Effects of forces

CORE VS EXTENDED GUIDE

• Core students study only the unlabelled sections.
• Extended students must study everything, including Extended Extended points.
• Extended = Core + Supplement.

Forces Change Size and Shape

A force can deform an object — changing its size, its shape, or both. When you squeeze a sponge, the force changes its shape. When you stretch a rubber band, the force increases its length, changing its size. The effect depends on the material and the magnitude of the force applied.

Forces do not always cause motion. A force applied to a rigid wall produces no visible movement, but the wall compresses by a tiny amount.

Therefore, every force either changes the motion of an object or changes its size and shape (or both). Stretching, compressing, bending, and twisting are all examples of forces producing a change in shape.

Load–Extension Graphs for Elastic Solids

A load–extension graph shows how the extension of an elastic solid changes as the applied force (load) increases. Extension means the increase in length beyond the original (natural) length — not the total length.

Exam TipAlways start the line at the origin; do not draw it starting above or below zero extension.

Reading a load–extension graph: The gradient of the straight-line section equals the load divided by the extension. A steeper gradient means a stiffer material (greater force needed per unit extension). The area under the graph represents the work done (energy stored) in stretching the solid.

When the load is removed from an elastic solid within the straight-line region, the solid returns to its original length. This is elastic behaviour.

Examiner InsightCIE often asks students to read values from a load–extension graph and then calculate extension for a different load using the proportional relationship. Always check whether the graph is still in the straight-line region before assuming proportionality. Exam cue: state "the graph is linear so extension is proportional to load" before extrapolating.

MisconceptionStudents often confuse extension with total length. Extension = new length − original length. If the original length is 10 cm and the stretched length is 14 cm, the extension is 4 cm, not 14 cm. Exam cue: always subtract the original length.

Exam TipEnsure the pointer is level with the ruler scale to avoid parallax error.

Extended Spring Constant and Limit of Proportionality

Key Equations

Spring constant equation:

$$k=\frac{F}{x}$$

Variables:

• $k$ = spring constant, in N/m (or N/cm if extension is in cm)
• $F$ = applied force (load), in N
• $x$ = extension, in m

SI unit of $k$: N/m

Rearrangements:

Starting from $k=\frac{F}{x}$:

Multiply both sides by $x$:

$$F=k x$$

Divide both sides by $k$:

$$x=\frac{F}{k}$$

ProportionalityForce is directly proportional to extension (when the spring is within the limit of proportionality). Doubling the extension doubles the force. The spring constant $k$ is constant only within this proportional region.

The spring constant is the force per unit extension. It measures the stiffness of a spring — a larger $k$ means a stiffer spring that requires more force for each metre of extension.

The limit of proportionality is the point beyond which the extension is no longer directly proportional to the applied force. On a load–extension graph, this is the point where the straight line begins to curve. Below this point, $F=kx$ applies and the graph is linear through the origin. Beyond this point, the spring constant is no longer constant and the equation $k=F/x$ no longer gives a fixed value.

MisconceptionThe limit of proportionality is not the same as the elastic limit. CIE does not require knowledge of the elastic limit for this syllabus. The limit of proportionality is identified purely from the graph — where the line stops being straight. Exam cue: label the exact point where the curve departs from the straight line.

Exam TipMark the limit of proportionality at the exact point the line departs from straight — not where it looks "obviously curved."

Worked Example

A spring extends by 8.0 cm when a force of 12 N is applied. Calculate the spring constant.

Converting extension to SI units

$$x=\frac{8.0}{100}$$

$$x=0.080\text{ m}$$

Finding the spring constant

Equation used: spring constant equation

$$k=\frac{F}{x}$$

Given

$$F=12\text{ N}$$

$$x=0.080\text{ m}$$

Substituting:

$$k=\frac{12}{0.080}$$

$$k=150\text{ N/m}$$

Resultant of Forces Along a Line

The resultant force is the single force that has the same effect as all the individual forces acting on an object combined. When forces act along the same straight line, the resultant is found by adding or subtracting.

Representation convention for this unit: Forces along a line are represented by arrows. The length of an arrow represents the magnitude of the force. The direction of the arrow represents the direction the force acts. When two arrows point the same way, add the magnitudes. When two arrows point in opposite directions, subtract the smaller from the larger. The resultant acts in the direction of the larger force.

