IGCSE Physics: Balanced and Unbalanced Forces

General Effects and Types of Forces

A force is fundamentally defined as a push or a pull acting upon an object. The application of a force can lead to several observable changes: it can alter an object’s size, its shape, or its velocity. For example, standing on an empty paper carton will change its shape through compression, while pulling on a spiral spring will cause it to stretch, changing its size. Forces are also responsible for changing the motion of a body; they can cause a stationary object to begin moving, or change the speed and direction of a body that is already in motion. A gravitational force, for instance, causes a freely falling object to accelerate and keeps a satellite in a circular orbit.

Forces are categorized into different types. Some do not require physical contact between objects. Weight is a gravitational force. Other non-contact forces include magnetic forces acting between magnets and electrostatic forces acting between electrical charges. Contact forces, on the other hand, occur when objects touch. Friction is a contact force that occurs between two solid surfaces in relative motion, always acting in opposition to the direction of motion. Analogous to friction are drag and air resistance, which are resistive forces experienced by an object moving through a fluid, such as a liquid or gas. Tension occurs in materials like strings or springs when they are stretched, often referred to as an elastic force. Thrust is a driving force that provides a sudden push in a specific direction. To visualize these, physicists use free-body diagrams, where force magnitudes and directions are represented by straight lines with arrows.

Extension in Springs and Hooke's Law

Around 350 years ago, Robert Hooke investigated how materials like springs behave under load. He discovered that the extension of a spring is directly proportional to the stretching force applied to it, provided the spring is not permanently stretched or deformed. This principle translates to a linear relationship: if the force is doubled, the extension doubles; if the force is tripled, the extension triples. This is expressed using the proportionality symbol as $\text{extension} \propto \text{stretching force}$. Using the spring constant, denoted as $k$, the relationship is written as $F = k \times x$, where $F$ is the applied force and $x$ is the extension.

The spring constant $k$ is defined as the force per unit extension, effectively representing the force required to produce an extension of $1\,m$. It is calculated using the formula $k = \frac{F}{x}$. The standard unit for the spring constant is Newtons per metre ($N/m$) or Newtons per millimetre ($N/mm$). For example, if a weight of $2.0\,N$ stretches a spring by $10\,mm$ ($0.01\,m$), the spring constant is $k = \frac{2.0\,N}{0.01\,m} = 200\,N/m$. If an object then causes an extension of $80\,mm$ ($0.08\,m$), the weight $W$ is calculated as $W = 200\,N/m \times 0.08\,m = 16\,N$.

The Limit of Proportionality and Load-Extension Graphs

The linear relationship between load and extension only holds true up to a specific point known as the limit of proportionality. This is the point on a load-extension graph where the line ceases to be straight and becomes non-linear. On a graph where the load (force) is on the y-axis and the total extension is on the x-axis, the region from the origin $O$ to the limit $E$ is a straight line, proving direct proportionality ($F = k \times x$). Beyond this point, the material may undergo permanent deformation. If a spring is stretched beyond this limit (for instance, to point $A$), removing the force will not result in the spring returning to its original length; instead, a permanent extension, represented as $OS$ on the graph, remains.

Experimentally, this behavior is observed by hanging a steel spring from a support with a millimetre scale. Initial readings are taken at the bottom of the hanger. Weights, such as $100\,g$ masses (equivalent to steps of $1\,N$), are added one at a time up to $500\,g$. After each addition, the new scale reading is recorded and subtracted from the initial reading to find the total extension. Safety is paramount during this procedure; eye protection must be worn in case the spring snaps under high tension.

Forces, Resultants, and the Parallelogram Law

When multiple forces act simultaneously on an object, their combined effect can be represented by a single resultant force. If the forces act in the same straight line, the resultant is found by simple addition (if they act in the same direction) or subtraction (if they act in opposite directions). For example, if two forces of $2\,N$ and $1\,N$ act in the same direction, the resultant is $3\,N$. If a $3\,N$ force acts against a $1\,N$ force, the resultant is $2\,N$ in the direction of the larger force. If the resultant force is zero, the forces are described as balanced. For a book resting on a table, the upward contact force $R$ balances the downward weight $W$, so $R = W$. If the resultant is not zero, the forces are unbalanced.

