How to Convert Linear Distance to Angle on a Diameter

A barrel must rotate farther to cover a greater linear distance.

When an object rolls along a horizontal surface, two different measurements describe the distance it travels. Its linear distance is its displacement as it moves from one point to another. Its angular displacement, however, describes its movement around its axis, with a displacement of pi radians corresponding with a single rotation. The object's diameter determines its circumference, which relates these two measures of displacement.

Instructions

1. Multiply the object's diameter by pi, which is approximately 3.142, to calculate its circumference. For example, if it has a diameter of 30 inches: 30 --- 3.142 = 94.26 in.

2. Divide the linear distance that the object travels by this circumference. For example, if the object covers 500 inches: 500 ÷ 94.26 = 5.304. This is the number of rotations that the object makes.

3. Multiply this answer by pi, which is the number of radians in each rotation: 5.304 --- 3.142 = 16.67 radians. This is the object's angular displacement.

Tips & Warnings

• To calculate displacement in degrees rather than radians, multiply the answer from Step 2 by 360, which is the number of degrees in a circle. With this example: 5.304 --- 360 = 1,909 degrees.

References

• Central Michigan University: Angular Motion; Andrzej Sieradzan
• Western Kentucky University: Angular Velocity vs. Linear Velocity; David Neal

Resources

• Georgia State University: Basic Rotational Quantities; Carl Nave
• Adam Mickiewicz University: Uniform Circular Motion