How to Calculate Angular Displacement: Formula, Units, and Solved Problems
Angular displacement is a fundamental concept in Physics, especially in understanding rotational motion. When an object moves along a circular path, its change in position can be described using angles rather than straight-line distances. This angular measure helps analyze the rotation of objects in a precise and consistent way.
What is Angular Displacement?
Angular displacement refers to the angle through which a point, line, or body rotates about a fixed axis. It measures the difference between an object's starting and ending positions, described in terms of angles rather than linear distance.
Unlike linear displacement, which measures movement in a straight line, angular displacement looks at how much an object has turned from its original position. This is especially useful when analyzing rotating wheels, gears, or any circular motion.
Formula for Angular Displacement
The standard formula used to calculate angular displacement is given by:
Δθ = Δs / r
Here, Δθ is angular displacement (in radians), Δs is the distance along the arc (in meters), and r is the radius of the circle (in meters).
This equation lets you convert a distance moved along a circular path into the equivalent change in angle, measured in radians.
Step-by-Step Approach to Solve Angular Displacement Problems
- Identify if the object is moving along a circular path and determine the center or axis of rotation.
- Measure or calculate the arc length (Δs) that the object has travelled.
- Find the radius (r) of the circle along which motion occurs.
- Apply the formula Δθ = Δs / r to find angular displacement in radians.
- If needed, convert radians to degrees (multiply the value in radians by 180/π).
Example Problem
Suppose a point moves 4 meters along the edge of a circle with a radius of 2 meters. To find the angular displacement:
Δθ = Δs / r = 4 / 2 = 2 radians
The point has turned through 2 radians from its initial position.
Key Formulas & Their Application
| Formula | Variable | Meaning | Unit |
|---|---|---|---|
| Δθ = Δs / r | Δθ | Angular Displacement | Radian (rad) |
| Δs = r × Δθ | Δs | Arc Length | Meter (m) |
Comparison: Angular vs Linear Displacement
| Angular Displacement | Linear Displacement |
|---|---|
| Measures the angle turned by a radius or line (typically in radians). | Measures direct straight-line distance from one point to another (in meters). |
| Useful in circular motion and rotations. | Useful in straight-line (translational) motion. |
| Δθ = Δs / r | Δx = xfinal – xinitial |
Applications and Practice
Understanding angular displacement is key for topics like rotational kinematics, gears, motors, and biomechanics. Practice solving problems by applying the core formula and identifying whether questions involve rotational or translational motion.
Try similar problems using different arc lengths and radii. Also, check your understanding with example cases from related topics, such as:
Next Steps for Deeper Learning
- Review the steps in calculating angular displacement and solve practice questions.
- Explore advanced topics such as Angular Acceleration and Angular Momentum.
- Study rotational motion concepts, including Rotational Kinetic Energy, for a complete understanding.
For structured lessons and more practice, refer to relevant sections on Vedantu for guided learning and topic mastery.
Summary
Angular displacement provides a clear way to describe how much an object has rotated compared to where it started, using the formula Δθ = Δs / r. Applying this formula helps analyze many problems in rotational motion and related fields.
Use Vedantu resources for stepwise learning on all key Physics concepts. Continue to practice with a focus on using formulas accurately for success in all rotational motion topics.
FAQs on What is Angular Displacement? Explained with Formulas, Units & Examples
1. What is the definition of angular displacement?
Angular displacement is the angle through which an object moves on a circular path around a fixed axis. It measures the change in position of a line (radius) joining the center of rotation to the object, expressed in radians. It is a vector quantity, having both magnitude and direction (along the axis of rotation).
2. What is the SI unit and symbol for angular displacement?
The SI unit of angular displacement is the radian (rad). The most common symbol used is θ (theta) or Δθ for change in angle.
3. What is the angular displacement formula?
Angular displacement (Δθ) can be calculated by:
- Δθ = θfinal – θinitial
- If arc length is known: Δθ = s / r, where s = arc length and r = radius.
4. Is angular displacement a scalar or vector quantity?
Angular displacement is a vector quantity. It has both magnitude (the angle swept) and direction (given by the right-hand rule, along the axis of rotation).
5. What is the difference between angular and linear displacement?
Key differences:
- Angular displacement measures the angle rotated by an object (in radians), while linear displacement measures the distance moved in a straight line (in metres).
- Linear displacement s = radius r × angular displacement (Δθ) when measured in radians.
6. How do you calculate angular displacement when given arc length and radius?
Use the formula Δθ = s / r, where:
- s = arc length the object travels
- r = radius of the circular path
- Resulting Δθ is in radians.
7. Why is angular displacement measured in radians and not degrees?
Angular displacement is measured in radians for consistency with SI units and because radians provide a direct link between arc length and radius (arc = radius × angle (in radians)). This also simplifies equations and calculations in physics.
8. Can angular displacement be negative? What does a negative value indicate?
Yes, angular displacement can be negative. A negative value indicates rotation in the opposite direction to the chosen positive sense (usually clockwise is negative and counter-clockwise is positive, by the right-hand rule).
9. What is the relationship between angular velocity and angular displacement?
Average angular velocity (ω) is defined as the rate of change of angular displacement over time: ω = Δθ / Δt, where Δθ is angular displacement and Δt is the time taken.
10. How would you represent angular displacement graphically?
Angular displacement vs time graph plots angle on the y-axis and time on the x-axis. A straight line indicates constant angular velocity; a curved line shows changing angular velocity or acceleration.
11. What are the most important formulas involving angular displacement for exams?
Key formulas to remember:
- Δθ = θfinal – θinitial
- Δθ = s / r
- Δθ = ω × t (if ω is constant)
- ω = Δθ / Δt
12. How is angular displacement relevant in rotational motion questions for JEE and NEET?
Angular displacement is a foundational concept for solving rotational motion problems in JEE, NEET, and other exams. It helps connect angular velocity, acceleration, and linear kinematics, and forms the basis for questions involving wheels, pulleys, and rotating bodies.
