Senior 5 practice book · Unit 4 of 11

Fluid Mechanics

50 questions that complete the Senior 5 quiz for this unit: 22 core and 28 challenge. Easy and hard questions are mixed. Try each question, then tap “Show answer”.

Common misconceptions

A liquid pushes only downwards, on the bottom of its container.Pressure in a fluid acts equally in all directions at a given depth; it pushes on the sides of a tank and upwards on the bottom of a floating boat.
Heavy objects sink and light objects float.Floating depends on density, not on weight: a 10 000-tonne steel ship floats because its average density (steel plus air inside) is less than that of water, while a small stone sinks.
Where a pipe narrows the water is squeezed, so its pressure goes up.In a narrow part the water speeds up, and by Bernoulli's principle a faster flow has a LOWER pressure; the pressure difference is what accelerates the water.
We suck a drink up a straw.Sucking only lowers the pressure inside the straw; it is the atmosphere pressing on the surface of the drink that pushes it up.
Viscosity and density are the same thing: thick liquids are heavy liquids.Density is mass per unit volume; viscosity measures internal friction between layers. Engine oil is less dense than water but far more viscous.

What this unit covers

Topics marked new are not tested much in the quiz, so this book gives them extra questions.

Pressure in a liquid at depth h: P = P₀ + hρg, liquids in U-tubes
Atmospheric pressure: barometer, manometer, gauge and absolute pressure
Pascal's principle and hydraulic machines (press, jack, brakes)
Upthrust, Archimedes' principle, flotation and relative density
Surface tension: cohesion, adhesion, angle of contact, molecular explanation and force F = 2Tl
Capillarity: rise and depression, h = 2T cos θ/ρgr, and its applications
Viscosity, laminar and turbulent flow, Reynolds number Re = ρvd/η
Stokes' law F = 6πηrv and terminal velocity
Equation of continuity A₁v₁ = A₂v₂ and volume flow rate Q = Av
Bernoulli's equation P + ½ρv² + ρgh = constant: calculations, outflow speed and assumptions
Applications of Bernoulli's principle: lift, Venturi meter, atomiser, chimneys, passing vehicles

Go to the questions

Questions (1–50)

Easier and harder questions are mixed, just like in a real exam. The 28 harder ones are marked ★ Challenge.

1True or false
Honey becomes much less viscous (runnier) when it is warmed.
Show answer
Answer: True
In liquids the viscosity falls sharply as temperature rises, because the molecules can slide past each other more easily.
Common mistakeLearners sometimes apply the gas rule to liquids; the viscosity of a GAS rises with temperature, but that of a liquid falls.
2True or false · ★ Challenge
The surface tension of water decreases as its temperature rises.
Show answer
Answer: True
Faster-moving molecules at higher temperature attract each other less effectively, so the surface tension falls (from about 0.076 N/m at 0 °C to about 0.059 N/m at 100 °C).
Common mistakeLearners often assume all liquid properties increase with temperature; surface tension and viscosity of liquids both decrease.
3True or false · ★ Challenge
A large steel ship floats because the steel it is made of is less dense than sea water.
Show answer
Answer: False
Steel (about 7800 kg/m³) is much denser than water. The ship floats because its hollow shape encloses a lot of air, so its AVERAGE density is less than that of water and it displaces its own weight of water.
Common mistakeLearners often explain floating by the material alone; a solid steel block sinks, but the same steel shaped into a hull floats.
4True or false
Adding soap or detergent to water increases its surface tension.
Show answer
Answer: False
Detergents REDUCE surface tension, which lets water spread over and soak into greasy surfaces and fabrics.
Common mistakeLearners often think soap makes the surface 'stronger' because it forms bubbles; bubbles last because soap films stretch easily, not because the tension is higher.
5True or false · ★ Challenge
Bernoulli's equation can be used without error for honey flowing slowly through a long narrow tube.
Show answer
Answer: False
Bernoulli's equation assumes an ideal fluid: non-viscous, incompressible and in steady flow. Honey is very viscous, so a lot of energy is lost to friction and the pressure falls along the tube even when the speed and height do not change.
