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Rotational Motion Calculator

Calculate torque, angular acceleration, moment of inertia, and analyze rotational dynamics for circular objects

Parameters

kgⓘ
mⓘ
Nⓘ
°ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Torque:
5.00;N⋅m5.00;N·m
Moment of Inertia:
0.50;kg⋅m20.50;kg·m²
Angular Acceleration:
10.00;rad/s210.00;rad/s²
Tangential Acceleration:
5.00;m/s25.00;m/s²
Angular Velocity (t=2s):
20.00;rad/s20.00;rad/s
Angular Displacement (t=2s):
20.00;rad20.00;rad

Examples

Example 1: Door Opening

A 2 kg door with radius 0.5 m is pushed with 10 N force at 90° angle.

  • Torque: 5.005.00
  • Moment of Inertia: 0.500.50
  • Angular Acceleration: 10.0010.00
  • Tangential Acceleration: 5.005.00

Example 2: Wheel Rotation

A 5 kg wheel with radius 0.3 m is turned with 20 N force at 45° angle.

  • Torque: 4.244.24
  • Moment of Inertia: 0.450.45
  • Angular Acceleration: 9.429.42
  • Tangential Acceleration: 2.832.83

Example 3: Heavy Disk

A 10 kg disk with radius 1.0 m is rotated with 50 N force at 30° angle.

  • Torque: 25.0025.00
  • Moment of Inertia: 10.0010.00
  • Angular Acceleration: 2.502.50
  • Tangential Acceleration: 2.502.50

Visualization

Rotational Motion and Torque

Rotational motion is the motion of an object around a fixed axis. It is described by angular quantities such as angular displacement, angular velocity, and angular acceleration, which are analogous to their linear counterparts.

Torque is the rotational equivalent of force and is defined as τ = r × F = rF sin θ, where r is the distance from the axis of rotation to the point where the force is applied, F is the force magnitude, and θ is the angle between r and F.

Moment of inertia (I) is the rotational equivalent of mass and depends on the mass distribution relative to the axis of rotation. For a point mass at distance r from the axis: I = mr². For a solid disk: I = ½mr².

Newton's second law for rotation states that τ = Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration. This is analogous to F = ma for linear motion.

Angular momentum (L) is the rotational equivalent of linear momentum and is given by L = Iω, where ω is the angular velocity. Angular momentum is conserved when no net external torque acts on the system.

Key Concepts

  • Torque: τ = rF sin θ (rotational force)
  • Moment of Inertia: I = mr² for point mass
  • Angular Acceleration: α = τ/I
  • Angular Velocity: ω = ω₀ + αt
  • Angular Displacement: θ = θ₀ + ω₀t + ½αt²
  • Angular Momentum: L = Iω (conserved)

Real-World Applications

  • Automotive engines and transmissions
  • Electric motors and generators
  • Gyroscopes and navigation systems
  • Centrifuges and separators
  • Wind turbines and propellers
  • Sports equipment (discus, hammer throw)

Explore Further

More mechanics tools

Physics Equations

Torque:
τ=rFsin⁡θ\tau = rF\sin\theta
Moment of Inertia (Point Mass):
I=mr2I = mr^2
Angular Acceleration:
α=τI\alpha = \frac{\tau}{I}
Angular Velocity:
ω=ω0+αt\omega = \omega_0 + \alpha t
Angular Displacement:
θ=θ0+ω0t+12αt2\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2
Angular Momentum:
L=IωL = I\omega

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Torque

Calculate the torque using the formula:

Equation:

τ=rFsin⁡θ\tau = rF\sin\theta

Calculation:

τ=(0.5)(10.0)sin⁡(90.0°)=5.00 N\cdotpm\tau = (0.5)(10.0)\sin(90.0°) = 5.00 \text{ N·m}

Explanation:

Torque is the rotational equivalent of force and depends on the force magnitude, distance, and angle.

2

Step 2: Calculate Moment of Inertia

Calculate the moment of inertia for a point mass:

Equation:

I=mr2I = mr^2

Calculation:

I=(2.0)(0.5)2=0.50 kg\cdotpm²I = (2.0)(0.5)^2 = 0.50 \text{ kg·m²}

Explanation:

Moment of inertia is the rotational equivalent of mass and depends on mass distribution.

3

Step 3: Calculate Angular Acceleration

Use Newton's second law for rotation:

Equation:

α=τI\alpha = \frac{\tau}{I}

Calculation:

α=5.000.50=10.00 rad/s²\alpha = \frac{5.00}{0.50} = 10.00 \text{ rad/s²}

Explanation:

Angular acceleration is the rate of change of angular velocity.

4

Step 4: Calculate Tangential Acceleration

The tangential acceleration is related to angular acceleration:

Equation:

at=αra_t = \alpha r

Calculation:

at=(10.00)(0.5)=5.00 m/s²a_t = (10.00)(0.5) = 5.00 \text{ m/s²}

Explanation:

Tangential acceleration is the linear acceleration of a point on the rotating object.

Frequently Asked Questions (FAQ)

What is torque?

Torque is the rotational equivalent of force. It is the product of the force magnitude, the distance from the axis of rotation, and the sine of the angle between the force and the radius vector.

How does moment of inertia differ from mass?

Moment of inertia depends not only on mass but also on how the mass is distributed relative to the axis of rotation. It's the rotational equivalent of mass in linear motion.

What is the relationship between torque and angular acceleration?

Torque equals the product of moment of inertia and angular acceleration: τ = Iα. This is the rotational equivalent of Newton's second law F = ma.

When is angular momentum conserved?

Angular momentum is conserved when no net external torque acts on the system. This is the rotational equivalent of conservation of linear momentum.

How does the angle affect torque?

Torque is maximum when the force is perpendicular to the radius (θ = 90°) and zero when the force is parallel to the radius (θ = 0° or 180°).

Practice MCQs

  1. Torque is maximum when the angle between force and radius is:
  2. The moment of inertia of a point mass m at distance r from the axis is:
  3. If the radius is doubled, the moment of inertia becomes:
  4. Angular acceleration is related to torque by:
  5. Angular momentum is conserved when:

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