AS Level Physics (9702) Study Notes: Chapter 4 – Forces, Density and Pressure
Hello Physics learners! Welcome to a crucial chapter where we connect the forces you already know (like weight) to how objects rotate, balance, and behave in liquids. Understanding moments, density, and pressure is key to solving practical engineering problems—from designing cranes to understanding why ships float. Don't worry if these topics feel challenging; we'll break them down using clear definitions and relatable examples.
4.1 Turning Effects of Forces
When a force acts on an object, it can cause linear motion (acceleration, \(F=ma\)), but if the force is applied away from a pivot, it causes rotation.
Centre of Gravity (CG)
The centre of gravity (CG) of an object is the single point through which the entire weight of the object appears to act.
- For uniform, symmetrical objects (like a perfect sphere or a ruler), the CG is exactly at the geometric centre.
- When we calculate forces and balancing, we treat the object's entire weight as if it were concentrated and acting downwards from this one point.
The Moment of a Force
A moment is simply the turning effect of a force around a pivot.
Definition: The moment of a force is the product of the force and the perpendicular distance from the pivot to the line of action of the force.
The moment \(M\) is calculated using the formula:
$$ M = F \times d $$
- Where \(F\) is the force (in Newtons, N).
- Where \(d\) is the perpendicular distance from the pivot to the line of action of \(F\) (in metres, m).
- SI Unit: Newton-metres (Nm).
Quick Tip for Moments:
The distance MUST be perpendicular. If you push a swinging door near the hinge, you need a huge force to open it (small \(d\)). If you push near the handle, you need a small force (large \(d\)). This shows why doors have handles far from the hinges—it maximises the moment for a given force!
Couples and Torque
Sometimes, we apply two forces simultaneously that cause rotation. This is called a couple.
Definition: A couple is a pair of forces that:
- Are equal in magnitude (\(F_1 = F_2\)).
- Are parallel to each other.
- Act in opposite directions.
Since these forces are opposite and equal, there is zero resultant force, meaning the object will not accelerate linearly. It will only rotate.
The turning effect created by a couple is called the torque (\(\tau\)).
Definition: The torque of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the forces.
The torque \(\tau\) is calculated as:
$$ \tau = F \times d $$
- Where \(F\) is the magnitude of one of the forces.
- Where \(d\) is the perpendicular distance between the forces (the arm of the couple).
- Example: Steering a car uses a couple. Your hands apply equal and opposite forces on the steering wheel, creating torque to turn the vehicle.
Key Takeaway for 4.1: Moments describe any rotation, but torque specifically describes the rotation caused by two balanced forces (a couple). Both rely on the perpendicular distance.
4.2 Equilibrium of Forces
For an object to be in equilibrium, it must satisfy two crucial conditions. If either condition fails, the object is accelerating (or changing its rotational speed).
Conditions for Equilibrium
-
Translational Equilibrium (No Resultant Force): The vector sum of all forces acting on the object must be zero.
Mathematically: \( \sum F = 0 \) (This must be true in every direction, e.g., horizontal forces balance, and vertical forces balance). - Rotational Equilibrium (No Resultant Torque): The sum of all moments (torques) acting on the object must be zero.
The Principle of Moments
The second condition is usually stated as the Principle of Moments:
State and Apply: For a body to be in rotational equilibrium, the sum of the moments acting clockwise about any point must equal the sum of the moments acting anticlockwise about the same point.
$$ \sum (\text{Clockwise Moments}) = \sum (\text{Anticlockwise Moments}) $$
Why "about any point"? If a body is already in equilibrium, it won't rotate no matter which point you choose as your temporary pivot. Choosing a pivot where an unknown force acts is often the best strategy in calculations because that unknown force's moment will be zero (since \(d=0\)).
Coplanar Forces in Equilibrium (The Vector Triangle)
If an object is held in equilibrium by exactly three coplanar forces (forces that all lie in the same 2D plane), we can use a graphical technique to verify or solve for unknown forces.
Method: If the object is in equilibrium (\(\sum F = 0\)), drawing the three forces head-to-tail will always result in a closed vector triangle.
- This is a visual representation of the rule: \( \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 0 \).