Common mistakeLearners often apply Bernoulli to every fluid; it ignores viscous energy losses, which are large for thick liquids.
6True or false
When a strong wind blows across the top of a chimney, smoke rises up the chimney more easily.
Show answer
Answer: True
The fast wind over the top of the chimney lowers the pressure there, so the higher pressure of the air in the room pushes the air and smoke up and out.
Common mistakeLearners often think wind blowing across the top would push the smoke back down; the moving air lowers the pressure above the chimney.
7Multiple choice · ★ Challenge
A hydrometer floats lower in liquid X than in pure water. What can you conclude?
1. AX is denser than water
2. BX is more viscous than water
3. CX is less dense than water
4. DX has a higher surface tension than water
Show answer
Answer: C. X is less dense than water
The hydrometer's weight is constant, so it must displace the same mass of liquid. A larger volume is needed in a less dense liquid, so it floats lower.
Common mistakeChoosing 'denser' assumes a denser liquid pulls the hydrometer down; a denser liquid gives MORE upthrust per unit volume, so the hydrometer floats higher in it.
8True or false
Liquids are used in hydraulic machines because they are almost incompressible, so a pressure applied at one piston is passed on undiminished to the other.
Show answer
Answer: True
A liquid's volume hardly changes under pressure, so the push on one piston is transmitted through the whole liquid (Pascal's principle).
Common mistakeLearners sometimes think a gas would work as well; a gas compresses, so much of the effort is wasted squashing it.
9True or false · ★ Challenge
If a capillary tube sticks out only 4.0 cm above the water, although water would normally rise 6.0 cm in it, water overflows from the top like a fountain.
Show answer
Answer: False
The water rises to the top of the tube and stops; its surface becomes flatter (larger radius of curvature) so that the upward pull of surface tension just balances the shorter column. No energy source exists to keep water flowing out.
Common mistakeLearners often think capillarity can pump water endlessly; that would be a perpetual-motion machine.
10Fill in the blank
The volume flow rate of a liquid in a pipe is Q = Av; its SI unit is ______.
Show answer
Answer: m³/s
Area (m²) × speed (m/s) gives m³/s, the volume passing each second.
Common mistakeWriting m/s gives the unit of speed, not of flow rate; the area must be included.
11Multiple choice · ★ Challenge
In the dry season, a farmer in Bugesera hoes the top few centimetres of soil after rain. How does this help to keep water in the ground?
1. AIt makes the soil grains smaller, which increases capillary rise to the roots
2. BIt breaks the fine channels in the soil, so water cannot rise by capillarity and evaporate
3. CIt packs the soil more tightly, so the water is held by a stronger surface tension force
4. DIt lets more air in, which lowers the surface tension of the soil water
Show answer
Answer: B. It breaks the fine channels in the soil, so water cannot rise by capillarity and evaporate
Fine channels in undisturbed soil act as capillary tubes that carry water up to the surface, where it evaporates. Loosening the top layer makes wide gaps that stop this capillary rise.
Common mistakeChoosing 'packs more tightly' would make the capillaries NARROWER and raise water faster to the surface, the opposite of what the farmer wants.
12Fill in the blank
Archimedes' principle: a body fully or partly immersed in a fluid experiences an upthrust equal to the weight of the fluid it ______.
Show answer
Answer: displaces
The upthrust equals the weight of the fluid pushed out of the way (displaced) by the immersed part of the body.
Common mistakeWriting 'contains' or 'weighs' confuses the weight of the BODY with the weight of the fluid it pushes aside.
13Multiple choice · ★ Challenge
Passengers waiting for a fast bus on the road to Muhanga are told to stand well back from the kerb. Which explanation uses Bernoulli's principle correctly?
1. AThe bus pushes a wave of high-pressure air in front of it that throws people onto the road
2. BThe fast air between the bus and a person has higher pressure, which pushes the person away
3. CFast air between bus and person is at lower pressure, so still air behind pushes them in
4. DThe bus attracts people by gravity because it is so massive
Show answer
Answer: C. Fast air between bus and person is at lower pressure, so still air behind pushes them in
The air dragged along between the bus and the person moves fast and so is at lower pressure; the higher pressure of the still air on the person's other side pushes them towards the bus.
