Torque and Rotational Motion Formulas, Concepts, and Worked Problems

This reference hub brings together the core equations and problem-solving steps for torque and rotational motion. Use it as a compact formula sheet when you need to identify the right equation quickly, then return to the worked examples when a problem combines torque, angular acceleration, and moment of inertia.

Torque and rotational motion at a glance

In fixed-axis rotational dynamics, the central relationship is the rotational equivalent of Newton’s second law: the net torque on a body is related to its moment of inertia and angular acceleration. That link is the foundation of most rotational motion problems you will solve in mechanics.

Core torque formula and variable meanings

• Usable torque relation: τ = rF sinθ
• When the force is perpendicular to the radius, the equation simplifies to τ = rF
• For rotational dynamics with a rigid body about a fixed axis, Στ = Iα
• τ = torque
• I = moment of inertia
• α = angular acceleration
• r = lever arm or moment arm, the perpendicular distance from the axis to the line of action of the force
• F = applied force

The lever-arm idea matters because only the perpendicular component of force contributes to turning the object. If the force is not perpendicular to the radius, the sine factor reduces the effective turning effect. This is why carefully drawing the axis, force direction, and angle is often the fastest way to avoid a setup error.

Moment of inertia formulas for common cases

Moment of inertia is the rotational analogue of mass: it measures how strongly an object resists changes in rotational motion. Two objects with the same mass can behave very differently if their mass is distributed differently about the axis of rotation. That is why axis choice matters and why many exam questions specify the pivot point explicitly.

Rotational kinematics equations used with torque

• Angular velocity relation: ωf = ωi + αt
• Angular velocity should usually be in radians per second
• Conversion from RPM: ω = 2π(RPM/60)
• Use angular kinematics when time, initial angular speed, final angular speed, and angular acceleration are linked in the same problem

If a problem gives speed in revolutions per minute, convert it before solving. Many students lose time by mixing RPM with rad/s, but the equations are written for angular quantities in consistent SI units. Once converted, the motion equation can be combined with the torque relation to find the required angular acceleration or torque.

Newton’s second law for rotation

The rotational analogue of Newton’s second law is Στ = Iα. For a rigid body rotating about a fixed axis, the net torque determines the angular acceleration, just as net force determines linear acceleration in translation. This is the conceptual bridge that connects force-based reasoning to rotational motion calculations.

Worked example: finding angular acceleration from torque

Suppose a rigid body has a moment of inertia of 4.0 kg·m² and the net torque acting on it is 12 N·m. What is its angular acceleration?

Use the fixed-axis relation:

Στ = Iα

Rearrange for angular acceleration:

α = τ / I

Substitute the values:

α = 12 / 4.0 = 3.0 rad/s²

The angular acceleration is 3.0 rad/s². The units work out as expected because torque divided by moment of inertia gives angular acceleration. In a sign-aware problem, the result could be positive or negative depending on your chosen direction convention.

Worked example: stopping a rotating object

Consider a solid sphere with mass 200 kg and diameter 6 m rotating at 180 RPM clockwise. You want to bring it to rest in 10 s. What torque is required?

First, convert the rotation rate to angular velocity. The radius is 3 m, and the initial angular speed is:

ωi = 2π(RPM/60) = 2π(180/60) = 6π rad/s

Because the object is rotating clockwise, choose a sign convention and treat clockwise as negative if that is your positive-axis setup. Then:

ωi = -6π rad/s

The final angular velocity is zero, so use:

ωf = ωi + αt

Rearrange:

α = (ωf - ωi) / t

Substitute:

α = (0 - (-6π)) / 10 = 3π/5 rad/s²

Now find the moment of inertia for a solid sphere:

I = 2/5 mr² = 2/5 × 200 × 3² = 720 kg·m²

Finally, use τ = Iα:

τ = 720 × (3π/5) = 432π N·m

The torque magnitude is 432π N·m, which is about 1.36 × 10³ N·m. If you tracked signs carefully, the torque direction would oppose the initial clockwise rotation.

How to solve rotational motion problems step by step

1. Identify the axis of rotation and decide on a sign convention.
2. List the known variables: torque, mass, radius, angular speed, time, or acceleration.
3. Choose the correct moment of inertia formula for the shape and axis.
4. Convert angular units before calculating, especially RPM to rad/s and degrees to radians.
5. Write the rotational equation first, then substitute the numbers.
6. Check whether the force is perpendicular to the radius; if not, use the perpendicular component.
7. Interpret the sign of the answer and make sure the result is physically reasonable.

Common mistakes with torque and rotational dynamics

• Using linear speed instead of angular speed.
• Forgetting to convert RPM to rad/s.
• Using the wrong moment of inertia for the chosen axis.
• Ignoring the sign of torque or angular acceleration.
• Assuming the force is perpendicular when it is not.

These mistakes usually come from rushing the setup, not from the algebra itself. A quick sketch and a unit check solve many of them before the calculation begins.

Quick formula recap

Keep this page as a revision bookmark. It is designed for quick formula lookup now and for later expansion with more common shapes, more sign-convention notes, and additional worked examples as you move deeper into mechanics.