Force in physics:
F = ma, newtons and how net force works

What a force does to a mass

A force is a push or a pull, and its measurable effect is a change in an object's velocity. The more mass the object has, the less it accelerates for the same push. Newton's second law states that relationship as a direct proportion between net force and acceleration, with mass as the constant.

One newton is defined as the force that gives a 1 kg mass an acceleration of 1 m/s². This definition is the reason the formula works with no conversion factors when the units are SI.

Direction matters as well as size. Force is a vector, so a 36 N force pointing east is different from one pointing north, and two equal forces in opposite directions cancel. Problems on slopes or with ropes at an angle first break each force into horizontal and vertical components, then apply F = ma along each direction separately.

The formula and its rearrangements

Convert units before you substitute. Grams must become kilograms, and an acceleration given in km/h per second has to be turned into m/s². Otherwise the answer will not be in newtons.

Worked example: a 12 kg crate

Working backward, a net force of 36 N on a mass of 12 kg gives 36 ÷ 12 = 3 m/s², and the same 36 N on a 4 kg mass gives 9 m/s². The lighter object accelerates three times faster.

Force, time and distance: what comes next

Once the acceleration is known, ordinary motion equations tell you what happens over time. A steady 3 m/s² starting from rest gives a speed of 3 × 4 = 12 m/s after 4 seconds, and the distance covered is ½ × 3 × 4² = 24 m.

Work and energy follow too. A 36 N force acting over those 24 m does 36 × 24 = 864 joules of work, which equals the gain in kinetic energy, ½ × 12 × 12² = 864 J. The two routes give the same answer, which is a handy check on your arithmetic.

Net force: the part people forget

The F in the formula is the vector sum of all forces acting along the line of motion, not any single force on its own. A person pushing a box with 50 N while friction pulls back with 14 N produces a net force of 36 N, and only that 36 N determines the acceleration.

• If net force is zero, acceleration is zero: the object stays at rest or keeps moving at a steady velocity.
• Forces along the same line are added with signs; forces at an angle need components.
• Gravity, friction, tension and air resistance all belong in the sum when they act on the object.

A free-body diagram, which is simply a sketch of the object with an arrow for each force on it, is the most reliable way to build the net force. Label each arrow with its size and direction, choose a positive direction, and add the forces with signs. Most mistakes in force problems come from skipping this step.

Weight is a force; mass is not

Mass is the amount of matter and is measured in kilograms. Weight is the gravitational force on that mass, and it uses the same formula with the free-fall acceleration g, about 9.81 m/s² near the Earth's surface.

A bathroom scale marked in kg is really measuring force and converting it using g. On the Moon, where g is about a sixth as large, the same person has the same mass and a much smaller weight.

The value of g varies a little by location, from about 9.78 m/s² at the equator to about 9.83 m/s² at the poles, and drops with altitude. For school problems 9.81, or sometimes 10, is used, so check which figure your question or exam expects.

Slips that cost marks and accuracy

• Using grams or tonnes without converting to kilograms first.
• Calling weight in kilograms a force and then adding it to forces in newtons.
• Forgetting friction, drag or tension when computing the net force.
• Giving a force with no direction when the problem needs a vector.
• Using the force on one object to find the acceleration of a different one.

For a system of connected objects, pick the object you are analysing and include only the forces acting on it. The equal and opposite partner of each force acts on the other body and does not enter that object's sum.

Checking a force answer

1. 1Write down which object the force acts on and list every force along its line of motion.
2. 2Convert every quantity to kg, m and s.
3. 3Find the net force or the acceleration, depending on what is unknown.
4. 4Check the unit: kg × m/s² is newtons.
5. 5Ask whether the size is plausible: 3,000 N to speed up a car at 2.5 m/s² is about the weight of a 300 kg object.

A calculator is handy for chains of problems, such as finding the force and then the acceleration over a given time, but the setting up of the net force is always the thinking part.

A practical example is a lift. For a 70 kg person standing in a lift accelerating upward at 1.2 m/s², the floor pushes up with 70 × (9.81 + 1.2) = 770.7 N, which is more than their standing weight of 686.7 N. That extra 84 N is why you feel heavier at the start of an upward ride.