10.1 Torque and Angular Acceleration

Torque is the rotational equivalent of force: instead of pushing something in a straight line, torque causes an object to rotate around an axis. It depends on both the force you apply and how far from the rotation axis you apply it. Understanding torque is essential for analyzing anything that spins, from opening a door to designing engines.

Net torque determines angular acceleration the same way net force determines linear acceleration. By summing up all the torques on an object and accounting for how its mass is distributed, you can predict exactly how fast it will speed up or slow down rotationally.

Understanding Torque and Angular Acceleration

Definition and effects of torque

Torque measures how effectively a force causes rotation. Two things matter: how hard you push (the force) and how far from the axis you push (the lever arm). A force applied far from the axis produces more torque than the same force applied close to it. That's why door handles are placed at the edge, not near the hinges.

The equation for torque is:

$\tau = rF\sin\theta$

• $r$ is the distance from the axis of rotation to the point where the force is applied
• $F$ is the magnitude of the applied force
• $\theta$ is the angle between the force vector and the position vector (lever arm direction)

The $\sin\theta$ factor means only the component of force perpendicular to the lever arm contributes to rotation. If you push directly along the lever arm ($\theta = 0°$), you get zero torque. Maximum torque happens when the force is fully perpendicular ($\theta = 90°$).

The direction of torque determines the direction of rotation. By convention, counterclockwise torques are positive and clockwise torques are negative.

Calculation of net torque

When multiple forces act on an object, each one can produce its own torque. The net torque is the sum of all individual torques:

$\tau_{net} = \sum \tau_i$

To find net torque, follow these steps:

1. Identify every force acting on the object
2. Determine the lever arm (perpendicular distance from the axis to each force's line of action)
3. Calculate each individual torque using $\tau = rF\sin\theta$
4. Assign signs: positive for counterclockwise, negative for clockwise
5. Add all the torques together

For example, picture a wrench on a bolt. If you push with 20 N at the end of a 0.3 m wrench at 90°, you get $\tau = (0.3)(20)\sin 90° = 6 \text{ N·m}$. If friction at the bolt exerts a 2 N·m clockwise torque, the net torque is $6 - 2 = 4 \text{ N·m}$ counterclockwise.

Angular acceleration from Newton's second law

Newton's second law has a rotational version:

$\tau_{net} = I\alpha$

This says that net torque equals the moment of inertia times the angular acceleration. It's directly analogous to $F_{net} = ma$.

Moment of inertia ($I$) measures how much an object resists changes in its rotation. It depends on both the total mass and how that mass is distributed relative to the axis. Mass concentrated far from the axis means a larger $I$, which means more torque is needed for the same angular acceleration. A solid disk and a hoop of the same mass and radius have different moments of inertia because their mass is distributed differently.

Solving for angular acceleration:

$\alpha = \frac{\tau_{net}}{I}$

This tells you that angular acceleration increases with greater net torque and decreases with larger moment of inertia. A spinning ice skater who pulls their arms in reduces their moment of inertia, which is why they spin faster (though that specific case involves angular momentum conservation, covered later in this unit).

Solving rotational dynamics problems

A consistent approach helps with these problems:

1. Draw a diagram showing the object, the axis of rotation, and all applied forces
2. Identify each force's magnitude, direction, and point of application
3. Calculate the lever arm and torque for each force, assigning correct signs
4. Sum the torques to find $\tau_{net}$
5. Use $\alpha = \frac{\tau_{net}}{I}$ to find angular acceleration (or rearrange to find whichever quantity is unknown)

Key equations to keep ready:

• $\tau = rF\sin\theta$
• $\tau_{net} = I\alpha$
• $\alpha = \frac{\tau_{net}}{I}$

Common problem types include:

• Finding net torque when multiple forces act at different points (e.g., a seesaw with unequal weights)
• Calculating angular acceleration given torque and moment of inertia (e.g., a flywheel starting from rest)
• Rotational equilibrium, where $\tau_{net} = 0$ and the object doesn't accelerate rotationally (e.g., a balanced beam or a stationary sign hanging from a pivot)

Watch for problems where the moment of inertia changes or where you need to combine rotational and translational motion. Always check your sign conventions and make sure your units are consistent (N·m for torque, kg·m² for moment of inertia, rad/s² for angular acceleration).