This note covers how forces combine, how a resultant force changes motion, what friction and drag do, how springs stretch, and why objects moving in circles need a force towards the centre. It is the core of mechanics, and most later work on energy, momentum and pressure builds on it.
Key ideas
Forces and resultant force
A force is a push or a pull, measured in newtons (N). It is a vector. A force can change the speed of an object, change its direction, or change its shape.
When several forces act on an object, they can be replaced by one resultant force. For forces along one line, add those in one direction and subtract those in the opposite direction.
- If the resultant force is zero, the object stays at rest or keeps moving at constant velocity.
- If the resultant force is not zero, the object accelerates in the direction of that force.
The acceleration is found from F = m × a. For the same force, a larger mass gives a smaller acceleration. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s2.
Friction and drag
Friction is a force between surfaces that opposes sliding or the tendency to slide. It acts opposite to the direction of motion and turns kinetic energy into thermal energy, so surfaces heat up. Friction can be reduced with lubricants such as oil. Drag (air resistance or fluid resistance) is the force that opposes motion through a gas or liquid, and it increases as the speed increases.
Falling and terminal velocity
When a skydiver jumps, the weight is the only large force at first, so the acceleration is about 9.8 m/s2. As speed rises, drag rises. The resultant force (weight minus drag) gets smaller, so the acceleration falls. When the drag equals the weight, the resultant force is zero. The acceleration is then zero and the skydiver falls at a constant speed called the terminal velocity. Opening a parachute increases the drag suddenly, so the skydiver slows until a new, lower terminal velocity is reached.
Springs and Hooke's law
A spring stretches when a force is applied. The extension is the new length minus the original length. For a spring that is not overstretched, extension is directly proportional to the force. This is Hooke's law. A graph of force against extension is a straight line through the origin, and its gradient is the spring constant, k, measured in N/m (or N/cm).
Beyond the limit of proportionality, the line curves and the extension grows faster than the force. If the spring is overstretched, it may not return to its original length when the force is removed. Rubber bands and some other materials do not obey Hooke's law at all.
To investigate a spring, hang masses one at a time, measure the new length each time with a ruler, subtract the original length for the extension, and plot a graph. Each 100 g mass adds a force of 0.98 N.
Circular motion (Extended)
An object moving in a circle at constant speed is accelerating, because its direction keeps changing. It needs a resultant force that points towards the centre of the circle, called the centripetal force. Examples are the tension in a string when you whirl a ball, gravity on a satellite, and friction on a car turning a bend. The force needed is larger for a bigger mass, a higher speed, or a smaller radius. If the force disappears, the object moves off in a straight line along the tangent.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Resultant force | The single force equal in effect to all the forces on an object combined. |
| F = m × a | Resultant force (N) = mass (kg) × acceleration (m/s2). |
| Friction | Force opposing sliding between two surfaces. |
| Terminal velocity | Constant speed reached when drag equals the driving force (such as weight). |
| Extension, x | Length of a spring under load minus its original length. |
| F = k × x | Hooke's law: force (N) = spring constant (N/m) × extension (m), up to the limit of proportionality. |
| Limit of proportionality | The point beyond which extension is no longer proportional to force. |
| Centripetal force | Resultant force towards the centre that keeps an object moving in a circle. |
Worked examples
Example 1 A car of mass 1200 kg has an engine force of 3000 N forwards. The resistive forces (friction and drag) total 800 N. Find the resultant force and the acceleration.
Step 1: resultant force = 3000 − 800 = 2200 N forwards.
Step 2: a = F ÷ m = 2200 ÷ 1200 = 1.83 m/s2.
Answer: 2200 N forwards; acceleration 1.83 m/s2.
Example 2 A spring of natural length 20.0 cm extends by 4.0 cm when a force of 6.0 N is applied. It stays within its limit of proportionality. Find the spring constant, and the length of the spring when the force is 9.0 N.
Step 1: convert the extension: 4.0 cm = 0.040 m.
Step 2: k = F ÷ x = 6.0 ÷ 0.040 = 150 N/m.
Step 3: for 9.0 N, x = F ÷ k = 9.0 ÷ 150 = 0.060 m = 6.0 cm.
Step 4: new length = 20.0 + 6.0 = 26.0 cm.
Answer: k = 150 N/m; length 26.0 cm.
Example 3 A skydiver of mass 80 kg (g = 9.8 N/kg) is falling. Find the drag force at terminal velocity, and the acceleration at an instant when the drag is 300 N.
Step 1: weight = m × g = 80 × 9.8 = 784 N.
Step 2: at terminal velocity the forces balance, so drag = 784 N.
Step 3: when drag = 300 N, resultant force = 784 − 300 = 484 N downwards.
Step 4: a = F ÷ m = 484 ÷ 80 = 6.05 m/s2.
Answer: 784 N at terminal velocity; 6.05 m/s2 when the drag is 300 N.
Common errors
- Saying that a zero resultant force means the object is at rest. It may be moving at constant velocity.
- Saying the skydiver stops accelerating because gravity stops. Weight stays the same; the drag has grown until the two balance.
- Using the total length of a spring instead of the extension in F = k × x.
- Leaving the extension in cm when k is in N/m.
- Drawing the centripetal force outwards. It always points towards the centre.
Exam tip: Start any F = m × a question by writing the resultant force, not just one of the forces. Draw a quick sketch with each force as an arrow so you do not miss friction or drag.
Check yourself
- A 5.0 kg trolley is pushed with a resultant force of 20 N. What is its acceleration?
Answer
a = 20 ÷ 5.0 = 4.0 m/s2. - A spring has k = 80 N/m. What force gives an extension of 0.15 m?
Answer
F = k × x = 80 × 0.15 = 12 N. - A parachutist is falling at a steady speed. The weight is 700 N. What is the size of the drag, and why?
Answer
The drag is 700 N. A steady speed means the resultant force is zero, so the upward drag equals the downward weight. - A ball is whirled on a string in a horizontal circle. The string breaks. In which direction does the ball move?
Answer
It moves in a straight line along the tangent to the circle at the point where the string broke, because the centripetal force has gone.