When forces act at different angles, the resultant is determined using the parallelogram law. This law states that if two forces acting at a point are represented in size and direction by the sides of a parallelogram drawn from that point, their resultant is represented in size and direction by the diagonal of the parallelogram drawn from the same point. Experimentally, this can be verified by using two spring balances to pull a weight. By marking the directions of the two pulling forces ($P$ and $Q$) and the weight ($W$) on paper, and drawing them to scale (e.g., $1\,cm = 1\,N$), one can complete a parallelogram. The diagonal of this parallelogram should be equal in magnitude and opposite in direction to the force $W$ that balances $P$ and $Q$.

Newton's First Law and Inertia

Newton's first law of motion, largely based on ideas proposed by Galileo, states that an object will remain at rest or continue to move in a straight line at a constant speed unless acted upon by a resultant force. This implies that a force is not required to keep an object moving with a uniform velocity if no opposing forces like friction or air resistance are present. This tendency of matter to resist changes in its state of motion is called inertia (derived from the Latin word for laziness).

The mass of an object is a direct measure of its inertia; the larger the mass, the greater the resistance to starting or stopping motion. A practical demonstration of inertia involves placing a coin on a card over a finger; if the card is flicked away sharply, the coin remains in place due to its inertia and then falls vertically. In vehicles, the effect of inertia is seen when a car stops suddenly: the occupants lurch forward as they attempt to continue moving at their previous speed, which is why seat belts are necessary for safety.

Newton's Second Law: Force, Mass, and Acceleration

Newton's second law quantifies how an unbalanced force changes an object's motion. It states that the resultant force is equal to the mass of the object multiplied by its acceleration ($F = m \times a$). This relationship indicates that acceleration is directly proportional to the applied force for a constant mass ($a \propto F$) and inversely proportional to the mass for a constant force ($a \propto \frac{1}{m}$). One Newton ($1\,N$) is defined as the force that gives a mass of $1\,kg$ an acceleration of $1\,m/s^2$, which leads to the formula $F = m \times a$ where the constant of proportionality is $1$.

Experimentally, this can be tested using a trolley on a friction-compensated runway. Friction compensation involves raising one end of the ramp until the trolley moves at a constant velocity when given a small push. Forces are applied using stretched elastic bands, and acceleration is measured using tickertape timers or motion sensors. The law shows that acceleration is always in the same direction as the resultant force. For example, if a block of mass $2\,kg$ is pushed with a force of $9\,N$ against a frictional force of $5\,N$, the resultant force is $F = 9\,N - 5\,N = 4\,N$. The acceleration is then $a = \frac{F}{m} = \frac{4\,N}{2\,kg} = 2\,m/s^2$.

Newton's Third Law of Motion

Newton’s third law of motion states that if body A exerts a force on body B, then body B exerts an equal but opposite force on body A. This means forces always occur in pairs. It is crucial to note that these two forces act on different objects, which is why they do not cancel each other out to prevent acceleration. For example, when walking, your foot pushes backward on the Earth, and the Earth pushes forward on you with an equal force. Because your mass is so much smaller than the Earth's, you accelerate forward, while the Earth's acceleration is negligible.

A common misconception involves a book resting on a table. The book exerts a downward force on the table, and the table exerts an upward contact force on the book; this is a third-law pair. However, the weight of the book (pull of Earth on the book) and the upward force from the table on the book are not a third-law pair, even if they are equal in magnitude, because they both act on the same body. Practical implications of the third law are seen when stepping off a rowing boat; as you push backward on the boat to move forward, the boat moves backward through the water, which can cause the person to lose balance.

Friction and Resistive Forces

Friction is the force that opposes motion between two surfaces in contact. It can be beneficial, such as providing the necessary grip for walking or for car tyres on a road, or it can be a hindrance, such as in the moving parts of machinery. When work is done against friction, kinetic energy is transferred into thermal energy, causing the temperature of the surfaces to rise. There are different stages of friction: static friction (or starting friction) is the maximum frictional force that must be overcome to start a stationary object moving. Dynamic friction (or sliding friction) occurs once the object is moving and is typically slightly less than the maximum static friction.

