Effects of forces Revision Notes

A force is a push or pull that can cause an object to accelerate, change direction, or deform. When a force acts on an object, it can change the object's size or shape by deforming it. Deformation means the object changes its dimensions or form due to the applied force.

Deformation occurs because the particles within the material are displaced from their original positions. The nature and extent of this displacement depend on the type of force and the material's properties.

• Force: a vector quantity that can cause changes in motion or shape.
• Deformation: change in size or shape of an object due to applied force.

Forces can deform objects in several ways, depending on how the force is applied: stretching, compressing, bending, and twisting.

Stretching occurs when forces pull an object apart, increasing its length.

Compression happens when forces push an object together, decreasing its length or volume slightly if the material is compressible (e.g., a spring). This is a pushing force acting inward on the object’s surface area, causing it to shorten or squash slightly if elastic properties allow it to do so without permanent damage. Bending involves forces applied at different points causing the object to curve or bend. This is common in beams or rods when a force is applied perpendicular to their length at one or more points, causing tension on one side and compression on the other. Twisting (torsion) occurs when forces cause an object to rotate about its axis, changing its shape by twisting.

• Stretching: pulling forces increase length.
• Compression: pushing forces decrease length or volume.
• Bending: forces cause curvature by tension and compression.
• Twisting: forces cause rotation about an axis, changing shape.

Deformation can be either elastic or plastic, depending on whether the object returns to its original shape after the force is removed.

Elastic deformation is temporary. When the force is removed, the object returns to its original size and shape. This occurs when the material is not stressed beyond its elastic limit. For example, a stretched spring returns to its original length when released.

Plastic deformation is permanent. The object does not return to its original shape after the force is removed because the material has been stressed beyond its elastic limit, causing permanent changes in the arrangement of particles. For example, bending a metal wire beyond a certain point causes it to remain bent.

• Elastic deformation: temporary, reversible shape change.
• Plastic deformation: permanent, irreversible shape change.
• Elastic limit: maximum force before permanent deformation occurs.

The extent to which a force changes an object's size or shape depends on several factors:

The magnitude of the force: larger forces generally cause greater deformation.

The material properties: some materials are more elastic and return to their original shape easily, while others deform permanently more readily. For example, rubber is highly elastic, while clay deforms plastically easily. The shape and size of the object: thinner or longer objects may deform more easily under the same force than thicker or shorter ones.

• Greater force causes greater deformation.
• Material elasticity determines reversibility of deformation.
• Object dimensions influence deformation extent.

A simple experiment to observe deformation is stretching a spring or elastic band and measuring its extension.

Apparatus: spring or elastic band, ruler or metre stick, weights, clamp stand.

Method: 1. Attach the spring vertically to a clamp stand. 2. Measure the original length of the spring. 3. Add a known weight to the spring and measure the new length. 4. Calculate the extension (new length - original length). 5. Repeat for different weights and record results. 6. Remove weights and observe if the spring returns to its original length (elastic deformation). 7. If the spring does not return fully, plastic deformation has occurred. Variables: Independent variable: applied force (weight). Dependent variable: extension of the spring. Control variables: spring type, temperature, initial length. Repeat measurements improve accuracy and help identify anomalies. Safety includes handling weights carefully to avoid dropping.

• Measure extension under varying forces.
• Observe elastic or plastic behaviour.
• Control variables for valid results.
• Repeat for reliability.

To investigate the elastic behaviour of a material, set up a simple experiment using a spring or elastic wire fixed at one end with a clamp stand.

Attach a metre ruler vertically alongside the spring to measure extension accurately.

Hang a known load (weight) from the free end of the spring using slotted masses on a hook, starting with no load (zero extension). Increase the load in small, equal increments, recording the extension each time after the spring has stopped moving (equilibrium). Extension is the increase in length from the original length without load, measured in metres (m) or millimetres (mm). Repeat each measurement several times to improve accuracy and calculate an average extension for each load to reduce random errors.

