Gravitation — review

Answerc) They land at the same instant

Gravity does pull the iron ball ten times harder — but the iron ball is also ten times harder to accelerate. The two effects cancel exactly, which is what you see when the mass cancels in g = GM/R². In air the wooden ball would lose to air resistance, but the chamber is a vacuum, so there is none.

Answerb) one ninth of what it was

The distance appears as r² in the denominator, so the force depends on the square of the distance, not the distance itself. Triple r and you divide by 3² = 9. Fig. 1 shows this collapse: by the time the objects are three times as far apart, the force bar has almost vanished.

Answerb) half of what it was

Take the two changes one at a time. Doubling a mass multiplies the force by 2, since the masses appear on top. Doubling the distance divides it by 2² = 4, since r² is underneath. Together: 2/4 = 1/2. The commonest error is to cancel them and answer "unchanged", which forgets that the distance is squared and the mass is not.

Answerb) Mass 60 kg, weight about 98 N

Mass is the amount of matter and does not change by travelling — it stays 60 kg anywhere in the universe. Weight is the force of gravity, W = mg, so on the Moon it is 60 × 1.63 ≈ 98 N, about a sixth of the 588 N the astronaut weighs on Earth.

Answerb) It is moving fast enough sideways that it keeps missing the Earth as it falls

The Moon IS falling toward the Earth, constantly. What saves it is its sideways velocity: by the time it has fallen, it has also moved along, and the Earth’s surface has curved away underneath by the same amount. A fall that never arrives is an orbit — which is the moment in Fig. 3 when the red path turns green.

Answerb) its velocity is zero and its acceleration is 9.8 m/s² downward

The stone stops rising for an instant, so its velocity is momentarily zero. Its acceleration is not: gravity never switches off, and it is still 9.8 m/s² downward the whole way up, at the top, and all the way down. If acceleration really were zero at the top, the stone would hang there forever instead of coming back.

Answerb) At the poles, because the radius is smallest there

Since g = GM/R², a smaller R gives a larger g. The Earth is slightly flattened, so the distance from the centre to a pole is less than the distance to the equator — which makes g largest at the poles. The mountain answer goes the wrong way: climbing increases R, so g falls.

Answerb) pascal (Pa)

Pressure is thrust divided by area, so its unit is one newton per square metre, and that is given the name pascal. The newton is the unit of the force itself, not of the force spread over an area.

Answerd) no unit at all

Relative density is one density divided by another, so the kg/m³ on the top cancels the kg/m³ on the bottom and nothing is left. A quantity that is a pure ratio has no unit — which is exactly what makes it a quick way to answer "will this float?" without carrying units around.

Answerb) the volume of the object

By Archimedes’ principle the upthrust equals the weight of the fluid displaced, and what an object displaces is set by its volume — not by how heavy it is or what it is made of. Two fully submerged balls of the same size feel the same upthrust even if one is cork and one is lead. Depth does not enter it either, once the object is completely under.

Answerb) About 60%

A floating object sinks until the water it displaces weighs exactly what the object weighs, and that happens when the submerged fraction equals the ratio of the densities: 600/1000 = 0.6, so 60% is under. Notice the size of the block never entered the calculation — only the densities did.

Answerb) the sharp knife concentrates the same force onto a smaller area, so the pressure is greater

Your hand supplies the same force either way. What changes is the area it acts over: pressure is thrust divided by area, so shrinking the area of the edge raises the pressure enormously. It is the same reason a nail has a fine point and a camel has wide feet — one concentrates pressure on purpose, the other spreads it.

Every object attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. In symbols, F = G m₁m₂ / r², where F is the force of attraction between the two objects, m₁ and m₂ are their masses, r is the distance between their centres, and G is the universal gravitational constant, 6.673 × 10⁻¹¹ N m² kg⁻², which has the same value everywhere in the universe. Marks are usually lost for writing r instead of r², and for not saying that r is measured centre to centre.

Because G is about 6.673 × 10⁻¹¹, which is fantastically small. Put ordinary human masses into F = G m₁m₂ / r² and the force comes out around a hundred-millionth of a newton — far too small to feel, and far smaller than the friction holding the people in place. Gravity only becomes noticeable when at least one of the masses is enormous, like a planet. That is why things fall toward the Earth rather than toward each other.