Sign convention: Taking rightward as positive, a force acting to the right is positive and a force acting to the left is negative. Always state the positive direction before calculating.

Exam TipAlways state both the magnitude and the direction of the resultant force in the final answer.

Newton's First Law — Objects and Resultant Force

An object remains at rest, or continues to move in a straight line at constant speed, unless a resultant force acts on it. This is Newton's first law of motion.

If the resultant force on an object is zero, its velocity does not change. A stationary object stays stationary. A moving object continues at the same speed in the same direction. Therefore, constant speed in a straight line does not require a force — it requires the absence of a resultant force.

A resultant force causes a change in velocity. This change may be a change in speed (the object speeds up or slows down), a change in direction (the object turns), or both.

MisconceptionStudents often believe a moving object needs a constant force to keep moving at constant speed. In fact, constant velocity requires zero resultant force. If a car travels at constant speed, the driving force equals the friction force, so the resultant is zero. Exam cue: "constant speed" means resultant force = 0, not driving force = 0.

Resultant Force Changes Velocity

A resultant force may change the velocity of an object by changing its speed, its direction of motion, or both. Velocity is speed in a stated direction, so any change in either speed or direction counts as a change in velocity.

Solid Friction Between Surfaces

Solid friction is the force between two surfaces that may impede [resist or oppose] motion and produce heating. Friction acts in the opposite direction to the motion (or to the direction the object would move if friction were absent).

When a box is pushed across a floor, friction between the box and floor opposes the push. Energy is transferred from the kinetic store to the thermal store of the surfaces, so both surfaces warm up. Friction does not only act on moving objects — static friction prevents an object from starting to move when a small force is applied.

Rough surfaces produce more friction than smooth surfaces because the irregularities on the two surfaces interlock more. Increasing the force pressing the surfaces together also increases friction.

Examiner InsightCIE commonly asks why a surface heats up during friction. The expected answer involves energy transfer: "friction causes energy to be transferred from the kinetic store to the thermal store of the surfaces, increasing their temperature." Exam cue: always link friction to energy transfer and heating, not just "slowing down."

Friction (Drag) in Liquids and Gases

Drag is the friction force that acts on an object moving through a fluid [a liquid or a gas]. Drag opposes the direction of motion, so it always acts to slow the object down.

Drag in a liquid acts on any object moving through it — for example, a boat moving through water or a ball sinking through oil. The liquid particles collide with the surface of the moving object, exerting a force that opposes the motion.

Drag in a gas is commonly called air resistance when the gas is air. A cyclist, a falling skydiver, and a moving car all experience air resistance opposing their motion through the air.

Drag increases as the speed of the object increases. At higher speeds, the object collides with more fluid particles per second and each collision involves a greater force. This is why a skydiver eventually reaches terminal velocity — the drag force increases until it equals the weight, so the resultant force becomes zero and the speed remains constant.

Exam TipAt terminal velocity, draw the weight and drag arrows exactly the same length to show zero resultant force.

Extended Newton's Second Law — F = ma

Key Equations

Newton's second law equation:

$$F=m a$$

Variables:

• $F$ = resultant force, in N
• $m$ = mass, in kg
• $a$ = acceleration, in m s⁻²

SI unit of $F$: N (1 N = 1 kg m s⁻²)

Rearrangements:

Starting from $F=ma$:

Divide both sides by $m$:

$$a=\frac{F}{m}$$

Divide both sides by $a$:

$$m=\frac{F}{a}$$

ProportionalityForce is directly proportional to acceleration when mass is constant — doubling the resultant force doubles the acceleration. Force is directly proportional to mass when acceleration is constant — doubling the mass requires double the force for the same acceleration. Acceleration is inversely proportional to mass when force is constant — doubling the mass halves the acceleration.

Acceleration is the change in velocity per unit time. The resultant force and the acceleration are always in the same direction. If the resultant force acts to the right, the object accelerates to the right.

The equation $F=ma$ applies only to the resultant force on the object, not to any single individual force (unless only one force acts).

