5.3 — Newton’s second law — F=ma

The second law

Net force on an object = mass × acceleration.

\[ \boxed{\vec F_\text{net} = m\vec a} \]

Three notes:

• Vector equation: direction of acceleration = direction of net force
• Bias: mass remains constant (classical mechanics, not relativity)
• Net force: sum of all forces on the object

Interpreting the equation

A given force produces a given acceleration:

• More force → more acceleration (proportional)
• More mass → less acceleration (inverse)

\[ a = F/m \]

Example: 10 N on 1 kg gives 10 m/s². Same force on 2 kg gives 5 m/s².

Component form

The vector equation splits into three scalar ones:

\[ F_x = m a_x,\quad F_y = m a_y,\quad F_z = m a_z \]

Problem-solving strategy:

1. Decompose all forces into components
2. Sum in each direction separately
3. Solve F/m in each direction

Free-body diagram (FBD)

FBD: a sketch showing only the object and forces acting on it.

Rules:

• Draw the object as a point
• Draw every force as an arrow from that point
• Only forces on this object — not forces this object exerts on others
• Choose meaningful coordinate axes

Example — pulling a box on a frictionless floor

A 10 kg box pulled horizontally by 50 N. Acceleration?

FBD:

• Weight: \vec W = -98\hat j\ N (down)
• Normal: \vec N = +98\hat j\ N (up)
• Pull: \vec F = 50\hat i\ N

Sum in y: -98 + 98 = 0 \Rightarrow a_y = 0. Box stays on the floor.

Sum in x: 50 = 10 a_x \Rightarrow a_x = 5\ m/s^2.

Example — frictionless inclined plane

A 5 kg box released on a 30° slope. Acceleration along slope?

Take x down the slope, y perpendicular.

Decompose weight:

• Along slope: W_x = mg\sin 30° = 24.5\ N
• Perpendicular: W_y = -mg\cos 30° = -42.4\ N

Normal: N = +42.4\ N (cancels).

Along slope: F_\text{net,x} = 24.5\ N \Rightarrow a_x = 4.9\ m/s^2.

Note: acceleration is independent of mass — because both net force and m scale with mass. For any mass, a = g\sin\theta.

Units

• Newton (N): 1\ N = 1\ kg \cdot m/s^2
• Dyne: 1\ dyne = 10^{-5}\ N — CGS unit
• Pound-force (lbf): 1\ lbf \approx 4.45\ N

A few notes and common mistakes

1. F is the net force, not one specific force. If you have four forces, sum them first, then F/m = a.

2. Don't confuse mass and weight. Mass is constant (kg); weight is the gravitational force (N, depending on g). §5.5 goes deeper.

3. Zero acceleration ≠ zero velocity. F_\text{net} = 0 means a = 0, but the object can move at constant velocity.

4. An object without force can't accelerate. If \vec a \neq 0, some force is present.

What you should be able to do

• Write the second law in vector and component form
• Draw a free-body diagram
• Apply the problem-solving strategy to simple problems
• Connect a = F/m with intuition

Preview of §5.4

The third law — every force has an opposite reaction. \vec F_{AB} = -\vec F_{BA}. Deeper than it looks, foundation of momentum conservation.

📚 See also: Halliday Vol 1, Ch 5, §5.3.

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