Interactive physics simulator
Angular Displacement
Explore how angles are swept, measured, and transmitted. Experiment with arc length sectors ($s = r\theta$), plot kinematics curves in real-time under constant acceleration, and analyze gear ratio transmissions.
Angular Displacement Lab
Configure parameters on the right and click Simulate to start loops or see real-time graphical plots.
Live Telemetry
- Radius (r)
- 0 px
- Arc Length (s)
- 0 px
- Displacement (θ)
- 0.00 rad
- Swept Angle (°)
- 0.0°
- Revolutions (rev)
- 0.00 rev
- State
- Static
- Initial speed (ω0)
- 0.0 rad/s
- Acceleration (α)
- 0.0 rad/s²
- Displacement (θ)
- 0.00 rad
- Current Speed (ω)
- 0.00 rad/s
- Rotations
- 0.00 rev
- Elapsed Time (t)
- 0.00 s
- Driver Radius (Ra)
- 0 px
- Driven Radius (Rb)
- 0 px
- Radius Ratio (Ra/Rb)
- 1.00
- Driver Angle (θa)
- 0.00 rad
- Driven Angle (θb)
- 0.00 rad
- Gear Direction
- Counter-rotational
Introduction to Angular Displacement
When an object rotates about a fixed axis, every particle of the object moves along a circular path. While particles located at different distances from the center travel different linear distances, they all rotate through the same angle. Angular displacement is defined as the angle (in radians, degrees, or revolutions) swept by a line segment connecting any point of a rotating body to the axis of rotation.
Key Rotational Concepts
1. Definition of the Radian ($s = r\theta$)
The standard SI unit of angular displacement is the radian (rad). It is a dimensionless unit defined as the ratio of the circular arc length ($s$) swept out by a point to the radius of rotation ($r$):
From this relationship, the linear distance traveled along the arc is:
Since a full circle has a circumference of $2\pi r$, one complete revolution represents an angular displacement of:
2. Sign Convention
By standard physics convention:
- Counter-Clockwise (CCW) rotation is designated as positive (+).
- Clockwise (CW) rotation is designated as negative (-).
This sign convention is consistent with the Right-Hand Rule: curl the fingers of your right hand in the direction of rotation, and your thumb points along the axis of rotation, defining the direction of the angular displacement vector.
3. Rotational Kinematics Equations
For rigid bodies rotating under a constant angular acceleration (α), we can write kinematics equations analogous to linear motion under constant acceleration:
4. Angular Displacement in Gears
In machine assemblies, torque and rotation are transmitted via interlocking gears or belts. If there is no slippage, the linear distance swept by the teeth of intermeshed Driver Gear A and Driven Gear B must be identical ($s_A = s_B$):
Consequently, the angular displacement is inversely proportional to the radius:
This formula is fundamental to gear ratio design: a smaller driven gear rotates faster and through a larger angle, while a larger driven gear turns slower but multiplies the torque.
Solved Numerical Examples
A record player turntable rotates through an angle of 150 radians. Convert this angular displacement into: (a) degrees, and (b) complete revolutions.
View Step-by-Step Solution
- Given: Angular displacement θ = 150 radians.
- To convert to degrees, multiply by 180 / π:
θ = 150 × (180 / π) ≈ 8,594.37°. - To convert to revolutions, divide by 2π (since 1 rev = 2π radians):
θ = 150 / (2π) ≈ 23.87 revolutions. - Results: The turntable has completed approximately 23.87 full rotations, equivalent to 8,594.37°.
A heavy flywheel starts from rest (ω<sub>0</sub> = 0) and rotates with a constant angular acceleration of α = 2.5 rad/s². Find: (a) the angular displacement (θ) after t = 4.0 seconds, and (b) the linear distance (s) traveled by a particle on the rim at a radius of r = 0.40 m.
View Step-by-Step Solution
- Given: ω0 = 0 rad/s, α = 2.5 rad/s², t = 4.0 s, r = 0.40 m.
- Use the second rotational kinematics equation: θ = ω0·t + 0.5·α·t².
- Calculation: θ = 0 · 4 + 0.5 · 2.5 · (4.0)² = 0.5 · 2.5 · 16 = 20.0 radians.
- Use the relationship between linear distance and angular displacement: s = r · θ.
- Calculation: s = 0.40 m · 20.0 rad = 8.0 meters.