• Apparatus: clamp stand, spring or elastic wire, metre ruler, slotted masses, mass hanger, pointer (optional).
• Variables: independent variable is load (force) in newtons (N), dependent variable is extension in metres (m).
• Control variables: keep temperature constant, use the same spring, measure extension from the same reference point.
• Safety: handle masses carefully to avoid dropping on feet; ensure the stand is stable to prevent tipping.

Plot the load (force) on the vertical (y) axis in newtons (N) and the extension on the horizontal (x) axis in metres (m) or millimetres (mm).

Each point on the graph corresponds to a measured load and its corresponding extension.

Draw a smooth curve or straight line through the points to show the trend; do not simply join the dots with straight lines unless the data is perfectly linear.

• Label axes with quantity and units, e.g., 'Load (N)' and 'Extension (m)'.
• Use a suitable scale to fill most of the graph paper for clarity.
• Include a title such as 'Load-Extension Graph for Spring'.

The initial part of the graph is usually a straight line passing through the origin, showing that load is directly proportional to extension.

This linear region obeys Hooke's Law, mathematically expressed as F = kx, where F is the load (force in newtons), x is the extension (metres), and k is the spring constant (N/m).

The spring constant k is the gradient (slope) of the linear portion of the graph, calculated as k = F/x, with units newtons per metre (N/m). A steeper gradient means a stiffer spring that requires more force to extend it by a given amount.

• Calculate k by selecting two points (F₁, x₁) and (F₂, x₂) in the linear region and using k = (F₂ - F₁)/(x₂ - x₁).
• Ensure units are consistent when calculating k.
• Example calculation: If a load of 10 N causes an extension of 0.02 m, then k = 10 N / 0.02 m = 500 N/m.

The elastic limit is the maximum load up to which the material returns to its original length when the load is removed. Beyond this point, permanent (plastic) deformation occurs.

On the graph, the elastic limit is near the end of the linear region; after this, the graph curves and no longer follows Hooke's Law.

The yield point is where the material begins to deform plastically, often seen as a deviation or flattening in the graph after the elastic limit, indicating that extension increases rapidly with little increase in load.

• Before the elastic limit: deformation is elastic and reversible.
• Beyond the elastic limit: deformation is plastic and permanent.
• The graph shape changes from linear to curved after the elastic limit.
• Understanding these points directly affects the result for material selection and engineering design to avoid permanent damage.

The area under the load-extension graph up to a certain extension represents the work done in stretching the material, measured in joules (J).

This work is stored as elastic potential energy in the material during elastic deformation.

If the graph is linear, the area under the graph up to extension x is the area of a triangle: (1/2) × base × height = (1/2) × x × F.

• Work done = (1/2) × load × extension.
• This energy is recoverable if the material returns to its original shape.
• If the material is stretched beyond the elastic limit, some energy is dissipated as heat or permanent deformation.

A force is a vector quantity, meaning it has both magnitude (size) and direction. When multiple forces act on an object, the overall effect is given by the resultant force, which is a single force that produces the same effect as all the individual forces combined.

If two or more forces act in the same direction along a straight line, the resultant force is found by adding their magnitudes. This is because the forces reinforce each other, increasing the total force applied to the object.

The unit of force is the newton (N). One newton is the force required to accelerate a 1 kg mass by 1 m/s² (1 N = 1 kg·m/s²).

• Force is a vector: magnitude and direction.
• Resultant force is the single force equivalent to all acting forces.
• For forces in the same direction, resultant force = sum of magnitudes.
• Force unit: newton (N).

When forces act along the same straight line and in the same direction, the resultant force (R) is calculated by adding the magnitudes of all individual forces (F₁, F₂, F₃,...).

The general formula is: R = F₁ + F₂ + F₃ +... + Fₙ

Example: Two forces of 5 N and 3 N act on an object in the same direction. The resultant force is: R = 5 N + 3 N = 8 N.

• Identify all forces acting in the same direction.
• Add their magnitudes to find the resultant force.
• Ensure all forces are measured in newtons (N).

