Newton's Laws of Motion
Newton's Laws describe the fundamental link between force and motion. They are the essential toolkit for solving nearly all classical mechanics problems by explaining why objects start moving, stop moving, or change direction.
Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- An object's velocity remains constant (including being stationary) unless a net external force acts on it. Motion itself does not require a force, but acceleration does.
- The net force on an object is the vector sum of all forces acting on it. This resultant force is directly proportional to the object's acceleration.Fnet = m × a
- Mass is the measure of an object's inertia, which is its inherent resistance to changes in velocity. It is a scalar quantity measured in kg.
- Forces always occur in pairs. If object A exerts a force on B, object B simultaneously exerts an equal and opposite force of the same type on A. These forces act on different objects and never cancel each other out.
› Why does this happen?
Why does an object keep moving without a force?
This is Newton's First Law, and it contradicts everyday experience because of invisible forces like friction and air resistance. Imagine sliding a book across a carpet – it stops quickly. Now imagine pushing a hockey puck on an ice rink. It glides for a very long time. The difference is the amount of friction. A force isn't needed for motion itself, but to *change* motion (to accelerate). In daily life, a constant push is often needed just to balance out the constant force of friction. If there were no friction, a single push would make an object move at a constant velocity forever.
What is 'inertia' in practical terms?
Inertia is simply an object's resistance to having its velocity changed. Imagine trying to push-start two cars: one is a small city car, the other is a heavy lorry. Both are stationary (velocity = 0 m/s). To get them moving, you need to change their velocity. It's much, much harder to get the lorry moving than the small car. The lorry has a greater resistance to changing its motion. We say it has more inertia. Mass is the way we measure inertia – the lorry has a much larger mass in kilograms.
Why don't action-reaction forces cancel each other out?
This is the most common confusion with Newton's Third Law. The key is that the two forces in a 'pair' always act on *different objects*. If you kick a football, your foot exerts a force on the ball (the 'action'). The ball simultaneously exerts an equal and opposite force on your foot (the 'reaction'). The force on the ball makes the ball accelerate. To work out what happens to you, you only look at the forces acting *on you*, which includes the force from the ball. Since the forces act on different objects, they can't be added together and can never cancel out.
Formulae
Fnet = m × a Use to relate the resultant force (Fnet) on an object of constant mass (m) to the acceleration (a) it experiences. It is the cornerstone for calculating motion dynamics.
Definitions
- Inertia
- The natural tendency of an object to resist a change in its state of motion (i.e., resist acceleration). Mass is the quantitative measure of inertia.
- Resultant Force
- The single force that is the vector sum of all individual forces acting on an object. It is the 'net' force that determines the object's acceleration.
- Newton's Third Law Pair
- Two forces of the same type that are equal in magnitude and opposite in direction, acting on the two different objects that are interacting. For example, the Earth's gravitational pull on you, and your gravitational pull on the Earth.
Worked example
A block of mass 5 kg is pulled along a rough horizontal surface by a rope. The tension in the rope is 30 N. The block accelerates from rest to a speed of 4 m/s over a distance of 10 m. What is the magnitude of the frictional force acting on the block?
- 1
First, determine the block's acceleration.
We are given initial velocity (u=0), final velocity (v=4 m/s), and distance (s=10 m).
The relevant kinematic equation is v2 = u2 + 2as.
- 2
Substitute the values:
42 = 02 + 2 × a × 10, which simplifies to 16 = 20a - 3
Solve for acceleration:
a = 16 / 20 = 4 / 5 = 0.8 m/s2 - 4
Now, apply Newton's Second Law, Fnet = m × a.
The net force is the forward tension minus the backward friction:
Fnet = Tension - Friction - 5
Substitute the known values into Newton's Second Law:
30 - Friction = 5 kg × 0.8 m/s2.
- 6
Calculate the right-hand side:
5 × 0.8 = 4 NSo, the net force is 4 N.
- 7
Finally, solve for the frictional force:
30 - Friction = 4, which gives Friction = 26 N.
Answer: 26 N
Common mistakes
- ×Confusing Newton's Third Law pairs with balanced forces. A book on a table has a downward weight force (from Earth) and an upward normal contact force (from the table). These are balanced forces on the book, NOT a Third Law pair. The Third Law pair to the book's weight is the book's gravitational pull on the Earth.
- ×Using weight (mg) instead of mass (m) in F=ma, or vice-versa. This introduces an error by a factor of g (approx. 10). Remember mass is in kg, weight is in N.
- ×Forgetting to calculate the resultant (net) force before using F=ma. The 'F' in the equation is always the vector sum of all forces, not just the driving force.
- ×Making sign errors. Always define a positive direction for your system at the start. Forces and accelerations in the opposite direction must be treated as negative.
- ×Assuming that if an object is moving, there must be a resultant force in the direction of motion. This is incorrect; a resultant force is only required for acceleration (a change in velocity).
No-calculator tips
- ✓Always draw a quick free-body diagram to visually account for all forces and their directions. This helps prevent misreading the problem and avoids sign errors.
- ✓For F=ma, if g=10 m/s2, multiplying or dividing by 10 is straightforward. Calculations like 5 kg × 0.8 m/s2 can be done as (5 × 8) / 10 = 40 / 10 = 4.
- ✓In multi-step problems, keep values as fractions (e.g., a = 4/5) for as long as possible to avoid arithmetic errors with decimals until the final step.