A bicycle pedal driver gear A of radius R<sub>A</sub> = 12.0 cm is intermeshed with a rear wheel driven gear B of radius R<sub>B</sub> = 4.0 cm. If the rider rotates the pedal through an angular displacement of θ<sub>A</sub> = 180°, find: (a) θ<sub>A</sub> in radians, and (b) the resulting angular displacement of the rear wheel gear B (θ<sub>B</sub>) in both radians and degrees.
View Step-by-Step Solution
- Given: RA = 12.0 cm, RB = 4.0 cm, θA = 180° = π rad ≈ 3.14 rad.
- Since the gear teeth interlock without slipping, the linear arc length traveled along both circumferences is equal: sA = sB ⇒ RA·θA = RB·θB.
- Solve for driven displacement: θB = θA · (RA / RB).
- In radians: θB = π · (12.0 / 4.0) = 3π radians ≈ 9.42 rad.
- In degrees: θB = 180° · (12.0 / 4.0) = 540°.
Conceptual Practice
What is the fundamental difference between linear displacement and angular displacement?
Show Explanation
Linear displacement measures the straight-line change in position of a point in space (meters), while angular displacement measures the angle of rotation swept by a line segment or rigid body about a fixed axis (radians or degrees). Points at different radii on a rotating wheel have different linear displacements but share the exact same angular displacement.
Why is the radian considered the natural and preferred SI unit for angles in physics, rather than degrees?
Show Explanation
A radian is defined directly as the ratio of arc length to radius ($ heta = s/r$), making it a dimensionless ratio. This definition simplifies equations relating linear and rotational motion (like $v = romega$ and $s = r heta$). If degrees were used, conversion factors containing $pi/180$ would clutter every physics and calculus equation.
If a wheel completes 5.5 revolutions, what is its net angular displacement, and does it depend on the direction?
Show Explanation
One full revolution is $2pi$ radians ($360^circ$). Therefore, 5.5 revolutions equal $5.5 imes 2pi = 11pi$ radians ($1,980^circ$). Yes, it depends on direction: counter-clockwise (CCW) is positive ($+11pi$ rad) and clockwise (CW) is negative ($-11pi$ rad).
In a two-gear transmission, why does the smaller gear rotate through a larger angular displacement than the larger gear?
Show Explanation
Because the teeth of both gears intermesh, they must travel the exact same linear distance (arc length) along their outer rims. Since the smaller gear has a smaller circumference (radius), it must sweep through a larger angle to match the same linear rim distance ($s = R_{ ext{large}} heta_{ ext{large}} = R_{ ext{small}} heta_{ ext{small}}$).
Frequently Asked Questions
What is angular displacement?
Angular displacement is the angle (in radians, degrees, or revolutions) through which a point or line has been rotated about a specified axis in a particular direction.
What is the formula for angular displacement?
The fundamental formula is θ = s / r, where s is the arc length traveled and r is the radius of rotation. In kinematics under constant acceleration, θ = ω<sub>0</sub>t + 0.5αt².
How do you convert radians to degrees?
Multiply the angle in radians by 180 / π (approximately 57.296).
How do you convert revolutions to radians?
Multiply the number of revolutions by 2π (approximately 6.283).
Is angular displacement a vector or scalar?
For large angles, angular displacement does not obey vector addition (commutative law), so it is treated as a scalar with direction (positive/negative). However, for infinitesimal rotations, it behaves as a vector pointing along the axis of rotation using the Right-Hand Rule.
What is a radian?
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. One full circle contains 2π radians.
What are the typical signs for rotation direction?
By convention, counter-clockwise (CCW) rotation is positive (+), and clockwise (CW) rotation is negative (-).
How does the radius affect angular displacement?
For a given arc length, angular displacement is inversely proportional to the radius (θ = s / r). A smaller radius results in a larger angular displacement for the same linear travel.
What is the relation between linear and angular displacement?
The linear distance s traveled along a circular path of radius r is given by s = r · θ, where θ must be in radians.
What happens to angular displacement if a gear is doubled in size?
If a driven gear's radius is doubled, its angular displacement is halved for the same rotation of the driver gear (θ<sub>B</sub> = θ<sub>A</sub> · R<sub>A</sub> / R<sub>B</sub>).
Can angular displacement be negative?
Yes. A negative angular displacement indicates clockwise rotation from the initial reference line.
How is angular displacement measured in real machines?
It is measured using rotary encoders, tachometers, gyroscopes, or potentiometers.