Rotational Motion Problems: Torque, Angular Momentum, and Rolling Without Slipping

Overview

Most rotation questions combine several familiar physics ideas. A wheel may have angular acceleration, a rope may exert torque, and the wheel may also translate as it rolls. The key is to separate the motion into the right model instead of searching for one formula that does everything.

Use this decision rule:

• Use rotational kinematics when angular acceleration is constant and the problem asks about angular displacement, angular speed, or time.
• Use torque and Newton’s second law for rotation when forces, lever arms, or angular acceleration are central: Στ = Iα.
• Use angular momentum when external torque is zero or negligible over the time interval, especially in collisions or changes in rotational shape: L = Iω.
• Use energy when the problem compares positions and speeds and friction or other energy transfers can be modeled clearly.
• Use rolling constraints when an object rolls without slipping: v_cm = Rω and, when the motion is accelerating, a_cm = Rα.

A reliable setup begins with a sketch. Mark the axis of rotation, show every force, label the radius or perpendicular lever arm, and choose positive clockwise or counterclockwise rotation. For broader problem-solving steps, compare this process with the step-by-step physics problem-solving method.

Core rotational equations

Checklist by scenario

Scenario 1: A force turns a wheel or pulley

1. Draw the free-body diagram and identify the rotational axis.
2. Choose a positive rotation direction.
3. Calculate each torque using τ = rFsinφ, where φ is the angle between the radius vector and force.
4. Write Στ = Iα.
5. If the force also accelerates a mass, write ΣF = ma for that mass.
6. Connect the motions with a rope or rolling condition, such as a = Rα.

Worked pulley problem. A 3.0 kg mass and a 2.0 kg mass hang on opposite sides of a pulley with radius 0.10 m and moment of inertia 0.20 kg·m². Assume the rope does not slip and use g = 9.8 m/s². Let the 3.0 kg mass move downward.

For the heavier mass, m₁g - T₁ = m₁a. For the lighter mass, T₂ - m₂g = m₂a. For the pulley, (T₁ - T₂)R = Iα. Because the rope does not slip, α = a/R.

Combining the equations gives:

a = (m₁ - m₂)g / (m₁ + m₂ + I/R²)

a = (1.0)(9.8) / (3.0 + 2.0 + 0.20/0.10²) = 0.392 m/s².

Thus α = a/R = 3.92 rad/s². The pulley’s rotational inertia reduces the acceleration compared with an ideal massless pulley. If your answer is larger than the acceleration from the ideal-pulley model, revisit the signs and the effective inertia.

Scenario 2: A solid object rolls down an incline

For rolling without slipping, use both translation and rotation. A useful energy equation is:

Mgh = ½Mv² + ½Iω², with ω = v/R.

For a solid cylinder, I = ½MR². If it rolls from rest through a vertical height h, then:

Mgh = ½Mv² + ¼Mv², so v = √(4gh/3).

For acceleration down an incline, the general result is:

a = gsinθ / (1 + I/MR²).

For a solid cylinder on a 30-degree incline, this gives a = 9.8(0.5)/(1 + 0.5) = 3.27 m/s². The static friction force points up the incline in this model because it provides the torque needed for rotation. Static friction does not automatically oppose the direction of motion; it opposes the tendency of the contact point to slip.

Scenario 3: Rotational equilibrium

An object is in rotational equilibrium when Στ = 0. If it is also stationary, require ΣF = 0. Choose a convenient pivot, often at a support, so forces through that point produce zero torque.

For a horizontal 4.0 m beam with a 200 N load 1.0 m from the left support and an unknown force at the right end, taking torques about the left support gives:

F(4.0 m) - (200 N)(1.0 m) = 0, so F = 50 N.

Then use the vertical force equation to determine the remaining support force. Always include the beam’s own weight if its mass is given; its weight acts at the center of mass.

Scenario 4: Angular momentum and short interactions

Use angular momentum when the external torque about your chosen axis is negligible. For a person stepping onto a rotating platform, conservation gives:

I_initialω_initial = I_finalω_final.

If a platform has I = 2.0 kg·m², rotates at 3.0 rad/s, and a person adds 0.50 kg·m², then:

ω_final = (2.0)(3.0)/(2.0 + 0.50) = 2.4 rad/s.

The angular speed decreases because the total moment of inertia increases. This is different from a torque problem over a longer interval: first ask whether external torque is negligible during the interaction.

What to double-check

• Lever arm: The torque magnitude is F times the perpendicular distance from the axis to the force’s line of action, not always rF.
• Moment of inertia: Use the correct shape and axis. A ring, disk, solid cylinder, rod, and point mass do not share the same expression.
• Units: Torque has units of N·m, angular speed has units of rad/s, and angular acceleration has units of rad/s². Radians are dimensionless in unit calculations, but should remain visible for clarity.
• Signs: Write a torque-sign convention before substituting numbers. Counterclockwise positive is common, but either choice works.
• Constraints: “No slipping” allows v = Rω. A slipping rope or tire needs a different model and may involve kinetic friction.
• Energy losses: If friction, air resistance, or deformation is important, do not silently assume mechanical energy is conserved.
• Axis consistency: Angular momentum and moment of inertia must refer to the same axis unless you explicitly use a theorem or coordinate transformation.

For connected force problems, review a Newton’s laws practice guide. For conservation problems involving collisions or impulses, see the momentum and collision problem guide.

Common mistakes

1. Using rotational kinematics without checking constant acceleration. Equations such as ω² = ω₀² + 2αθ require constant α.
2. Confusing force with torque. A large force close to the axis may produce less torque than a smaller force applied farther away.
3. Assuming tension is the same on both sides of a massive pulley. Different tensions are often required to produce the pulley’s angular acceleration.
4. Setting friction to zero in a rolling problem. Static friction may be essential for torque even though it can do no net work in ideal rolling.
5. Conserving angular momentum when an external torque acts. A motor, brake, axle friction, or support force can invalidate the conservation step about a selected axis.
6. Mixing degrees and radians. Rotational equations use radians. Convert angle data before using it in equations.
7. Skipping a limiting-case check. If the pulley’s inertia approaches zero, your result should approach the ideal-pulley result. If the radius becomes very large or small, check whether the predicted angular acceleration behaves sensibly.

When to revisit

Return to this checklist before a mechanics test, after completing a new set of rotational motion problems, or whenever your course changes the allowed equation sheet or notation. Revisit it especially after an error involving signs, moments of inertia, or rolling constraints. A useful review cycle is to solve one torque problem, one equilibrium problem, one rolling problem, and one angular-momentum problem without looking at the solutions. Then compare not only the final answer but also your model choice, diagram, units, and assumptions.

Before submitting homework or preparing for an exam, use this final action list:

1. Draw and label the system.
2. Choose the axis and positive direction.
3. List forces, torques, and known quantities.
4. Decide whether the problem is primarily kinematics, torque, energy, angular momentum, or a combination.
5. Write equations symbolically before inserting numbers.
6. Check units, signs, limiting cases, and whether the result answers the question asked.
7. Record the specific mistake or decision that slowed you down so the next practice problem targets it.

For a broader collection of equations, consult the physics formula sheet and equation finder. The goal is not to memorize every rotational formula separately; it is to recognize the physical condition that makes each equation appropriate.