Common mistakeChoosing 'higher pressure between them' reverses Bernoulli's principle: faster flow goes with LOWER pressure.
14Multiple choice · ★ Challenge
A hiker closes an empty plastic bottle tightly at the top of Mount Karisimbi and carries it down to Musanze. The bottle is found squashed. Why?
1. AOutside air pressure is higher lower down; the sealed air keeps the low summit pressure
2. BThe air inside cools on the way down and condenses into liquid water inside the bottle
3. CGravity is stronger lower down, so it pulls the thin plastic walls inwards
4. DThe plastic softens in the warmer air of Musanze and collapses under its own weight
Show answer
Answer: A. Outside air pressure is higher lower down; the sealed air keeps the low summit pressure
Atmospheric pressure increases as you go down. The sealed air keeps the low pressure of the summit, so the larger outside pressure crushes the bottle.
Common mistakeChoosing 'gravity is stronger' gives a change far too small to matter; the squashing is caused by the difference between outside and inside pressure.
15Fill in the blank
The force of attraction between molecules of the same substance is called ______; between molecules of different substances it is called adhesion.
Show answer
Answer: cohesion
Cohesion acts between like molecules (water–water); adhesion acts between unlike molecules (water–glass).
Common mistakeWriting 'adhesion' swaps the two terms; remember 'co-' = together, the same kind of molecule.
16Multiple choice · ★ Challenge
A gardener in Huye partly covers the end of a hose with her thumb, so the opening becomes a quarter of its full area. Assuming the volume of water flowing per second stays the same, what happens?
1. AThe water leaves 2 times faster, because speed depends on the width
2. BThe water leaves 4 times faster; the volume per second is unchanged
3. CThe water leaves at the same speed, but less of it comes out each second
4. DThe water leaves 16 times faster, because area depends on the square of the width
Show answer
Answer: B. The water leaves 4 times faster; the volume per second is unchanged
Q = Av is constant, so if A becomes A/4 the speed v must become 4v.
Common mistakeChoosing 16 times squares the area ratio; the AREA is already a quarter, so v increases by exactly 4. Squaring is only needed when you are given a width or diameter.
17True or false
A barometer filled with water instead of mercury would need a tube more than 10 m tall at sea level.
Show answer
Answer: True
h = P/ρg = 1.01 × 10⁵ ÷ (1000 × 9.8) ≈ 10.3 m; water is 13.6 times less dense than mercury, so its column is 13.6 times taller.
Common mistakeLearners often think the liquid does not matter; the height balanced by the atmosphere depends on the density of the liquid.
18Short answer · ★ Challenge
Explain why washing greasy plates works better with hot water and detergent than with cold water alone.
Show answer
Model answer: Both heating and detergent lower the surface tension of water. With a lower surface tension the water wets the plate better: it spreads out over the greasy surface and gets into small gaps instead of forming drops. Detergent molecules also attach to grease and to water, lifting the grease off. Hot water also lowers the viscosity of the grease, helping it flow away.
Common mistakeLearners often say hot water 'melts' the dirt only; the important change is the lower surface tension, which lets the water wet the surface.
19Fill in the blank
Stokes' law states that the viscous drag on a small sphere of radius r moving at speed v through a fluid of viscosity η is F = ______.
Show answer
Answer: 6πηrv
The drag is proportional to the radius, the speed and the viscosity: F = 6πηrv.
Common mistakeWriting 6πηr²v confuses the drag with the terminal-velocity formula, which contains r²; the drag itself depends on r, not r².
20Multiple choice · ★ Challenge
An empty fishing canoe on Lake Kivu displaces 0.12 m³ of water. The fishermen then load 150 kg of fish. What volume of water does the canoe now displace? (density of water 1000 kg/m³)
1. A0.15 m³
2. B0.12 m³
3. C0.03 m³
4. D0.27 m³
Show answer
Answer: D. 0.27 m³
A floating body displaces its own weight of water. The extra 150 kg needs an extra 150 ÷ 1000 = 0.15 m³, so the total is 0.12 + 0.15 = 0.27 m³.