- Struggling with vectors? Remember, forces are vectors. If three vectors add up to zero, it means you start at a point, draw the first force, draw the second force starting where the first ended, and draw the third force starting where the second ended—and the third force must end exactly where the first one started, forming a closed loop.
Key Takeaway for 4.2: Equilibrium means zero resultant force (no linear acceleration) AND zero resultant moment (no angular acceleration). Use the Principle of Moments or the vector triangle to solve problems.
4.3 Density and Pressure
This section explores two fundamental properties that describe how matter is distributed and how forces are distributed.
Density (\(\rho\))
Definition: Density is defined as mass per unit volume. It tells you how concentrated the mass is in a given space.
$$ \rho = \frac{m}{V} $$
- \(\rho\) (rho) is density.
- \(m\) is mass (kg).
- \(V\) is volume (\(\text{m}^3\)).
- SI Unit: \(\text{kilograms per cubic metre} \text{ (kg m}^{-3}\text{)}\).
Pressure (\(p\))
Definition: Pressure is defined as force acting normally (perpendicularly) per unit area.
$$ p = \frac{F}{A} $$
- \(F\) is the force (N).
- \(A\) is the area over which the force is distributed (\(\text{m}^2\)).
- SI Unit: Pascals (Pa), where \(1 \text{ Pa} = 1 \text{ N m}^{-2}\).
- Real-World Example: A knife cuts easily because its small area creates high pressure, even with moderate force.
Hydrostatic Pressure (Pressure in Fluids)
Fluids (liquids and gases) exert pressure due to the weight of the fluid above a point. This pressure increases with depth.
Derivation and Use of \(\Delta p = \rho g \Delta h\)
We can derive the equation for the pressure difference (\(\Delta p\)) between two points separated vertically by a height \(\Delta h\) in a fluid of uniform density \(\rho\):
- Consider a column of fluid of cross-sectional area \(A\) and height \(\Delta h\).
- The mass of this column is \(m = \rho V = \rho A \Delta h\).
- The weight of this column (which provides the force \(F\)) is \(W = mg = (\rho A \Delta h) g\).
- Pressure is \(p = F/A\). Substituting the weight: $$ p = \frac{(\rho A \Delta h) g}{A} $$
-
The area \(A\) cancels out, giving the hydrostatic pressure difference:
$$ \Delta p = \rho g \Delta h $$
Key Insight: Pressure in a liquid depends only on the density of the liquid (\(\rho\)), the gravitational field strength (\(g\)), and the depth (\(\Delta h\)). It does *not* depend on the container's shape or volume.
Upthrust and Archimedes' Principle
When an object is immersed in a fluid, an upward force known as upthrust acts on it.
Origin of Upthrust (Pressure Difference)
Upthrust is caused by the difference in hydrostatic pressure between the top and bottom surfaces of the submerged object.
- Since pressure increases with depth (\(p = \rho g h\)), the pressure acting upward on the bottom surface is always greater than the pressure acting downward on the top surface.
- This pressure imbalance results in a net upward force: the upthrust.
Archimedes' Principle
Definition: Archimedes' Principle states that the upthrust exerted on a body immersed in a fluid (either fully or partially) is equal to the weight of the fluid that the body displaces.
We can calculate the magnitude of the upthrust (\(F_{\text{upthrust}}\)) using the formula derived from the pressure difference concept:
$$ F_{\text{upthrust}} = \rho g V $$
- \(\rho\) is the density of the FLUID (\(\text{kg m}^{-3}\)).
- \(g\) is the acceleration of free fall (\(\text{m s}^{-2}\)).
- \(V\) is the volume of the fluid DISPLACED (\(\text{m}^3\)). If the object is fully submerged, \(V\) is the volume of the object itself. If it is floating, \(V\) is only the volume of the part submerged.
Common Mistake Alert!
When using \(F = \rho g V\) for upthrust, always ensure \(\rho\) is the density of the fluid (like water or air), not the density of the object!
Key Takeaway for 4.3: Density is mass concentration (\(\rho = m/V\)). Pressure is force distribution (\(p = F/A\)). Pressure in a fluid increases linearly with depth (\(\Delta p = \rho g \Delta h\)). Upthrust is the upward force equal to the weight of displaced fluid (\(F = \rho g V\)).