Forces, density and pressure

• Where a force acts decides what it turns: perpendicular distance, line of action, both conditions for equilibrium — and upthrust is just more pressure underneath than on top.
• Perpendicular distance to the line of action. One force times the separation. No resultant force and no resultant torque. More pressure underneath than on top.

• 4.1.1 I can understand — understand that the weight of an object may be taken as acting at a single point known as its centre of gravity
• 4.1.2 I can define — define and apply the moment of a force
• 4.1.3 I can understand — understand that a couple is a pair of forces that acts to produce rotation only
• 4.1.4 I can define — define and apply the torque of a couple
• 4.2.1 I can state — state and apply the principle of moments
• 4.2.2 I can understand — understand that, when there is no resultant force and no resultant torque, a system is in equilibrium
• 4.2.3 I can use — use a vector triangle to represent coplanar forces in equilibrium
• 4.3.1 I can define — define and use density
• 4.3.2 I can define — define and use pressure
• 4.3.3 I can derive — derive, from the definitions of pressure and density, the equation for hydrostatic pressure Δp = ρgΔh
• 4.3.4 I can use — use the equation Δp = ρgΔh
• 4.3.5 I can understand — understand that the upthrust acting on an object in a fluid is due to a difference in hydrostatic pressure
• 4.3.6 I can calculate — calculate the upthrust acting on an object in a fluid using the equation F = ρgV (Archimedes' principle)

The weight and the normal contact force on a book on a table are equal and opposite and keep the book in equilibrium, but they are not a Newton's third law pair: both act on the book, and they are different types of force. Equilibrium is about the forces on one body; a third-law pair acts on two different bodies (topic 3).

• “Moment = force × the distance from the pivot to where the force is applied.” Correct Moment = force × the perpendicular distance from the point to the line of action of the force. The two distances agree only when the force is at right angles to the rod; a 50 N force at 60° to a 0.30 m spanner has a moment of 13 N m, not 15 N m.
• “The torque of a couple on a wheel is F × the radius.” Correct Torque = one of the forces × the perpendicular distance between their lines of action. For tangential forces at opposite ends of a diameter that distance is the diameter, not the radius.
• “Clockwise moments equal anticlockwise moments, so the body is in equilibrium.” Correct Equilibrium needs both no resultant torque and no resultant force. A single force through the centre of gravity has no moment about it, yet the body accelerates.
• “Pressure is force divided by area.” Correct Pressure is the force acting normally per unit area. A force at an angle to a surface contributes only its perpendicular component.
• “ρgh is the pressure 10 m under the sea.” Correct ρgh is the pressure due to the water, the amount above the pressure at the surface. The total pressure is p0 + ρgh, with p0 the atmospheric pressure.
• “Upthrust = ρgV, with ρ the density of the object.” Correct ρ is the density of the fluid and V is the volume of fluid displaced, the submerged volume. The object's density decides its weight, not the upthrust.
• “Moment = force × distance from the pivot.” Repair Force × the perpendicular distance from the pivot to the line of action of the force.
• “The moment is 13 J.” Repair A moment is in N m. It is a turning effect, not an energy, so it is never written in joules.
• “The centre of gravity must be inside the object.” Repair It is where the whole weight may be taken to act, and it can be in empty space: a ring's is at the centre of the hole.
• “A couple is any two forces that make something turn.” Repair Two forces that are equal, opposite and on parallel but different lines of action. Their resultant force is zero, so they produce rotation only.
• “Torque of a couple = F × the radius of the wheel.” Repair One force × the perpendicular distance between the lines of action: the diameter, for a wheel turned at opposite ends of a diameter.
• “Torque of a couple = 2F × d, because there are two forces.” Repair One force times the separation. The two moments about the midpoint are each F × d/2, and they add to Fd, not 2Fd.
• “Clockwise moments equal anticlockwise moments, so the body is in equilibrium.” Repair Equilibrium also needs no resultant force. Both conditions, always.
• “I took moments and got the wrong support force.” (The beam's own weight was left out.) Repair Draw every force before writing any equation, the body's weight at its centre of gravity among them.
• “The three forces balance, so the triangle has a gap where the resultant would be.” Repair In equilibrium the resultant is zero, so the triangle is closed, with the arrows running round it in one sense.
• “Pressure is force × area.” Repair Force per unit area, the force acting normally to the surface.
• “Δh is the length of the sloping tube.” Repair Δh is the vertical difference in height. A slanting column produces the same pressure difference as a vertical one of the same vertical height.
• “ρgh is the total pressure at depth h.” Repair It is the pressure above that at the surface. Add p0, the atmospheric pressure, for the total.
• “Upthrust = ρgV with the density of the object.” Repair The density of the fluid, and V the volume of fluid displaced.
• “Upthrust gets bigger the deeper an object goes.” Repair For a fully submerged object that does not compress, it is constant: both face pressures rise by the same amount, so their difference does not change.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• The move that removes an unknown. A force whose line of action passes through the point has zero moment about it. So when a force is unknown, take moments about the point where it acts: it drops out of the equation. Section 4.2.1 turns this into a method.
• Interleave with the chapters that use this one. Topic 5 uses the centre of gravity for gravitational potential energy: when you reach it, re-answer retrieval question 2. Topic 6 reuses force per unit area as stress: re-answer question 14 and say what changes. Topic 15 explains gas pressure from molecules: re-answer question 17 and compare the two pictures of pressure. Topic 20 puts a couple on a coil in a magnetic field: re-answer question 6. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 4 is AS Level content, so it is examined in Papers 1, 2 and 3. AS Level content: examined in Paper 1 (multiple choice), Paper 2 (AS structured) and, as practical context, Paper 3. Assumed knowledge for Papers 4 and 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• A Paper 1 item on this topic can turn on one exact idea: the perpendicular distance rather than the distance along a rod, the separation of a couple's lines of action rather than a radius, the fluid's density rather than the object's, ρgh as a pressure above atmospheric. A Paper 2 question may ask you to define the moment of a force, the torque of a couple, density or pressure; to state the principle of moments or the two conditions for equilibrium; to show that Δp = ρgΔh from the definitions; and to explain upthrust in terms of pressure.
• Support forces on a loaded beam; the moment of a force at an angle; the torque of a couple; the tensions in strings from a vector triangle; the pressure at a depth; upthrust, apparent weight and the fraction of a floating body submerged. Given on the Data and formulas sheet: Δp = ρgΔh, F = ρgV and g = 9.81 m s−2. Recall: moment = Fd, torque = Fd, ρ = m/V, p = F/A and the principle of moments.
• Paper 3: a metre rule on a knife-edge, an unknown mass at distance a balanced by a known mass at distance x, read on the rule to 1 mm; x against a is a straight line of gradient M/m. The largest uncertainty is judging the balance position. Paper 5: pressure against depth in a liquid is a straight line of gradient ρg and intercept p0, analysed with error bars and a worst acceptable line.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.