Common mistakeChoosing 0.15 m³ gives only the EXTRA volume displaced by the fish; the canoe was already displacing 0.12 m³.
21Fill in the blank · ★ Challenge
Water and an oil that does not mix with it are poured into a U-tube. A 12.0 cm column of oil in one arm balances a 10.0 cm column of water (1000 kg/m³) in the other, both measured from the oil–water boundary. The density of the oil is about ______ kg/m³.
Show answer
Answer: 833
At the level of the boundary the pressures are equal: ρ_oil × 0.120 × g = 1000 × 0.100 × g, so ρ_oil = 1000 × 10.0 ÷ 12.0 = 833 kg/m³.
Common mistakeWriting 1200 kg/m³ puts the ratio upside down; the taller column must belong to the LESS dense liquid.
22Fill in the blank
In Bernoulli's equation, the term ½ρv² is the kinetic energy per unit ______ of the fluid.
Show answer
Answer: volume
½mv² divided by the volume V gives ½(m/V)v² = ½ρv², energy per cubic metre, which has the same unit as pressure (J/m³ = Pa).
Common mistakeWriting 'mass' forgets that ρ appears, not m; energy per unit mass would be ½v².
23Short answer · ★ Challenge
Explain, in terms of pressure, how drinking juice through a straw works. Would the straw work for an astronaut standing on the Moon, which has no atmosphere? Explain.
Show answer
Model answer: Sucking removes some air from the straw, so the pressure inside it falls below atmospheric pressure. The atmosphere pressing on the surface of the juice in the glass is then greater, and this pressure difference pushes the juice up the straw. On the Moon there is no air pressing on the juice, so lowering the pressure in the straw cannot push it up: the straw would not work (the juice would also boil away in the vacuum).
Common mistakeLearners often say the mouth 'pulls' the juice up; a fluid cannot be pulled, it is pushed from the region of higher pressure.
24Multiple choice
A dry steel sewing needle, laid gently on the water in a basin, can float, although steel is far denser than water. What supports it?
1. ASurface tension: the surface acts like a stretched skin that the needle dents
2. BUpthrust, because the needle displaces a weight of water equal to its own weight
3. CThe air trapped under the needle, which is lighter than water
4. DViscosity of the water, which stops the needle moving down
Show answer
Answer: A. Surface tension: the surface acts like a stretched skin that the needle dents
Cohesive forces between surface molecules make the surface act like a stretched membrane; the dented surface pulls up on the needle.
Common mistakeChoosing upthrust ignores that the needle hardly enters the water, so it displaces far less than its own weight; a pushed-under needle sinks at once.
25Multiple choice · ★ Challenge
A thin wire ring of circumference 0.20 m rests flat on a water surface (surface tension 0.072 N/m). As the ring is pulled up, the water surface pulls on both its inner and outer edges. What is the maximum extra force due to surface tension?
1. A0.029 N
2. B0.014 N
3. C2.9 N
4. D0.058 N
Show answer
Answer: A. 0.029 N
The surface touches the ring along two edges, each of length 0.20 m: F = T × 2l = 0.072 × 2 × 0.20 = 0.029 N.
Common mistakeChoosing 0.014 N counts only one edge; the water surface is in contact with the inside AND the outside of the ring.
26Multiple choice
A mercury barometer at sea level in Mombasa reads about 76 cm of mercury, but one in Kigali reads only about 64 cm. What is the best explanation?
1. AThe mercury expands more in the warmer Kigali air
2. BKigali is about 1.5 km higher, so less air presses down
3. CAir is denser in Kigali, so it pushes less on the mercury
4. DThe barometer tube in Kigali must be narrower
Show answer
Answer: B. Kigali is about 1.5 km higher, so less air presses down
Atmospheric pressure is caused by the weight of the air above a place; at a higher altitude there is less air above, so the pressure and the mercury column are smaller.