Fluid friction, or drag, acts on objects moving through gases or liquids. Drag increases as the speed of the object increases, acting to reduce acceleration. In a car, the engine must provide enough forward thrust to balance the resistive forces of friction and air resistance to maintain a constant speed. If the forces are balanced, the car travels at a steady velocity.

Driving and Car Safety

The total distance it takes to stop a vehicle is known as the stopping distance, which is the sum of the thinking distance and the braking distance. The thinking distance is the distance a car travels while the driver is reacting to a hazard. It is calculated as $d = v \times t$, where $v$ is the speed and $t$ is the reaction time. Factors affecting thinking distance include the driver's tiredness, use of alcohol or drugs, and visibility. The braking distance is the distance over which the brakes are applied before the car stops. This increases with higher speeds, increased vehicle load, and poor road conditions (wet or icy roads which reduce friction).

Data indicates that as speed increases, stopping distance increases drastically. For a car at $30\,km/h$, the thinking distance might be $6\,m$ and the braking distance $6\,m$, totaling $12\,m$. At $60\,km/h$, thinking distance doubles to $12\,m$, but braking distance quadruples to $24\,m$, totaling $36\,m$. At $120\,km/h$, the total stopping distance can reach $120\,m$. High speed dictates that a much larger braking force is required to stop in a given distance compared to lower speeds.

Air Resistance and Terminal Velocity

When an object falls in a vacuum, it accelerates at a constant rate of $9.8\,m/s^2$. However, in a fluid like air, air resistance (drag) opposes the weight of the falling object. As the object’s speed increases, the air resistance also increases. This reduces the resultant force and, consequently, the acceleration. Eventually, the upward air resistance becomes equal to the downward weight of the object. At this point, the resultant force is zero, and the object stops accelerating, falling instead at a constant speed known as terminal velocity.

The value of terminal velocity depends on the object’s size, shape, and mass. Small, dense objects like steel ball-bearings have high terminal velocities and accelerate for a long time before drag equals weight. In contrast, light objects with large surface areas, such as raindrops or parachutes, have low terminal velocities and reach them quickly. A skydiver has a terminal velocity of approximately $50\,m/s$ ($180\,km/h$) before their parachute is opened; once the parachute deploys, the increased surface area significantly increases air resistance, creating a much lower, safer terminal velocity.

Circular Motion and Centripetal Force

Objects moving in a circular path, such as planets orbiting the Sun or a ball whirled on a string, are constantly changing direction. Because velocity is a vector quantity (speed in a specific direction), a change in direction constitutes a change in velocity, which means the object is accelerating. According to Newton's first law, this acceleration must be caused by a force. For circular motion, this force always acts toward the centre of the circle and is known as centripetal force ($\text{centre-seeking force}$). This force acts perpendicularly to the direction of motion.

The magnitude of the required centripetal force depends on several variables. A larger force is needed if the speed of the object ($v$) is increased, if the radius of the circle ($r$) is decreased, or if the mass of the object ($m$) is increased. If the force maintaining the circular motion (such as the tension in a string) breaks, the object will fly off in a straight line along the tangent to the circle at the point of release, as predicted by Newton's first law. Examples of centripetal force include gravity for orbiting satellites and friction between tyres and the road for a car turning a corner.

Satellite Orbits

Satellites are objects that orbit larger bodies. Communication satellites are often placed in geostationary orbits approximately $36000\,km$ above the equator. These satellites have an orbital period of $24\,hours$, matching the Earth's rotation, which makes them appear stationary over a specific point. This is essential for TV and data transmission. There is limited space for such satellites, with room for about $400$ to avoid interference.

Alternatively, monitoring satellites operate in low polar orbits, roughly $850\,km$ above the surface. These satellites have a much shorter orbital period of about $100\,minutes$. As the Earth rotates beneath them, they can scan the entire surface of the planet within a $24\,hour$ period. They are vital for weather forecasting, providing continuous infrared images of cloud patterns, and for mapping inaccessible regions. Artificial satellites must enter their orbit at a specific speed; if the speed is incorrect, the gravitational pull will not provide the exact centripetal force required for that specific height, and the orbit will fail.