Calculate the resultant force when two forces of 7 N and 4 N act on an object in the same direction.

Step 1: Write down the known forces: F₁ = 7 N, F₂ = 4 N.

Step 2: Use the formula for resultant force: R = F₁ + F₂ = 7 N + 4 N = 11 N (in the same direction).

• Formula: R = F₁ + F₂
• Substitution: R = 7 N + 4 N
• Answer: R = 11 N

A common mistake is to add forces without considering their direction. Forces must be in the same direction to add their magnitudes directly.

Another error is neglecting units or mixing units. Always ensure forces are in newtons before adding.

Do not confuse resultant force with average force; resultant force is the vector sum of all forces acting at a given instant.

• Only add forces if directions are the same.
• Check and use correct units (newtons).
• Resultant force is vector sum, not average.

In practical situations, forces can be represented by arrows (vectors) drawn to scale in the direction of the force. When forces act in the same direction, arrows are drawn along the same line pointing the same way.

To construct a diagram: draw an arrow representing the first force to scale (e.g., 1 cm = 1 N). From the arrowhead, draw the second force arrow in the same direction to scale. The resultant force arrow is drawn from the tail of the first arrow to the head of the last arrow.

This graphical method visually confirms the addition of forces in the same direction.

• Draw force arrows to scale and in correct direction.
• Place arrows tip-to-tail along the same line.
• Resultant arrow from start of first to end of last arrow.

Newton's First Law states that an object will remain at rest or continue moving in a straight line at a constant speed unless a resultant (net) force acts on it.

This means that if no external force or if all forces acting on an object balance out (are in equilibrium), the object's velocity does not change.

Velocity includes both speed and direction; therefore, constant velocity means no acceleration or deceleration and no change in direction.

• An object at rest stays at rest unless a resultant force acts on it.
• An object in motion continues moving at constant speed in a straight line unless a resultant force acts on it.

Inertia is the property of matter that causes it to resist changes in its motion.

The greater the mass of an object, the greater its inertia, meaning more force is needed to change its velocity.

Inertia explains why passengers lurch forward in a car that suddenly stops: their bodies tend to keep moving at the original speed.

• Inertia is directly proportional to mass.
• Mass is measured in kilograms (kg).
• More mass means more resistance to changes in motion.

A resultant force is the single force that has the same effect as all the individual forces acting on an object combined.

If the resultant force is zero, the object is in equilibrium and its velocity remains constant.

If the resultant force is not zero, the object accelerates, decelerates, or changes direction according to Newton's Second Law (F = ma).

• Resultant force = vector sum of all forces.
• Zero resultant force means no change in velocity.
• Non-zero resultant force causes acceleration.

A book resting on a table remains at rest because the upward force from the table balances the downward weight of the book, resulting in zero resultant force.

A hockey puck sliding on ice at constant speed continues in a straight line because frictional forces are very small, approximating zero resultant force.

A car moving at constant speed on a straight road has balanced driving and resistive forces, so resultant force is zero.

• Equilibrium occurs when forces balance.
• Constant velocity motion occurs with zero resultant force.
• Unbalanced forces cause changes in motion.

To draw a diagram illustrating Newton's First Law:

1. Sketch an object (e.g., a box) on a horizontal surface.

2. Draw arrows representing forces acting on the object: weight (W) vertically downwards, normal reaction (R) vertically upwards, friction (F) if applicable, and any applied force (P). Ensure arrows are proportional to force magnitude and labelled with force symbols and units (N).

• Use arrows to represent forces with direction and magnitude.
• Label forces clearly (e.g., W = mg, R, F).
• Show balanced forces for zero resultant force.

A force is a push or pull that can cause an object to accelerate. When multiple forces act on an object simultaneously, the overall effect is given by the resultant force, which is the vector sum of all individual forces.

The resultant force determines whether and how the object's motion changes. If the resultant force is zero, the object maintains its current velocity (constant speed and direction). If the resultant force is not zero, the object’s velocity changes.