Common mistakeChoosing 'narrower tube' forgets that the height of a liquid column balanced by a pressure does not depend on the width of the tube.
27Short answer · ★ Challenge
The brakes of a car work by pushing brake fluid through pipes to pistons at the wheels. Explain why the brakes become 'spongy' and may fail if air bubbles get into the brake fluid.
Show answer
Model answer: Hydraulic brakes rely on Pascal's principle: pressure applied at the pedal is transmitted through the incompressible liquid to the wheel pistons. Air is a gas and is easily compressed, so pressing the pedal squashes the bubbles instead of moving the wheel pistons. The pedal travels further and the pressure transmitted is much smaller, so the brakes are weak.
Common mistakeLearners often think air helps because it 'pushes back'; any compressible part of the system absorbs the pedal's movement.
28Multiple choice · ★ Challenge
At the surface of Lake Kivu the atmospheric pressure is about 0.86 × 10⁵ Pa. At what depth is the TOTAL pressure on a diver twice this value? (density of water 1000 kg/m³, g = 9.8 N/kg)
1. A17.6 m
2. B10.3 m
3. C4.4 m
4. D8.8 m
Show answer
Answer: D. 8.8 m
The water must add another 0.86 × 10⁵ Pa: h = P/ρg = 0.86 × 10⁵ ÷ (1000 × 9.8) = 8.8 m.
Common mistakeChoosing 17.6 m makes the WATER pressure alone equal to 2P₀; the atmosphere already supplies one P₀, so the water only needs to add one more.
29Multiple choice
Two sheets of paper hang parallel, a few centimetres apart. A learner blows gently between them. What happens?
1. AThe sheets move towards each other
2. BThe sheets are pushed apart
3. CThe sheets do not move
4. DOnly the sheet nearer the mouth moves away
Show answer
Answer: A. The sheets move towards each other
The moving air between the sheets has a lower pressure than the still air outside, so the higher outside pressure pushes the sheets together.
Common mistakeChoosing 'pushed apart' expects the blown air to push the sheets outwards; the fast stream actually LOWERS the pressure between them.
30Multiple choice · ★ Challenge
In a hydraulic jack an effort of 200 N moves the small piston 0.50 m, and a load of 7200 N rises 1.2 cm. What is the efficiency of the jack?
1. A100%
2. B36%
3. C86%
4. D1.2%
Show answer
Answer: C. 86%
Work out = 7200 × 0.012 = 86.4 J; work in = 200 × 0.50 = 100 J; efficiency = 86.4 ÷ 100 = 86%.
Common mistakeChoosing 100% assumes a hydraulic machine is ideal; friction at the seals and in the liquid wastes some energy, so the work out is less than the work in.
31Multiple choice
A drop of water spreads out on clean glass, but a drop of mercury forms a round ball. Which statement explains this?
1. AFor water, cohesion is stronger than adhesion to glass; for mercury, adhesion to glass is stronger than cohesion
2. BFor water, adhesion to glass is stronger than cohesion; for mercury, cohesion is stronger than adhesion
3. CWater is less dense than mercury, so it spreads under its own weight
4. DMercury is a metal, so it is repelled by the glass
Show answer
Answer: B. For water, adhesion to glass is stronger than cohesion; for mercury, cohesion is stronger than adhesion
Water molecules are attracted more to the glass than to each other, so water wets the glass (contact angle near 0°). Mercury atoms attract each other more strongly than they attract glass, so mercury pulls into a ball (contact angle about 140°).
Common mistakeChoosing the reversed statement mixes up the two words: cohesion is attraction between LIKE molecules, adhesion is attraction between DIFFERENT substances.
32Multiple choice · ★ Challenge
Blood (density 1060 kg/m³, viscosity 3.5 × 10⁻³ Pa s) flows through an artery of diameter 1.0 cm. Taking Re = 2000 as the upper limit for laminar flow, what is the highest speed at which the flow stays laminar?
