Torque and Linear Momentum

What linear momentum, torque and angular momentum share. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The word momentum is used in physics with three distinct meanings: linear momentum, the ability of a force to rotate a body, and the tendency of a system to rotate. Although they share a name, each concept describes a different facet of motion. Understanding what unites them and what separates them is essential to move from linear mechanics to rotational mechanics.

🟢 Essential level

Imagine pushing a shopping cart. If you push it straight, you give it quantity of motion: the more mass and speed it has, the harder it is to stop. That is $p$. Now imagine opening a door: the farther from the hinges you push, the easier it is to turn, even with less force. That is $\tau$. Finally, a spinning top that rotates fast has a large $L$ and therefore takes time to stop. All three share the idea of contained motion, but linear momentum goes in a straight line, $\tau$ produces rotation, and angular momentum describes the rotation itself.

The key that unites them is that all three express contained motion and the tendency to persist. What separates them is the type of motion they describe: $p$ quantifies the advance, $\tau$ quantifies the ability to rotate, and $L$ quantifies how much rotation is already happening. For this reason, understanding these three ideas is necessary to analyze any mechanical system, from a bicycle to a planet in orbit.

🔵 Formal level

Basic formulas

The $p$ of a particle is defined as the product of its $m$ and $v$.

Linear momentum

The $\tau$ measures the rotational capacity of a $F$ and depends on the $r$ to the axis and the angle of application.

Torque

Real student questions

Why is $\tau$ not the same as force? Because force pushes or pulls, but $\tau$ measures the ability of that force to rotate a body, which depends on where it is applied.

Are linear momentum and angular momentum the same thing? No. Linear momentum describes straight-line motion. Angular momentum describes rotation about a point, and can exist even when a body does not spin about itself, such as a planet in orbit.

What happens if $\tau$ is zero? Then angular momentum is conserved. The body maintains its angular velocity.

Why is opening a door far from the hinges easier? Because the same force produces a larger $\tau$ when the distance to the axis is larger.

Final synthesis

Linear momentum, $\tau$, and angular momentum share the idea of contained motion and the structure of a kinematic quantity multiplied by a distance or inertia. They are distinguished by the type of motion they describe: translation, rotational capacity, or actual rotation. Mastering these three ideas allows tackling mechanics problems from the particle to the rigid body.