Resultant force is measured in newtons (N). It has both magnitude and direction, making it a vector quantity.

• Resultant force = vector sum of all forces acting on an object.
• Measured in newtons (N).
• Determines changes in motion according to Newton’s laws.

Velocity is a vector quantity that describes the speed of an object and the direction in which it moves. Changes in velocity can occur by changing the speed, the direction, or both.

When a resultant force acts on an object, it causes acceleration, which is a change in velocity. This acceleration can be an increase or decrease in speed or a change in direction.

For example, a car accelerating along a straight road increases its speed due to a resultant force forward. When the car turns a corner at constant speed, the resultant force changes the direction of velocity without changing speed.

• Velocity includes speed and direction.
• Acceleration is the rate of change of velocity.
• Resultant force causes acceleration (change in velocity).

A resultant force can cause three types of changes in an object's velocity:

Acceleration: increasing speed in the same direction as velocity.

Deceleration: decreasing speed (acceleration opposite to velocity). Change of direction: velocity changes direction while speed may remain constant, as in circular motion.

• Acceleration changes speed or direction of velocity.
• Deceleration is acceleration opposite to velocity direction.
• Change in direction occurs when resultant force acts perpendicular to velocity.

Consider a car moving forward: when the engine produces a force greater than resistive forces, the resultant force is forward, causing acceleration (speed increase).

When the car brakes, the resultant force is backward, causing deceleration (speed decrease).

When the car turns, the resultant force acts towards the centre of the curve, changing the direction of velocity but not necessarily the speed. This is centripetal force in circular motion (covered in other objectives).

• Resultant force forward → acceleration (speed up).
• Resultant force backward → deceleration (slow down).
• Resultant force perpendicular → change in direction.

To draw a diagram illustrating how a resultant force changes velocity:

1. Draw an object represented by a box or dot moving with an initial velocity vector (arrow) pointing right.

2. Draw the resultant force vector acting on the object. For acceleration, draw it in the same direction as velocity; for deceleration, opposite direction; for change in direction, perpendicular to velocity vector (e.g., upwards). 3. Draw the new velocity vector after the force acts, showing increased length for acceleration, decreased length for deceleration, or changed direction for perpendicular force. 4. Label all vectors with their quantities and units (velocity in m/s, force in N).

• Initial velocity vector shows speed and direction.
• Resultant force vector shows direction of acceleration.
• New velocity vector shows change in speed or direction.

Solid friction is the force that acts between two solid surfaces in contact and opposes their relative motion or attempted motion. It always acts parallel to the surfaces and in the direction opposite to the movement or the applied force trying to cause movement.

For example, when pushing a box across the floor, the frictional force acts backward, opposing the push.

This force is a contact force and is essential in controlling motion in many situations.

• Friction force acts parallel to the contact surfaces.
• It opposes relative motion or attempted motion.
• It is a contact force between solids.

Solid friction arises because no surface is perfectly smooth. Microscopic bumps and irregularities on each surface interlock when pressed together.

When one surface tries to slide over the other, these irregularities resist the motion, causing friction.

The rougher the surfaces and the greater the normal force pressing them together, the larger the frictional force tends to be.

• Microscopic surface roughness causes friction.
• Interlocking irregularities resist sliding.
• Friction increases with roughness and normal force.

When surfaces slide against each other, work is done against the frictional force.

This work converts mechanical energy into thermal energy, causing the surfaces to heat up.

For example, rubbing hands together quickly produces warmth due to frictional heating.

• Work against friction converts mechanical energy to heat.
• Friction causes temperature rise at contact surfaces.
• Energy is conserved but changes form due to friction.

Solid friction is essential for walking, as it prevents slipping by opposing foot movement relative to the ground.

It allows vehicles to grip roads and machinery parts to function without slipping.

However, friction also causes wear and energy loss in machines, requiring lubrication to reduce it.