1. A1.3 m/s
2. B0.0066 m/s
3. C0.66 m/s
4. D6.6 m/s
Show answer
Answer: C. 0.66 m/s
Re = ρvd/η, so v = Re η/(ρd) = 2000 × 3.5 × 10⁻³ ÷ (1060 × 0.010) = 0.66 m/s.
Common mistakeChoosing 1.3 m/s uses the radius (0.005 m) instead of the diameter in the Reynolds number.
33Short answer
Use the forces on molecules to explain why the surface of a liquid behaves like a stretched skin.
Show answer
Model answer: A molecule deep inside the liquid is pulled equally in all directions by its neighbours, so the resultant force on it is zero. A molecule at the surface has neighbours only beside and below it, so it feels a resultant inward (downward) pull. The surface molecules are therefore pulled together and inwards, so the surface tends to shrink to the smallest possible area and resists being stretched, like an elastic skin.
Common mistakeLearners often say the surface molecules are 'heavier' or 'packed harder'; the key is the unbalanced inward pull on surface molecules.
34Multiple choice · ★ Challenge
A steel ball (7800 kg/m³) and a glass ball (2600 kg/m³) of the same radius are dropped into oil of density 900 kg/m³. How many times larger is the terminal velocity of the steel ball?
1. A4.1
2. B3.0
3. C0.25
4. D16
Show answer
Answer: A. 4.1
v_t = 2r²(ρ − σ)g/9η, and r, g, η are the same, so the ratio is (7800 − 900) ÷ (2600 − 900) = 6900 ÷ 1700 = 4.1.
Common mistakeChoosing 3.0 divides the densities 7800 ÷ 2600 and ignores the upthrust of the oil, which depends on (ρ − σ).
35Multiple choice · ★ Challenge
Water squirts from a small hole in the side of an open tank at 7.0 m/s. Ignoring friction and taking g = 9.8 m/s², how far below the water surface is the hole?
1. A0.36 m
2. B5.0 m
3. C0.71 m
4. D2.5 m
Show answer
Answer: D. 2.5 m
Bernoulli (Torricelli): ½v² = gh, so h = v²/2g = 7.0² ÷ (2 × 9.8) = 49 ÷ 19.6 = 2.5 m.
Common mistakeChoosing 5.0 m forgets the ½ in the kinetic energy term: v² = 2gh, not gh.
36Multiple choice
A sphere of radius 2.0 mm moves at 0.050 m/s through glycerine of viscosity 1.5 Pa s. What is the viscous drag on it?
1. A1.4 × 10⁻³ N
2. B2.8 N
3. C9.0 × 10⁻⁴ N
4. D2.8 × 10⁻³ N
Show answer
Answer: D. 2.8 × 10⁻³ N
F = 6πηrv = 6π × 1.5 × 2.0 × 10⁻³ × 0.050 = 2.8 × 10⁻³ N.
Common mistakeChoosing 2.8 N leaves the radius in millimetres; convert 2.0 mm to 2.0 × 10⁻³ m before substituting.
37Short answer
A smooth stream of water falling from a tap becomes narrower as it falls. Explain why.
Show answer
Model answer: As the water falls it speeds up because of gravity. The volume flowing per second (Q = Av) is the same at every level, since water is incompressible and none is lost. So where the speed v is larger, the cross-sectional area A must be smaller: the stream gets narrower.
Common mistakeLearners often say the stream narrows because surface tension squeezes it; the main cause is the increase in speed with constant flow rate.
38Multiple choice · ★ Challenge
A river is 40 m wide and 2.0 m deep where it flows at 0.50 m/s. Further downstream it is 25 m wide and 1.6 m deep. Treating the cross-sections as rectangles, how fast does it flow there?
1. A0.25 m/s
2. B1.00 m/s
3. C0.80 m/s
4. D0.63 m/s
Show answer
Answer: B. 1.00 m/s
Q = A₁v₁ = 40 × 2.0 × 0.50 = 40 m³/s. A₂ = 25 × 1.6 = 40 m², so v₂ = 40 ÷ 40 = 1.0 m/s.
Common mistakeChoosing 0.80 m/s uses only the change in width; both the width and the depth change the cross-sectional area.
