Calculating Linear Speed of a Ball Whirling in a Horizontal Circle

Understanding the linear speed of an object in circular motion is crucial in various physical applications. This article will guide you through a detailed example of calculating the linear speed of a ball tied to a cord and making circular motion.

Understanding Circular Motion Basics

In circular motion, the linear speed (or tangential speed) of an object is the rate at which it covers distance along the circumference of the circle. The formula for calculating linear speed is given by:

v frac{d}{t}, where v is the linear velocity, d is the distance traveled, and t is the time taken.

Step-by-Step Calculation

Consider a ball with a mass of 400 grams that is tied to a cord and whirled in a horizontal circle of radius 0.6 meters. The ball makes five complete revolutions in 2 seconds.

Step 1: Calculate the Distance Traveled in One Revolution

The distance traveled in one complete revolution is the circumference of the circle. The formula for the circumference is:

C 2πr, where C is the circumference and r is the radius of the circle.

Given: r 0.6 m

Calculating the circumference:

C 2π × 0.6 ≈ 3.77 meters

Step 2: Calculate the Total Distance for Five Revolutions

If the ball makes five complete revolutions, the total distance d will be:

d 5 × C 5 × 3.77 ≈ 18.85 meters

Step 3: Calculate the Time Taken

The time taken for these five revolutions is given as t 2 seconds.

Step 4: Calculate the Linear Speed

Substituting d and t into the linear speed formula:

v frac{d}{t} frac{18.85 , meters}{2 , seconds} ≈ 9.425 , meters/second

Alternative Methods for Calculation

There are several other methods to calculate the linear speed using different formulas:

Using Angular Velocity and Radius

The linear speed can also be calculated using the formula:

v ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius.

The angular velocity ω can be calculated as:

ω 5 × 2π/2 5π rad/s

Given: r 0.6 m

Substituting these values into the formula:

v 5π × 0.6 m/s 3π m/s ≈ 9.4 m/s

Using Time per Revolution

The time per revolution is given by T frac{2}{5} 0.4 seconds. The linear velocity formula can be used as:

v (frac{2πr}{T})

Substituting the values:

v (frac{2π × 0.6}{0.4} 9.42 m/s)

Conclusion

The ball's linear speed is approximately 9.43 m/s. This example demonstrates the step-by-step process and alternative methods for calculating the linear speed in circular motion.