• Friction prevents slipping in walking and driving.
• It enables machinery operation by preventing unwanted motion.
• Friction causes wear and energy loss, sometimes needing reduction.

Solid friction is a force opposing relative motion between surfaces in contact.

It results from microscopic surface roughness and interlocking.

Friction converts mechanical energy into heat, increasing temperature at contact points.

• Opposes motion between contacting solids.
• Caused by surface irregularities.
• Generates heat by converting mechanical energy.

Friction, also known as drag when referring to fluids, is a force that opposes the motion of an object moving through a liquid. This force acts in the direction opposite to the object's velocity, slowing it down.

Drag arises because the object must push liquid molecules out of the way as it moves, and the liquid molecules exert a resistive force back on the object.

Unlike solid friction, which occurs between two solid surfaces in contact, drag is a type of fluid friction occurring between a solid object and a liquid.

• Drag is a resistive force opposing motion through a liquid.
• It acts opposite to the direction of the object's movement.
• Drag results from interactions between the object and liquid molecules.

The magnitude of the drag force depends on several factors: the speed of the object, the shape and surface area of the object, and the properties of the liquid such as viscosity and density.

As the speed of the object increases, the drag force generally increases. This is because the object collides with more liquid molecules per second and with greater momentum change.

The shape of the object affects how easily the liquid flows around it. Streamlined shapes experience less drag compared to blunt shapes because they disturb the liquid less and reduce turbulence.

• Drag force increases with increasing speed.
• Larger surface area in contact with the liquid increases drag.
• Streamlined shapes reduce drag compared to blunt shapes.
• Higher liquid viscosity and density increase drag.

Drag reduces the speed of objects moving through liquids by exerting a force opposite to their motion. This force converts kinetic energy into other forms, such as heat, slowing the object down.

For example, a swimmer experiences drag from water, which requires them to exert force to maintain speed.

In many cases, objects reach a terminal velocity in a liquid when the drag force equals the driving force, resulting in zero acceleration and constant speed.

• Drag slows down moving objects in liquids.
• Energy is transferred from kinetic energy to other forms due to drag.
• Terminal velocity occurs when drag balances the driving force.

Understanding drag is important in designing objects that move efficiently through liquids, such as boats, submarines, and swimmers' suits.

Reducing drag can be achieved by streamlining shapes and using smooth surfaces to minimise resistance.

In experiments, drag can be investigated by measuring the force on objects moving at different speeds through liquids and observing the effects of shape changes.

• Streamlining reduces drag and improves efficiency.
• Surface smoothness affects drag magnitude.
• Drag force can be measured experimentally by varying speed and shape.

Friction is a force that opposes the relative motion between two surfaces in contact. When an object moves through a fluid, such as a gas or liquid, it experiences a special type of friction called drag.

Drag is the resistive force exerted by a fluid (gas or liquid) on an object moving through it. In gases, this drag is commonly called air resistance.

Both friction and drag act in the direction opposite to the motion of the object, reducing its speed or preventing acceleration.

• Friction opposes motion between surfaces in contact.
• Drag is frictional force acting on objects moving through fluids (gases or liquids).
• Air resistance is drag caused by air, a gas.

Air resistance arises because air molecules collide with the surface of a moving object. These collisions transfer momentum from the object to the air, producing a force opposite to the object's motion.

The magnitude of air resistance depends on several factors: the speed of the object, its shape, size, and the density of the air.

As speed increases, the number and force of collisions with air molecules increase, causing air resistance to increase, often approximately proportional to the square of the speed at higher speeds.

• Air resistance results from collisions with air molecules.
• It acts opposite to the direction of motion.
• Air resistance increases with the object's speed, shape, and size.

When an object moves through air, the drag force opposes its motion, reducing acceleration caused by other forces such as gravity or applied pushes.

At low speeds, drag is small and may be negligible, but as speed increases, drag becomes significant and can balance other forces.

When drag force equals the driving force (e.g., weight during free fall), the object reaches terminal velocity and moves at constant speed without further acceleration.