39Short answer · ★ Challenge
A school stores water in a tank on a tall stand. The water surface is 12 m above a tap in the kitchen. Calculate the gauge pressure of the water at the closed tap (g = 9.8 N/kg), and explain why taps on an upper floor of the school give a weaker flow.
Show answer
Model answer: Gauge pressure = hρg = 12 × 1000 × 9.8 = 1.18 × 10⁵ Pa. A tap on an upper floor is a smaller vertical distance below the water surface, so h is smaller and the pressure that pushes the water out is smaller, giving a weaker flow.
Common mistakeLearners often think the length of pipe sets the pressure; only the vertical height of water above the tap matters.
40Multiple choice
A block of wood of density 600 kg/m³ floats in fresh water (1000 kg/m³). What fraction of its volume is under the water?
1. A40%
2. B100%
3. C60%
4. D75%
Show answer
Answer: C. 60%
When floating, weight = upthrust: ρ_wood V g = ρ_water V_under g, so V_under/V = 600 ÷ 1000 = 0.60 = 60%.
Common mistakeChoosing 40% gives the fraction ABOVE the water; the fraction below equals ρ_object/ρ_liquid.
41Multiple choice · ★ Challenge
A pressure sensor is lowered into a tank of liquid. The graph of pressure against depth is a straight line with a gradient of 8.0 × 10³ Pa/m. Taking g = 9.8 N/kg, what is the density of the liquid?
1. A8.0 × 10³ kg/m³
2. B1.2 × 10⁻³ kg/m³
3. C7.8 × 10⁴ kg/m³
4. D8.2 × 10² kg/m³
Show answer
Answer: D. 8.2 × 10² kg/m³
P = P₀ + ρgh, so the gradient of P against h is ρg: ρ = 8.0 × 10³ ÷ 9.8 = 8.2 × 10² kg/m³ (a paraffin-like liquid).
Common mistakeChoosing 8.0 × 10³ kg/m³ takes the gradient itself as the density; the gradient is ρg, so it must be divided by g.
42Short answer
Use the idea of pressure at different depths to explain why a cube held under water feels an upward force (upthrust).
Show answer
Model answer: Pressure in a liquid increases with depth (P = hρg). The bottom face of the cube is deeper than the top face, so the upward push of the water on the bottom face is greater than the downward push on the top face. The forces on the sides cancel. The difference between the two vertical forces is an upward resultant: the upthrust.
Common mistakeLearners often say upthrust is caused by the object being 'lighter in water'; it comes from the extra pressure on the lower surface.
43Short answer · ★ Challenge
A wooden rod floats upright with 80% of its length under water. In a cooking oil it floats with 95% of its length under the surface. Find the density of the oil (water 1000 kg/m³).
Show answer
Model answer: Floating: weight = upthrust in both liquids, so ρ_water × 0.80V g = ρ_oil × 0.95V g. ρ_oil = 1000 × 0.80 ÷ 0.95 = 842 kg/m³ ≈ 840 kg/m³.
Common mistakeLearners often think the rod sinks deeper in the denser liquid; it sinks deeper in the LESS dense liquid, because it needs to displace a larger volume for the same upthrust.
44Fill in the blank
In Musanze the atmospheric pressure is about 8.1 × 10⁴ Pa. A mercury barometer there (density of mercury 13 600 kg/m³, g = 9.8 N/kg) reads ______ cm of mercury.
Show answer
Answer: 61
h = P/ρg = 8.1 × 10⁴ ÷ (13 600 × 9.8) = 0.61 m = 61 cm.
Common mistakeWriting 76 cm copies the sea-level value; the reading falls with altitude because the pressure falls.
45Short answer · ★ Challenge
A ball bearing falls through a tall tube of oil. A learner records the time it takes to fall through each successive 10.0 cm section: 0.90 s, 0.70 s, 0.62 s, 0.61 s, 0.62 s. (a) Find the terminal velocity. (b) Where should the timing marks have been placed to measure it, and why? (c) Suggest one improvement to the experiment.
