Understanding Circumference and Rotational Motion in Disk Dynamics

Introduction to Disk Rotation and Circumference

Understanding the relationship between the radius of a disk, its rotation rate, and the distance traveled by a point on its rim is essential in various scientific and engineering applications. In this article, we will explore how to calculate the distance traveled by a point on the outside rim of a disk during a specified period of time.

The Problem Statement

A disk of radius 50 cm rotates at a constant rate of 100 RPM (revolutions per minute). We need to determine the distance in meters that a point on the outside rim of the disk will travel during a 30-second rotation period.

Calculating the Circumference

The circumference of a circle is given by the formula ( C 2pi R ), where ( R ) is the radius of the circle. Given that the radius ( R 50 ) cm, the circumference of the disk is:

[ C 2pi times 50 approx 314 text{ cm} ]

Calculating Rotations and Distance Traveled

In 30 seconds, the disk rotates half a minute, which means it completes:

[ text{Rotations} frac{100 text{ RPM}}{2} 50 text{ rotations} ]

Therefore, the distance traveled by a point on the rim during this time can be calculated by multiplying the number of rotations by the circumference:

[ text{Distance} 50 times 314 approx 15700 text{ cm} ]

Converting this distance to meters:

[ text{Distance} frac{15700 text{ cm}}{100} 157 text{ meters} ]

Confirming the Calculation

The calculation can be verified as follows:

[ text{Distance} 50 times 2pi times 50 approx 5000pi text{ cm} ]

Converting cm to meters:

[ text{Distance} approx frac{5000pi}{100} 157.08 text{ meters} ]

This confirms that the point on the rim of the disk will travel approximately 157.08 meters.

Summary

Through the detailed calculation, we have determined that the distance traveled by a point on the outside rim of a disk with a radius of 50 cm, rotating at 100 RPM, during a 30-second period is approximately 157 meters. This problem showcases the practical application of both geometry and basic physics principles.

Note: The calculations are rounded to two decimal places for simplicity.