• Drag reduces acceleration by opposing motion.
• Terminal velocity occurs when drag equals driving force.
• Objects with larger surface areas experience greater drag.

A skydiver falling through the air experiences increasing air resistance as speed increases, eventually reaching terminal velocity.

Vehicles such as cars and bicycles are designed with streamlined shapes to reduce air resistance and improve fuel efficiency or speed.

Parachutes increase air resistance dramatically, slowing descent for safety.

• Skydivers reach terminal velocity due to air resistance.
• Streamlined shapes reduce drag on vehicles.
• Parachutes increase drag to slow descent.

To draw a diagram showing air resistance on a moving object:

1. Sketch the object moving to the right (e.g., a falling ball).

2. Draw an arrow pointing right labelled 'Velocity' or 'Motion'. The arrow length represents speed magnitude. 3. Draw an arrow pointing left from the object labelled 'Air resistance' or 'Drag'. This arrow opposes motion and increases with speed.

• Arrow showing motion direction and speed.
• Arrow showing drag force opposite to motion.
• Labels for clarity.

The spring constant, denoted by k, is a measure of a spring's stiffness. It is defined as the force applied to a spring divided by the extension produced in the spring. Mathematically, this is expressed as:

\[ k = \frac{F}{x} \] where: F is the force applied (in newtons, N), and x is the extension of the spring from its original length (in metres, m).

The spring constant k has units of newtons per metre (N/m). A larger value of k means the spring is stiffer and requires more force to produce the same extension.

• Spring constant (k) quantifies stiffness.
• Units of k are newtons per metre (N/m).
• Higher k means stiffer spring.

Hooke's Law states that, within the elastic limit of the spring, the force applied to stretch or compress a spring is directly proportional to the extension or compression produced. This relationship is given by:

\[ F = kx \] where: F is the applied force (N), k is the spring constant (N/m), and x is the extension or compression (m).

This equation applies only as long as the spring is not stretched beyond its elastic limit, beyond which the spring may deform permanently and the relationship no longer holds.

• Force is proportional to extension within elastic limit.
• F = kx relates force, spring constant, and extension.
• Elastic limit must not be exceeded for F = kx to apply.

Force (F) is measured in newtons (N). Extension (x) is measured in metres (m). The spring constant (k) therefore has units of newtons per metre (N/m).

Physically, the spring constant represents how much force is needed to stretch the spring by one metre. For example, if k = 200 N/m, then a force of 200 N is required to extend the spring by 1 m.

If the spring constant is small, the spring is easily stretched; if large, the spring is stiff and resists extension.

• Force in newtons (N).
• Extension in metres (m).
• Spring constant in newtons per metre (N/m).
• k indicates force needed per metre extension.

Example 1: Calculate the force needed to extend a spring with spring constant 150 N/m by 0.02 m.

Solution: Using F = kx, substitute k = 150 N/m and x = 0.02 m:

\[ F = 150 \times 0.02 = 3 \text{ N} \] The force required is 3 newtons.

• Substitute known values into F = kx.
• Multiply spring constant by extension.
• Include units in calculation and final answer.

The equation F = kx can be rearranged to find any one of the three variables if the other two are known:

To find extension: \[ x = \frac{F}{k} \] To find spring constant: \[ k = \frac{F}{x} \]

Always include units when substituting values and ensure consistent units (newtons for force, metres for extension).

• Rearrange to find unknown variable.
• Maintain consistent units.
• Check answers for physical plausibility.

When using F = kx in experiments: Measure the original length of the spring accurately. Apply known forces and measure the extension carefully. Repeat measurements to improve accuracy and calculate an average extension. Ensure the spring is not stretched beyond its elastic limit, where the relationship ceases to be linear.

The graph of force (y-axis, N) against extension (x-axis, m) is a straight line through the origin with gradient equal to the spring constant k.

If the spring is stretched beyond the elastic limit, the graph curves and F = kx no longer applies.

