Newton’s laws

Choosing the system and reference frame

A system is the body or collection of bodies you choose to study. Forces exerted by objects outside the system are external; interactions between parts of the system itself are internal. A rope can exert an external force on a block, but the same interaction can be internal when we study the block and rope together.

List the interactions before writing symbols: Earth’s gravitation, contact with a surface, a rope’s pull, air resistance. Each interaction corresponds to a force exerted on the system. The resultant ΣF is the vector sum of the external forces, not an additional force to add to those already present.

Newton’s laws in their usual form apply in inertial reference frames. A laboratory fixed to Earth is a sufficient approximation for many short-duration, small-scale school problems. In a braking vehicle, the vehicle’s reference frame is accelerating; the feeling of being thrown forwards does not require a real forwards force on the passenger.

First law: the natural state needs no push

If the resultant is zero, a body maintains its velocity vector: rest or uniform straight-line motion. This is the first law, associated with inertia. A force is needed to change velocity, not to “feed” the existence of motion. In everyday life, resistive forces hide this distinction.

A box dragged at constant speed experiences a force from the hand and balancing resistive forces. If the hand stops pulling and friction remains, the resultant points opposite to the velocity and the box slows down. It is not an absence of force that stops it; it is the presence of a resistive resultant.

Translational mechanical equilibrium means ΣF = 0, so a = 0 for constant mass. It does not necessarily mean constant position. Moreover, this lesson treats the body as a point or considers translation only: to prevent an extended object from rotating, the moments of forces must also be analysed, which requires another equilibrium condition.

Second law: a vector equation

For constant mass, ΣF = ma. Along perpendicular axes, this equation becomes ΣF_x = ma_x and ΣF_y = ma_y. Acceleration points in the direction of the resultant, not necessarily in the direction of velocity. The newton unit confirms the connection: N = kg·m/s².

One equation per axis prevents quantities pointing in different directions from being added as if they were positive numbers. In horizontal motion with no vertical acceleration, a_y = 0, but a_x may be non-zero. The vertical forces balance without requiring the horizontal forces to balance.

Mass measures resistance to a change in velocity: with the same resultant, doubling mass halves acceleration. For gravitation near the surface, P = mg, with g = 9,8 m/s² in this lesson. Equality of inertial and gravitational mass means that, without air, different bodies have the same falling acceleration. Equal weight is not a necessary condition for equal acceleration.

Third law: pairs acting on different bodies

If A exerts a force on B, B simultaneously exerts another force on A with the same magnitude and line of action, but the opposite direction. Write the agents in the names: force of the foot on the ground and force of the ground on the foot. This makes it clear that the two members of the pair act on different bodies.

The pairs do not cancel in the equation for a single body. When walking, the foot pushes the ground backwards and the ground exerts forwards friction on the foot. The person’s diagram includes the force of the ground on the person. The force of the person on the ground belongs in Earth’s diagram.

If we study the two bodies as a system, these forces become internal and their sum is zero. Even so, each body can accelerate differently because their masses differ. In an interaction between a light cart and a heavy one, the paired forces have equal magnitudes; the accelerations need not be equal.

Normal force and tension: consequences of constraints

The normal force N is perpendicular to the contact surface. It is not synonymous with weight. On a horizontal support, with only weight and normal force vertically and a_y = 0, we obtain N = mg. If an additional force pulls the body upwards, the normal force decreases; if the body accelerates vertically upwards, the normal force may exceed mg.

Tension is the force a stretched rope exerts when pulling a body. An ideal rope is massless and inextensible; in a continuous rope passing over an ideal frictionless pulley, tension has a uniform magnitude. A real rope may have mass and behave differently. A slack rope does not push: tension may become zero.

On a plane inclined at angle θ, choose one axis parallel and another perpendicular to the plane. Weight resolves into mg sin θ down the slope and mg cos θ into the support. With no other perpendicular forces and no acceleration in that direction, N = mg cos θ. Sine and cosine arise from the geometry of the components, not from a rule to memorise without a drawing.

Friction: two regimes that must not be confused

Friction between surfaces opposes relative sliding or its tendency. It can help a body move forwards: when a vehicle starts, the ground may exert forwards friction on the driven wheels. You must observe the contact and tendency to slide, rather than claiming that friction always opposes the body’s velocity.

In the simple dry-friction model, static friction adjusts up to the limit f_s,max = μ_sN. Before sliding, we write f_s ≤ μ_sN; the maximum value is not always imposed. When sliding occurs, we use approximately f_k = μ_kN, where μ has no unit.

This model depends on the surfaces and operating regime. It does not universally describe deforming tyres, lubrication or air resistance. To solve a problem: draw the forces, determine N from the perpendicular axis, test whether static friction can balance the applied force, and use the kinetic regime only when sliding occurs. Finally, check whether the predicted acceleration is consistent with the initial assumption.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The resultant is another force to draw.

It is the sum of the real forces already represented.

The normal force always equals weight.

It is obtained from the equation perpendicular to the contact.

Static friction is always μsN.

That is the limit; the value adjusts below it.

Action and reaction cancel on the same body.

They act on different bodies.