Angular Displacement Formula Calculator

\[ \theta = \frac{s}{r} \]

1. What is the Angular Displacement Formula Calculator?

Definition: This calculator computes the angular displacement (\(\theta\)) of an object in circular motion, defined as the ratio of the arc length (\(s\)) to the radius (\(r\)) using the formula \(\theta = \frac{s}{r}\).

Purpose: It is used in physics and engineering to determine the angle through which an object rotates, applicable in rotational motion, mechanics, and geometry.

2. How Does the Calculator Work?

The calculator uses the angular displacement formula:

Formula: \[ \theta = \frac{s}{r} \] where:

\(\theta\): Angular displacement (rad, deg)
\(s\): Arc length (m, cm, ft, in)
\(r\): Radius (m, cm, ft, in)

Unit Conversions:

Arc Length and Radius:
- 1 m = 1 m
- 1 cm = 0.01 m
- 1 ft = 0.3048 m
- 1 in = 0.0254 m
Angular Displacement (Output):
- 1 rad = 1 rad
- 1 deg = \( \frac{180}{\pi} \) rad \(\approx 57.2958 \, \text{rad}\)

The angular displacement is calculated in radians (rad) and can be converted to the selected output unit (rad, deg).

Steps:

Enter the arc length (\(s\)) and radius (\(r\)) with their units (default: \(s = 1 \, \text{m}\), \(r = 0.5 \, \text{m}\)).
Convert inputs to SI units (m).
Validate that the radius is greater than 0.
Calculate the angular displacement in radians using the formula.
Convert the angular displacement to the selected output unit.
Display the result, rounded to 4 decimal places.

3. Importance of Angular Displacement Calculation

Calculating angular displacement is crucial for:

Physics: Analyzing rotational motion, such as the motion of wheels, pendulums, or celestial bodies.
Engineering: Designing mechanical systems like gears, motors, and turbines where rotational angles are critical.
Education: Teaching the principles of circular motion and geometry in physics and mathematics.

4. Using the Calculator

Examples:

Example 1: Calculate the angular displacement for \(s = 1 \, \text{m}\), \(r = 0.5 \, \text{m}\), output in rad:
- Enter \(s = 1 \, \text{m}\), \(r = 0.5 \, \text{m}\).
- Angular displacement: \(\theta = \frac{1}{0.5} = 2 \, \text{rad}\).
- Output unit: rad (no conversion needed).
- Result: \( \text{Angular Displacement} = 2.0000 \, \text{rad} \).
Example 2: Calculate the angular displacement for \(s = 12 \, \text{in}\), \(r = 24 \, \text{cm}\), output in deg:
- Enter \(s = 12 \, \text{in}\), \(r = 24 \, \text{cm}\).
- Convert: \(s = 12 \times 0.0254 = 0.3048 \, \text{m}\), \(r = 24 \times 0.01 = 0.24 \, \text{m}\).
- Angular displacement in rad: \(\theta = \frac{0.3048}{0.24} = 1.27 \, \text{rad}\).
- Convert to output unit (deg): \(1.27 \times \frac{180}{\pi} \approx 72.7735 \, \text{deg}\).
- Result: \( \text{Angular Displacement} = 72.7735 \, \text{deg} \).

5. Frequently Asked Questions (FAQ)

Q: What is angular displacement?
A: Angular displacement is the angle through which an object rotates about a fixed axis, measured as the ratio of the arc length to the radius.

Q: Why must the radius be greater than zero?
A: Zero or negative radius would result in undefined or meaningless angular displacement, as radius represents the distance from the center of rotation.

Q: What units can angular displacement be measured in?
A: Angular displacement is typically measured in radians (rad), but it can be converted to degrees (deg) for convenience, where \( 1 \, \text{rad} \approx 57.2958 \, \text{deg} \).