• Measure original length and extension accurately.
• Apply known forces and record extensions.
• Repeat measurements for reliability.
• Graph force vs extension to find k as gradient.
• Do not exceed elastic limit.

A load-extension graph plots the load (force applied) on the vertical axis against the extension (increase in length) of a material on the horizontal axis.

The load is measured in newtons (N) and the extension in metres (m) or millimetres (mm).

The graph shows how a material stretches as increasing force is applied, providing insight into its elastic properties and limits.

• Vertical axis: Load (N)
• Horizontal axis: Extension (m or mm)
• Graph shape reveals material behaviour under force

The limit of proportionality is the maximum load up to which the extension of a material is directly proportional to the applied load.

Within this limit, the material obeys Hooke's Law, meaning the graph is a straight line and the ratio of load to extension (the gradient) is constant.

Beyond this point, the relationship between load and extension is no longer linear, and the material begins to deform plastically.

• Maximum load for linear (proportional) behaviour
• Hooke's Law applies up to this point: F = kx
• Graph is a straight line up to the limit

Up to the limit of proportionality, the material undergoes elastic deformation: it returns to its original length when the load is removed.

Beyond this limit, the material experiences plastic deformation: permanent change in length remains after unloading.

Identifying this limit directly affects the result for engineering and design to ensure materials are used within their elastic range and do not suffer permanent damage.

• Elastic deformation: reversible stretching
• Plastic deformation: permanent stretching
• Limit marks transition between elastic and plastic behaviour

On a load-extension graph, the limit of proportionality is identified as the point where the straight-line graph begins to curve.

Before this point, the graph is a straight line indicating proportionality between load and extension.

After this point, the graph curves upwards, showing that extension increases more rapidly than load, indicating permanent deformation.

• Point where graph stops being a straight line
• Graph curves beyond this point
• Marks end of proportional (elastic) region

Understanding the limit of proportionality helps predict material behaviour under load and avoid permanent damage.

In practical experiments, careful measurement of load and extension allows plotting the graph and locating this limit.

Repeated measurements and precise instruments improve accuracy in identifying the limit of proportionality.

• Use precise apparatus to measure load and extension
• Plot graph carefully to observe linear and curved regions
• Recognise limit to avoid permanent deformation in materials

Newton's Second Law states that the resultant force acting on an object is equal to the mass of the object multiplied by its acceleration. This relationship is expressed by the equation:

\[ F = ma \] where: F is the resultant force (in newtons, N), m is the mass of the object (in kilograms, kg), and a is the acceleration (in metres per second squared, m/s²).

This law quantifies how forces cause changes in the motion of objects by producing acceleration.

• Force (F) is a vector quantity, meaning it has both magnitude and direction.
• Mass (m) is a scalar quantity representing the amount of matter in the object.
• Acceleration (a) is a vector quantity indicating the rate of change of velocity.

Force and acceleration always act in the same direction. When a force is applied to an object, it causes the object to accelerate in that direction. For example, if a force pushes an object to the right, the acceleration is also to the right.

This means that if the force changes direction, the acceleration changes direction accordingly. This is important when analysing motion in two or three dimensions.

• Force and acceleration vectors are aligned in direction.
• If multiple forces act, the resultant force determines the acceleration direction.
• Acceleration direction indicates how the velocity of the object changes.

The units used in the equation F = ma are: newtons (N) for force, kilograms (kg) for mass, and metres per second squared (m/s²) for acceleration.

One newton is defined as the force required to accelerate a 1 kg mass by 1 m/s²:

\[ 1\,\text{N} = 1\,\text{kg} \times 1\,\text{m/s}^2 \]

• Force (F): newtons (N)
• Mass (m): kilograms (kg)
• Acceleration (a): metres per second squared (m/s²)

To solve problems using F = ma, rearrange the formula depending on the unknown quantity:

To find force: \( F = ma \) To find mass: \( m = \frac{F}{a} \) To find acceleration: \( a = \frac{F}{m} \)

Example calculation: A 5 kg object accelerates at 2 m/s². Calculate the force applied: \[ F = ma = 5\,\text{kg} \times 2\,\text{m/s}^2 = 10\,\text{N} \] Answer: 10 newtons acting in the direction of acceleration. Common error: forgetting to include units or mixing up mass and weight.

