Forces

A force is a push or pull that can change the motion of an object. Newt the physics frog will help us build the idea from the ground up: first with no forces at all, and then with some of the most common forces.

Forces are vector quantities. If vector arrows or component addition feel rusty, start with the Vectors page first.

If no force acts on an object, its velocity remains constant. A short push will change Newt’s velocity, but otherwise Newt moves at constant velocity until the next push.

Newton’s Second Law

The net force $\sum \vec{F}$ is the vector sum of every force acting on the object. If the forces are perfectly balanced, we again have constant velocity motion. Otherwise, the object accelerates in the same direction as the net force:

The denominator is the mass of the object. If we’re being philosophically precise, this is also called inertial mass. For the same net force, a more massive object accelerates less, i.e. it has more inertia. Mass is measured in kilograms, and force is measured in newtons: $1\ \text{N} = 1\ \text{kg}\cdot\text{m/s}^2$.

To see the combined effect of two individual forces, let’s place Newt on frictionless ice. He and the sled have a mass of $1\ \text{kg}$, and each crate adds another kilogram. Drag the rope handles to pull him left or right (or both).

Only the net force matters: either push alone could be large, yet the acceleration depends on their difference. Continue playing with Newt below to see some of the other common forces.

Springs

Springs can be compressed or stretched. In both cases, it’s as if springs are attempting to return to their natural length. An ideal spring exerts a force that is proportional to the displacement from its natural length, $\vec{x} = L - L_0$,

The constant $k$, called the spring constant, measures the stiffness. A larger displacement or a stiffer spring produces a larger restoring force.

This spring picture will become our microscopic model for contact forces. Atoms in a solid do not pass freely through one another. Push them a little too close and electric interactions push back; pull neighboring bits slightly apart and they tug back together. For small distortions, that push or pull often behaves like a tiny spring.

Gravity

Gravity is rather complicated and quite magical; the Earth doesn’t need to be touching you to an exert a force. Near Earth’s surface, we usually model it as a steady downward force:

The magnitude of this force $mg$ is known as the weight of the object. In the following diagrams, the force of gravity is then drawn as a weight arrow. When you speak of “gravity”, you should now be specific: do you mean the acceleration due to gravity, the force due to gravity, or just the general phenomena known as gravity?

Normal Force

The normal force is the contact force perpendicular to a surface. Unfortunately, there is no corresponding abnormal force. Normal just means perpendicular. Newt compresses the surface very slightly, and the surface’s microscopic spring-like structure pushes back.

On a flat surface with no vertical acceleration, the normal force usually balances weight. On a ramp or in an accelerating system, the normal force may be smaller or larger than $mg$.

Tension

Tension can also be modeled as a spring-like contact force, but now the microscopic pieces are stretched instead of compressed. A rope can pull because neighboring bits of rope are tugged slightly apart, and those internal bonds pull back toward their relaxed spacing.

Newt has an unusually useful rope: his tongue.

A slack rope, string, cable, or tongue does not apply a tension force. Once it becomes taut, it can pull only along its own length.