Because g = GM/R², and R is not constant. The Earth is not a perfect sphere — it is slightly flattened at the poles — so the distance from the centre to the surface is smaller at the poles than at the equator. A smaller R means a larger g, so g is greatest at the poles and least at the equator. Height has the same effect: climbing a mountain increases your distance from the centre, so g falls. The value 9.8 m/s² is a convenient average, not an exact figure for any particular place.

First, what they are: mass is the quantity of matter in the object, while weight is the force with which gravity pulls on it. Second, their units: mass is in kilograms, weight is in newtons, because weight is a force. Third, whether they change: mass is the same everywhere in the universe, while weight changes with location because it depends on g — the same object weighs about a sixth as much on the Moon. A fourth difference worth knowing: mass is a scalar, weight is a vector, since it acts in a direction (toward the centre of the Earth).

Archimedes’ principle: when an object is immersed wholly or partly in a fluid, it experiences an upward force equal to the weight of the fluid it displaces. Two applications: a submarine dives and surfaces by changing its average density, flooding its ballast tanks with water to sink and blowing them out with compressed air to rise; and a hydrometer measures a liquid’s density by how deep it floats, since it settles lower in a thinner liquid — the lactometer is the version built for testing whether milk has been watered down.

Pressure is thrust divided by area. The camel is far heavier than the person, so its thrust on the sand is greater — but its feet are broad, and spreading that weight over a large area keeps the pressure low enough that the sand supports it. A narrow heel concentrates a much smaller weight onto a tiny area, so the pressure is far higher and the sand gives way. The lesson is that sinking depends on pressure, not on weight alone.

Floating depends on the average density of the whole object, not on the density of the material. The ship encloses a large volume of air, so its total mass divided by its total volume is less than the density of water — it displaces a weight of water greater than its own weight, and the upthrust holds it up. The solid block has the same mass packed into a far smaller volume, so it displaces too little water to be supported and it sinks.

1. FormulaF = G m₁ m₂ / r²
2. SubstituteF = (6.673 × 10⁻¹¹ × 20 × 40) ÷ 2²
3. Top20 × 40 = 800
4. Bottom2² = 4
5. DivideF = 6.673 × 10⁻¹¹ × (800 ÷ 4) = 6.673 × 10⁻¹¹ × 200

AnswerF = 1.33 × 10⁻⁸ N

Notice how tiny that is for two masses you could pick up. It is G being of the order 10⁻¹¹ that makes it so — which is exactly why you never feel the pull of the person next to you.

1. Velocityv = u + g t
2. Substitutev = 0 + 9.8 × 3
3. Distanceh = u t + ½ g t²
4. Substituteh = 0 + ½ × 9.8 × 3²
5. Simplifyh = ½ × 9.8 × 9

Answerv = 29.4 m/s and h = 44.1 m

Two answers, two formulae, two different units. Quoting 29.4 m for the distance is the usual slip — that number is the velocity. "Dropped" is what tells you u = 0; if the stone had been thrown down, u would not be zero.

1. FormulaW = m g
2. On EarthW = 10 × 9.8
3. On the MoonW = 10 × 1.63

AnswerW = 98 N on Earth, 16.3 N on the Moon

The mass stays 10 kg in both places — only the weight changes, because only g changed. Giving the answer in kilograms is wrong: weight is a force, so it is measured in newtons. And 16.3 is about a sixth of 98, which is the one-sixth rule you can check your answer against.

1. Formularelative density = density of substance ÷ density of water
2. Rearrangedensity of substance = relative density × density of water
3. Substitute= 10.8 × 10³

Answerdensity of silver = 1.08 × 10⁴ kg/m³ (10 800 kg/m³)

Relative density has no unit, so the kg/m³ in the answer comes entirely from the density of water you multiplied by. Since 10.8 is greater than 1, silver sinks in water — a quick sanity check on the answer.

1. ThrustF = m g = 100 × 9.8
2. F = 980 N
3. FormulaP = F ÷ A
4. SubstituteP = 980 ÷ 0.5

AnswerP = 1960 Pa

The thrust here is just the block’s weight, so the first move is always mg — starting from 100 and dividing by 0.5 gives 200, which is a mass per area and not a pressure. Stand the same block on a smaller face and F would be unchanged while P rose, which is the whole point of the thrust-and-pressure section.