Newton's Second Law connects the cause of motion (force) to the change in motion (acceleration). It states that the acceleration of a system's center of mass is directly proportional to the net force exerted on it, and inversely proportional to its mass. In equation form:
Rearranged to highlight what actually depends on what, it reads a = ΣF / m. Acceleration is the output; net force and mass are the inputs. Force is measured in newtons (N), mass in kilograms (kg), and acceleration in meters per second squared (m/s²). One newton is exactly the force needed to accelerate one kilogram at one meter per second squared.
The two proportionalities are the heart of the law. Hold mass constant and double the net force — acceleration doubles. Hold force constant and double the mass — acceleration halves. These aren't separate rules; they both fall straight out of a = ΣF / m.
Change the net force and the mass. Watch how the acceleration — and the length of the acceleration arrow — responds. This is Newton's Second Law.
The ΣF in the equation is the net force — the vector sum of every force acting on the system. A single force rarely acts alone. To use the Second Law you first draw a free-body diagram, add all the force vectors, and only then divide by mass.
A 10 kg crate is pushed to the right with a 60 N force. Friction exerts 20 N to the left. Find the crate's acceleration.
Just like the First Law, the Second Law applies independently in each direction. You write one equation for the x-direction and a separate one for the y-direction: ΣFₓ = maₓ and ΣFᵧ = maᵧ. Solving two-dimensional problems almost always means breaking forces into components first, then applying the law axis by axis.
ΣFₓ = maₓ. Add every horizontal force component, set equal to mass times horizontal acceleration.
ΣFᵧ = maᵧ. Often maᵧ = 0 when an object slides along a flat surface — vertical forces balance.
A 5 kg sled is pulled by a rope at an angle. The horizontal component of tension is 30 N and friction is 10 N. Vertically, the sled stays on the ground. Find the horizontal acceleration.
Because ΣF = ma and mass is always positive, acceleration always points in the same direction as the net force — never the direction of the velocity. This is the source of one of the biggest misconceptions in the course.
When you solve a problem and get a negative acceleration, that sign is telling you a direction, not "slowing down." Always interpret the sign relative to the axis you chose. If net force comes out negative, the acceleration points in the negative direction of your coordinate system.