Deformation and load–extension graphs

A force can deform an object by changing its shape or size. An elastic solid returns to its original shape and size when the deforming force is removed, provided it has not been deformed beyond its elastic limit.

When a solid is stretched, its extension is the increase in length from its original, unloaded length. For example, if its original length is $$10\text{ cm}$$ and its stretched length is $$13\text{ cm}$$, its extension is $$3\text{ cm}$$.

Measuring load and extension

To investigate how a load stretches an elastic solid:

1. Suspend the solid securely and measure its original length with no load attached.
2. Add a known load and measure the new length. Subtract the original length to find the extension.
3. Add further known loads, measuring the length and calculating the extension after each addition.
4. Plot the paired values on a graph, with load on the horizontal axis and extension on the vertical axis. Label both axes with their quantities and units.

Measurements can also be taken as the loads are removed. The final measurement shows whether the solid has returned to its original length.

Interpreting a load–extension graph

A straight-line section through the origin shows that extension is proportional to load over that range. For example, doubling the load doubles the extension. If the graph stops following that straight line, this proportional relationship no longer holds.

A return to the original length after unloading is evidence that the solid has returned from its stretch. If it remains longer than its original length, it has not returned to its original size.

Resultant of collinear forces

Several forces can act on an object at the same time. Their resultant force is the single force that has the same combined effect as all of them acting together.

When forces are collinear, they act along the same straight line. To find their resultant, choose one direction as positive and the opposite direction as negative. Then add the forces, including their signs.

Forces in the same direction contribute together; forces in opposite directions oppose one another. The sign of the answer shows the direction of the resultant relative to the direction chosen as positive.

Calculating a collinear resultant

Suppose forces of $$8,\mathrm{N}$$ and $$2,\mathrm{N}$$ act to the right, while a force of $$3,\mathrm{N}$$ acts to the left. Choosing right as positive gives $$8 + 2 - 3 = 7,\mathrm{N}$$. The resultant is therefore $$7,\mathrm{N}$$ to the right.

If the forces in opposite directions are equal, the resultant is zero: the forces balance along that line.

Resultant force and motion

The resultant force is the overall force on an object after all the forces acting on it have been combined, taking their directions into account. It determines whether the object’s motion changes.

Zero resultant force

When the resultant force is zero, the object’s velocity does not change:

• An object at rest remains at rest.
• A moving object continues in a straight line at constant speed.

A zero resultant force does not necessarily mean that no forces act on the object. Forces may act but balance one another, giving an overall force of zero.

Non-zero resultant force

When the resultant force is non-zero, the object’s velocity changes. It may speed up, slow down, change direction, or experience a change in both speed and direction.

Velocity includes direction as well as speed. For example, an object moving around a bend changes direction, so its velocity changes even if its speed stays constant.

Solid friction

Solid friction is a force between two solid surfaces in contact. It acts along the surfaces and opposes their relative motion, or their tendency to move relative to each other.

For example, if a box is pushed across a floor, friction from the floor acts against the box’s sliding motion. Friction can also act before the box starts moving: when the box is pushed but remains at rest, friction opposes its tendency to slide.

Friction can therefore make motion harder to start or continue. When surfaces slide against each other, friction can also produce heating at the contacting surfaces.

Fluid drag

An object moving through a fluid experiences a frictional force that opposes its motion relative to the surrounding fluid. This force is called drag.

A fluid can be a liquid or a gas. A swimmer moving through water experiences drag, as does a cyclist moving through air. Drag acting on an object moving through air is called air resistance.

The direction of drag depends on the object’s motion relative to the fluid, not simply on its motion relative to the ground. For example, if an object moves forward through still air, air resistance acts backwards on it.

Hooke's law

When a force stretches an elastic solid, its extension is the increase in its length. For a solid obeying Hooke’s law, the applied force is proportional to the extension: doubling the force doubles the extension, provided the limit of proportionality is not exceeded.

This relationship is written as:

Here, $$F$$ is the applied force in newtons, $$x$$ is the extension in metres, and $$k$$ is the spring constant. The spring constant is the force per unit extension:

The unit of spring constant is newtons per metre ($$\mathrm{N/m}$$). A larger spring constant means that more force is needed to produce the same extension.

Hooke's law and the limit of proportionality

On a load–extension graph, load is the applied force. The proportional relationship appears as an initial straight line through the origin. The limit of proportionality is the point where this straight-line relationship ends.

Up to this limit, the ratio $$F/x$$ is constant. Beyond it, force and extension are no longer proportional, so that ratio is no longer constant.

Newton's second law

An object accelerates when there is a resultant force on it. The resultant force is the combined effect of all the forces acting on the object. Its acceleration is in the same direction as the resultant force.

Newton’s second law relates the resultant force to the object’s mass and acceleration:

Here, $$F$$ is the resultant force in newtons ($$\mathrm{N}$$), $$m$$ is the mass in kilograms ($$\mathrm{kg}$$), and $$a$$ is the acceleration in metres per second squared ($$\mathrm{m/s^2}$$).

For a given mass, a larger resultant force produces a larger acceleration. For a given resultant force, a larger mass produces a smaller acceleration: more force is needed to give a more massive object the same acceleration.

Force in circular motion

An object moving in a circle continually changes direction. Its velocity therefore changes, even if its speed stays constant. A resultant force is needed to produce this change in velocity.

For circular motion at constant speed, the resultant force points towards the centre of the circle. At every point, the object’s instantaneous motion is along a tangent to the circle, so the force is perpendicular to its motion. As the object moves around the circle, the direction of the force changes so that it continues to point towards the centre.

This inward resultant force changes the direction of the velocity, not its speed. If the speed also changes, the resultant force has an additional component along or opposite to the direction of motion.

Force, speed, radius and mass

The force needed for circular motion depends on the object’s speed and mass, and on the radius of its path:

• Speed: for the same mass and radius, a greater speed requires a greater force.
• Radius: for the same mass and speed, a greater force produces a smaller radius.
• Mass: for the same speed and radius, a greater mass requires a greater force.