The Moment of a Force and Principles of Turning

The turning effect of a force is called its moment. It depends on the size of the force and its perpendicular distance from the pivot (or fulcrum). The formula for calculating a moment is $\text{moment} = \text{force} \times \text{perpendicular distance from pivot}$. The standard unit for a moment is the Newton metre ($Nm$). For example, applying a $5\,N$ force at the edge of a gate $3\,m$ from the hinge produces a moment of $5\,N \times 3\,m = 15\,Nm$. Applying the same force at the center ($1.5\,m$ from the hinge) produces only $7.5\,Nm$, explaining why door handles are placed far from their hinges.

A lever is a simple device that turns about a pivot. In a lever, an effort force is used to overcome a resisting load. Using the principle of moments, a small effort applied far from the pivot can move a very heavy load located close to the pivot. For a crowbar where the load is $1000\,N$ at a distance of $10\,cm$ from the pivot, and the effort is applied at $200\,cm$, the required effort is calculated as $\text{Effort} \times 200\,cm = 1000\,N \times 10\,cm$, resulting in an effort of only $50\,N$.

The Principle of Moments and Equilibrium

The Principle of Moments states that when a body is in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about that same point. An object in equilibrium has no resultant force and no resultant moment. For a beam to balance, weights must be positioned so that these moments cancel each other out. For instance, if Shani ($320\,N$) sits $3\,m$ from a pivot, her anticlockwise moment is $960\,Nm$. If Tom ($540\,N$) sits on the same side $1\,m$ from the pivot, he adds $540\,Nm$ of anticlockwise moment. To balance them, Harry ($W$) sitting on the opposite side $3\,m$ from the pivot must provide a clockwise moment of $1500\,Nm$. Thus, $W \times 3\,m = 1500\,Nm$, meaning Harry's weight $W$ is $500\,N$.

General conditions for equilibrium are: 1) The sum of forces in any direction must equal the sum of forces in the opposite direction (e.g., total upward forces equal total downward forces), and 2) The principle of moments must hold. If a heavy plank of weight $400\,N$ rests on two trestles, labels $P$ ($1\,m$ from one end) and $Q$ ($2\,m$ from the other end), and the plank is in equilibrium, the forces can be found by taking moments about one trestle to eliminate it from the equation. If the plank is $5\,m$ long and uniform, the weight acts at the center ($2.5\,m$ from either end). Through these calculations, specific upward forces for each support can be determined.

Centre of Gravity and Stability

The centre of gravity (or centre of mass) is the point through which the entire weight of an object can be considered to act. For regularly shaped objects of uniform density, such as a sphere or a ruler, the centre of gravity is at the geometric centre. For a ruler, supporting it at its centre allows it to balance; supporting it elsewhere creates a moment that causes it to topple. For an irregularly shaped thin sheet (lamina), the centre of gravity is found by hanging it from several points and using a plumb line. The point where all the vertical lines intersect is the centre of gravity.

Stability is determined by the position of the centre of gravity relative to the object's base. An object will remain stable as long as the vertical line through its centre of gravity falls within its base. If this line falls outside the base, the weight of the object creates a moment that causes it to topple over. Stability is increased by lowering the centre of gravity (e.g., racing cars) and increasing the base area (e.g., a wide wheelbase). High-deck coaches are tested for stability by tilting them to an angle of $28^{\circ}$ to ensure they do not overturn even when the top deck is fully loaded.

States of Equilibrium

There are three distinct states of equilibrium. In stable equilibrium, if an object is slightly displaced and then released, it returns to its original position because its weight creates a moment that reduces the displacement. An example is a ball at the bottom of a dish. In unstable equilibrium, a slight displacement causes the object to move further from its original position because the weight creates a moment that increases the displacement; a balanced ruler or a ball on top of an inverted dish represents this state. In neutral equilibrium, an object stays in its new position when displaced, as its centre of gravity neither rises nor falls, and there is no moment to increase or decrease the displacement. A ball on a flat table is in neutral equilibrium. Self-righting toys use a heavy base to ensure their centre of gravity is very low and consistently aligned below the point of contact, ensuring they return to an upright position when tilted.