MisconceptionStudents sometimes use $F=ma$ with the weight or the applied force alone, rather than the resultant force. Always calculate the resultant force first, then substitute into $F=ma$. Exam cue: if multiple forces act, find the resultant before using Newton's second law.

Worked Example

A trolley of mass 2.5 kg experiences a resultant force of 7.5 N. Calculate its acceleration.

Finding the acceleration

Equation used: Newton's second law, rearranged for acceleration

$$F=m a$$

Rearranging for $a$:

$$a=\frac{F}{m}$$

Given

$$F=7.5\text{ N}$$

$$m=2.5\text{ kg}$$

Substituting:

$$a=\frac{7.5}{2.5}$$

$$a=3.0{\text{ m s}}^{-2}\text{ in the direction of the resultant force}$$

Extended Circular Motion Due to a Perpendicular Force

An object moving in a circular path requires a force that is always directed perpendicular to its motion — towards the centre of the circle. This force continuously changes the direction of the velocity without changing the speed (if the force magnitude stays constant).

Because the direction of motion changes continuously, the velocity changes continuously, so the object accelerates even if its speed stays constant. The force causing this circular motion is the resultant force acting towards the centre.

Three qualitative relationships describe how force, speed, radius, and mass are connected for circular motion:

These relationships can be understood through cause and effect:

• A greater force towards the centre causes a greater change in direction per unit time, which results in either a higher speed for the same radius or a smaller radius for the same speed.
• A more massive object resists changes in velocity more (greater inertia), so a greater force is needed to maintain the same circular motion.

Examiner InsightCIE tests these three relationships as separate statements. Questions typically describe one variable changing and ask for the effect on another, with the remaining variables held constant. Exam cue: always state which quantities are kept constant before describing the effect.

Exam TipThe force arrow must point towards the centre, not along the direction of motion.

PRACTICAL: Load–Extension Investigation

Aim & Principle: To investigate how the extension of a spring varies with the applied load, verifying that extension is directly proportional to force within the limit of proportionality (the relationship $F=kx$).

Variables

• Independent variable (IDV): applied load (force), in N — varied by adding slotted masses in equal increments (e.g. 1.0 N steps up to 8.0 N)
• Dependent variable (DV): extension of the spring, in mm or cm — measured using a metre rule
• Control variables: same spring used throughout; masses added gently to avoid oscillation; temperature kept constant (thermal expansion could affect spring properties); ruler position fixed throughout

Method

1. Clamp the spring securely to the top of a clamp stand so it hangs vertically. Attach a pointer (e.g. a small piece of card) to the bottom of the spring.

2. Place a metre rule vertically beside the spring. Record the position of the pointer on the ruler — this is the original length, ${l}_{0}$.

3. Hang a weight hanger from the bottom of the spring. Add the first slotted mass (e.g. 100 g = 1.0 N). Wait for the spring to come to rest.

4. Record the new pointer position, $l$. Calculate the extension: $x=l-{l}_{0}$.

5. Repeat by adding masses one at a time up to the maximum planned load. Record the pointer position after each addition.

6. Unload the masses one at a time and record pointer positions during unloading to check for permanent deformation.

7. Plot a graph of load (y-axis) against extension (x-axis).

Measurement technique

Use a metre rule with 1 mm resolution. Read the pointer position at eye level to avoid parallax error. Ensure the pointer is thin and aligned directly against the ruler scale. Check for zero error on the ruler before starting.

Key Observation & Explanation: The extension increases in equal steps as equal loads are added, producing a straight-line graph through the origin. This confirms that extension is directly proportional to load within the proportional region. Beyond a certain load, the graph curves — extension increases more rapidly per unit load — indicating the limit of proportionality has been reached. The spring constant $k$ equals the gradient of the straight-line section.

SafetyWear eye protection in case the spring snaps under high load. Place a soft surface (e.g. a cushion or sand tray) beneath the masses to prevent damage or injury if masses fall. Do not exceed the maximum safe load for the spring.

Recording & Processing:

Load / N	Pointer position / mm	Extension / mm
0.0	(original length ${l}_{0}$)	0
1.0	...	...
2.0	...	...
...	...	...

Calculate extension for each row: extension = pointer position − original length. Calculate the spring constant from the gradient of the straight-line portion.