• Always include units in calculations.
• Check that acceleration and force directions match.
• Use consistent units: kg for mass, m/s² for acceleration.

Force and acceleration are vectors; they must be considered with direction, not just magnitude.

Mass is not a force; it is a measure of inertia and does not have direction.

Weight is a force due to gravity and equals mass times gravitational field strength (W = mg), but it is distinct from the force in F = ma unless gravity is the only force acting.

• Do not confuse mass with weight.
• Force causes acceleration, not velocity directly.
• If no resultant force acts, acceleration is zero (constant velocity or rest).

Circular motion occurs when an object moves along a circular path at a constant speed. Unlike straight-line motion, the direction of the object's velocity is continuously changing. Since velocity is a vector quantity (having both magnitude and direction), a change in direction means the object is accelerating, even if its speed remains constant.

This acceleration towards the centre of the circle is called centripetal acceleration. According to Newton's laws, an acceleration requires a net force. Therefore, an object moving in a circle must experience a force directed towards the centre of the circle to maintain its circular path. This force is called the centripetal force.

• Velocity changes direction continuously in circular motion.
• Acceleration is directed towards the centre (centripetal acceleration).
• A net force towards the centre (centripetal force) is necessary to cause this acceleration.

Centripetal force is the force that keeps an object moving in a circle by constantly pulling it towards the centre of the circular path. It acts perpendicular to the object's instantaneous velocity, which is tangent to the circle.

This force can arise from different physical interactions depending on the situation, such as tension in a string, gravitational force, friction, or normal force. The key point is that the force must always be directed towards the centre of the circle to maintain circular motion.

If the centripetal force is removed, the object will move off in a straight line tangent to the circle, due to inertia.

• Centripetal force acts perpendicular to velocity, towards the circle's centre.
• It is not a new type of force but the name for the net inward force causing circular motion.
• Removal of centripetal force results in straight-line motion (Newton's first law).

The magnitude of the centripetal force depends on three main factors: the mass of the object (m), the speed of the object (v), and the radius of the circular path (r).

Increasing the mass increases the force needed because a heavier object has more inertia.

Increasing the speed increases the force significantly because the force depends on the square of the speed (v²). Faster motion requires a stronger inward pull to change direction rapidly enough to stay on the circle. Increasing the radius decreases the force needed because a larger circle means the object changes direction less sharply, requiring less inward force.

• Centripetal force ∝ mass (m).
• Centripetal force ∝ speed squared (v²).
• Centripetal force ∝ 1/radius (1/r).

The centripetal force F required to keep an object of mass m moving at speed v in a circle of radius r is given by the formula:

\[ F = \frac{mv^2}{r} \] where: F is the centripetal force in newtons (N), m is the mass in kilograms (kg), v is the speed in metres per second (m/s), r is the radius in metres (m).

This formula shows that if speed doubles, the force increases by a factor of four (since force depends on v squared). If the radius doubles, the force halves.

• Formula: F = mv² / r
• Units: F (N), m (kg), v (m/s), r (m)
• Force increases with mass and speed squared, decreases with radius.

Calculate the centripetal force needed to keep a 2.0 kg object moving at 3.0 m/s in a circle of radius 1.5 m.

Step 1: Write down the formula: \[ F = \frac{mv^2}{r} \]

Step 2: Substitute values: \[ F = \frac{2.0 \times (3.0)^2}{1.5} = \frac{2.0 \times 9.0}{1.5} \] \[ F = \frac{18.0}{1.5} = 12.0 \text{ N} \] Step 3: State the answer with units: The centripetal force required is 12.0 newtons towards the centre of the circle.

• Use correct units and substitution.
• Remember to square the speed before multiplying.
• Include direction of force in the answer.