Show answer
Model answer: (a) The last three times are constant (about 0.617 s), so v_t = 0.100 ÷ 0.617 = 0.16 m/s. (b) Only below the first two sections (about 20 cm below the surface), because the ball is still accelerating at first and reaches a steady speed only after the drag has grown to balance weight minus upthrust. (c) Any one: time over a longer distance to reduce the timing error; repeat and average; use light gates or a video; keep the ball away from the walls of the tube; keep the oil at a constant temperature.
Common mistakeLearners often average all five times; the first two sections are before terminal velocity and give too low a speed.
46Short answer · ★ Challenge
In an experiment, ethanol (density 790 kg/m³, contact angle 0°) rises 1.4 cm in a clean glass tube of internal DIAMETER 0.80 mm. Calculate the surface tension of ethanol (g = 9.8 N/kg).
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Model answer: r = 0.40 mm = 4.0 × 10⁻⁴ m. From h = 2T cos θ/ρgr with cos 0° = 1: T = hρgr/2 = 0.014 × 790 × 9.8 × 4.0 × 10⁻⁴ ÷ 2 = 0.022 N/m.
Common mistakeLearners often put the diameter in place of the radius, which doubles the answer; the formula uses r.
47Short answer
A learner opens a kitchen tap slowly, then fully. Describe how the stream of water changes, and use the Reynolds number to explain the change.
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Model answer: When the tap is opened slightly, the stream is smooth, clear and steady: laminar (streamline) flow. When it is opened fully, the stream becomes rough, cloudy and noisy, with swirls: turbulent flow. The Reynolds number Re = ρvd/η increases with the speed v; when it becomes large (above about 2000–3000), the flow changes from laminar to turbulent.
Common mistakeLearners often say turbulence happens because the water is 'thicker'; it is the higher SPEED (larger Re) that causes it.
48Short answer · ★ Challenge
A pump in the basement of a Kigali apartment block keeps water at 3.0 × 10⁵ Pa in a pipe moving at 2.0 m/s. The same pipe (same diameter) rises 10 m to a higher floor. Find the water pressure there (density 1000 kg/m³, g = 9.8 N/kg, ignore friction).
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Model answer: Same diameter, so by continuity the speed is still 2.0 m/s and the ½ρv² terms cancel. Bernoulli: P₂ = P₁ − ρgh = 3.0 × 10⁵ − 1000 × 9.8 × 10 = 3.0 × 10⁵ − 0.98 × 10⁵ = 2.0 × 10⁵ Pa.
Common mistakeLearners often think the water must slow down as it rises; with the same pipe area the speed is the same, and it is the PRESSURE that drops.
49Short answer
Distinguish between gauge pressure and absolute pressure. A tyre gauge at a garage in Kigali reads 2.0 × 10⁵ Pa where the atmospheric pressure is 0.85 × 10⁵ Pa. What is the absolute pressure of the air in the tyre?
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Model answer: Gauge pressure is the pressure above atmospheric pressure (what most gauges read); absolute pressure is the total pressure, measured from a vacuum. Absolute = gauge + atmospheric = 2.0 × 10⁵ + 0.85 × 10⁵ = 2.85 × 10⁵ Pa.
Common mistakeLearners often take the gauge reading as the total pressure; a flat tyre still contains air at atmospheric pressure, though the gauge reads zero.
50Short answer · ★ Challenge
Air (density 1.2 kg/m³) flows over the top surface of a light aircraft's wings at 70 m/s and under them at 60 m/s. The total wing area is 20 m². Find the lift force, and decide whether it can support the aircraft of mass 1500 kg (g = 9.8 N/kg).
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Model answer: ΔP = ½ρ(v_top² − v_bottom²) = 0.5 × 1.2 × (4900 − 3600) = 780 Pa. Lift = ΔP × A = 780 × 20 = 1.56 × 10⁴ N. Weight = 1500 × 9.8 = 1.47 × 10⁴ N, which is less than the lift, so yes, the wings can support it (and it can climb).
Common mistakeLearners often calculate ½ρ(v_top − v_bottom)², squaring the difference; the speeds must be squared first and then subtracted.