⚡Newton's Second Law
F = ma
Newton's Second Law: The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass.
ΣF = ma
This is arguably the most important equation in all of classical mechanics. Let's unpack it: - ΣF (sigma F): the net force — the vector sum of ALL forces acting on the object (in N) - m: mass in kg (not weight!) - a: acceleration in m/s²
The direction of acceleration is the same direction as the net force.
Mass vs. Weight
Mass is the amount of matter in an object (kg). It's the same everywhere in the universe.
Weight is the gravitational force on an object (N):
W = mg
where g = 9.8 m/s² ≈ 10 m/s² near Earth's surface.
A 70 kg person has the same mass on the Moon as on Earth, but weighs about 6× less on the Moon because g_moon ≈ 1.6 m/s².
Newton's Second Law Playground
Apply forces to the block and watch it accelerate. Change mass and force to see F = ma in action. The FBD, v-t graph, and numbers all update in real time.
Think About It
If you push a 2 kg box with 10 N and there's 4 N of friction, what is the net force? What is the acceleration?
✏️ Worked Example
Problem: A 5 kg box on a frictionless surface is pushed by a 20 N force to the right. Find the acceleration.
✏️ Worked Example 2: Atwood Machine
Problem: Two masses (3 kg and 5 kg) hang from a massless pulley. Find the acceleration of the system.
📐 Key Equations
Newton's Second Law
Σ F⃗ = ma⃗W = mga = (Σ F)/(m)⚠️ Common Mistakes
Misconception: You can use the total weight of the system to find the acceleration of one part of it.
✓ Correct thinking: Apply F = ma to each object separately using only the forces acting on that object. The mass you use must match the object you are analyzing.
Why: Using system-level forces with individual masses (or vice versa) gives wrong results, especially in pulley and multi-body problems.
Misconception: Mass and weight are the same thing.
✓ Correct thinking: Mass (kg) is the amount of matter; it is constant everywhere. Weight (N) is the gravitational force W = mg, which depends on the local gravitational field g.
Why: Plugging a weight value (in N) into an equation that calls for mass (in kg) is a unit error that invalidates the calculation.
Misconception: If an object is accelerating upward, the normal force must be greater than its weight.
✓ Correct thinking: That is actually correct — but students often forget the reverse: if an object accelerates downward (or is in free fall), N < mg or N = 0.
Why: Always write Newton's Second Law (ΣF = ma) for the vertical direction, substituting the correct sign of a, rather than assuming N equals mg.
📝 Practice Problems
Try these problems. Check your answer when ready.
A net force of 36 N acts on a 9 kg object. What is the acceleration?
a = (Σ F)/(m)What is the weight of a 70 kg person on Earth (g = 10 m/s²)? What would their weight be on the Moon where g = 1.6 m/s²?
A 12 kg box is pushed horizontally with 48 N. Friction exerts 12 N opposing motion. Find (a) net force, (b) acceleration.
An elevator of mass 800 kg accelerates upward at 2 m/s². What is the tension in the cable supporting it? (g = 10 m/s²)
T - mg = maA 4 kg and an 8 kg block are connected by a string on a frictionless surface. A 24 N force pulls the 8 kg block. Find (a) acceleration of the system and (b) tension in the string between them.
In an Atwood machine, mass A = 6 kg and mass B = 4 kg hang over a massless frictionless pulley. Find (a) the acceleration and (b) the tension in the rope. (g = 10 m/s²)
a = ((m_A - m_B)g)/(m_A + m_B)A 5 kg block rests on a 30° frictionless incline. What is the acceleration of the block along the incline? (g = 10 m/s²)
a = g\sinθFinished reading through this lesson?