vertical circle motion. In this scenario, the tension in the string varies with the object's position due to the influence of gravity.

Highlight: In vertical circular motion, the centripetal force is the vector sum of the tension in the string and the component of the object's weight towards the center of the circle.

Two key equations are presented for the tension at different points in the vertical circle:

1. At the top of the circle: T = Fc - w
2. At the bottom of the circle: T = Fc + w

Where T is tension, Fc is centripetal force, and w is the object's weight.

An example is provided to illustrate these concepts:

Example: A 1-kg stone is whirled in a vertical circle with a 0.5 m string at 5 m/s. The tension in the string is calculated at both the top (40.2 N) and bottom (59.8 N) of the circle.

This example demonstrates how the tension varies depending on the position in the vertical circle, providing a practical application of the formulas and concepts introduced.