Elasticity and net force

Hang a load from a spring. The spring stretches and settles at a new length. Why does it not keep stretching forever? And how can its extension tell us the force when Earth also acts on the load?

Squeeze a rubber ball, then bend a soft wire, and let go. The ball usually regains its shape, while the wire may stay bent. A change in an object's shape or size is deformation. If the change disappears when the load is removed, it is elastic deformation. The part that remains is plastic, or permanent, deformation. A real object may deform elastically at first, then acquire a permanent change under a larger load.

A spring starts with length l₀ and has length l after stretching. Does its response depend on its whole length or on how much that length changed? What matters is the extension Δl = l − l₀, measured in metres. The elastic restoring force acts to reduce deformation: a stretched spring pulls its ends towards each other. In its original undeformed state Δl = 0, but the full length is not zero. Full length cannot replace extension.

Add small loads to the same spring. Within a certain range, doubling the load doubles the extension. To compare springs, define the spring constant k as restoring force divided by extension in that linear range. Its unit is N/m. A stiffer spring needs more force for the same extension; under the same force, it stretches less. The constant describes the spring, including both its material and its construction.

Write the observed proportionality as Fупр = kΔl: Hooke's law for the magnitude of force during stretching in the linear elastic range. Multiplying k in N/m by Δl in m gives newtons. Double the extension in this range and the force doubles; at zero deformation the force is zero. We cannot extend this straight-line relationship to arbitrarily large loads. Even elastic recovery does not guarantee that force is proportional to extension.

If the same force always produces the same extension, the spring can become a measuring instrument. Apply known forces and mark the pointer positions in newtons, then compare an unknown force with those marks. This is a spring force meter. Check zero before reading and wait for a steady indication. Exceeding its measuring range may permanently deform the spring, making the old scale inaccurate.

A load on a spring experiences gravity downwards and spring force upwards. Their combined effect determines how its motion changes. The net force R is the vector sum of the forces acting on one chosen object, measured in newtons. Here we either model the object as a point particle, ignoring size and rotation, or consider forces along one line. We cannot add forces on different objects this way: they change the motion of different objects.

Two people pull the same trolley along one line in the same direction with forces of 2 N and 3 N. The actions reinforce each other: R = F₁ + F₂ = 5 Н, in the same direction. If one force becomes zero, the other remains; doubling both doubles the result. Adding magnitudes this way works only when the directions agree. For forces at an angle, simply adding numbers is not enough.

What if the trolley is pulled in opposite directions? A 3 N force to the left cancels part of the 7 N force to the right, leaving R = 4 Н to the right. The magnitude is the larger force minus the smaller, and the direction is that of the larger force. Equal opposite forces along one line give R = 0. Inertia then means unchanged velocity, not necessarily a stop. Separate lines of action can still cause rotation, which this simple model does not describe.

A spring of constant 100 N/m lengthens from 10 to 12 cm within the range of Hooke's law. Find the restoring force.

First find the extension: 12 − 10 = 2 см = 0,02 м.

Then Fупр = kΔl = 100·0,02 = 2 Н.

The force acts to restore the original length. Do not substitute the full 12 cm into the formula.

The restoring force has a magnitude of 2 N.

A spring responds to a change in length. Hooke's law uses extension, not full length.

The proportionality Fупр = kΔl is established experimentally and applies only in the linear elastic range.

The combined effect depends on force directions and which object they act on. A zero net force preserves the existing velocity.

Friction is often among the forces on a moving object. Next we will explore when it hinders motion and when it helps.