FORCES 2. Mass and the effect of gravity on it - weight

calculations and various phenomena explained, plus work done and GPE

Doc Brown's Physics Revision Notes

Suitable for GCSE/IGCSE Physics/Science courses or their equivalent

 This page will answer questions such as ...

 What is weight and how do we calculate it?

 What is the difference between mass and weight?

 Why does gravity vary from planet to planet?

 Why can a feather and iron bar fall at the same rate in a vacuum?

Introduction to gravity

A gravitational attractive forces acts between all objects of any mass, no matter how close or far apart they are.

Gravity is universal and its force exists wherever there is mass.

You experience gravity it as you jump up vertically against it's force and are then pulled back down to earth by the same force.

Gravity makes everything fall towards the surface of an object e.g. like a planet, and it is gravity that gives everything 'weight' (explained below).

The force of gravitational attraction between two masses increases by two factors:

(i) The bigger the two masses involved

With its bigger mass, an elephant is more strongly attracted to the earth than you!

The greater a planet's mass, the greater the strength of the gravitational field around it.

(ii) The closer the two objects are together

The further you go above the Earth's surface the weaker the gravitational attraction between you and your planet.

(Note: Although not needed for GCSE: F m1 x m2/d2, F = force of attraction between the two objects, m = masses of objects, d = distance from centres of gravity, and you can relate this arithmetical proportionality equation to the two rules above)


What is mass? What is weight and how do you calculate it?

Mass is the amount of matter in an object (all the atoms added together) and is constant unless you change the object in someway to remove atoms or add atoms.

The standard unit of mass is the kilogram (kg), in chemistry or physics laboratory you often weigh things out as grams (1 g = 1 kg/1000). In chemistry calculations you tend to work in g, in physics calculations is often kg.

Mass is NOT a force, but the mass of an object is constant no matter where it is in the whole universe e.g.

you may be 40 kg here on Earth, in outer space, up in a satellite space station, on planet Mars or frozen on Pluto!

One consequence of gravity is that you experience weight.

You should appreciate immediately that your mass is constant at a given instance in time wherever you are in the universe, but the same cannot be said for weight.

So what is weight? Why can it vary for a given mass?

Quite simply, weight is the force of gravity acting on an object of given mass. Weight is in effect the 'pulling' force an object experiences in a gravitational field e.g. you experience the Earth's gravitational field as your weight even if it says kg on your bathroom scales!

weight = a force in newtons

Weight varies with the mass of the object and the strength of the gravitational field at the point where the object is.

Weight is directly proportional to mass and directly proportional to gravity too.

The formula to calculate this force, that is to calculate the weight of an object, is quite simple.

weight in newtons = mass of object in kilograms x gravitational field strength

W (N) = m (kg) x g (N/kg)

Easily rearranged: W = m x g, m = W g, g = W m (learn the triangle, always helps!)

Note that all three are variables, weight is proportional to mass for a give g, but g varies from planet to planet because the mass of the planets vary. g also varies with the distance you are from

On the surface of planet Earth the force of gravity on objects is 9.8 N/kg (the Earth's 'g' value'), so a mass of 1 kilogram experiences an attractive force of about 10 newtons. However, on the surface of the moon, the gravitational field force is only 1.6 N/kg (the moon's 'g' value'), so 1 kg on the moon only experiences a force of 1.6 N. On the moon you would feel much lighter and could leap around with your Earth designed muscles to much greater heights - you may have seen how the astronauts on the moon had to be careful to not overdo things!

Although you would seem 'lighter' on the moon, your mass will be still the same! Weighing machines like bathroom scales are calibrated to the strength of the Earth's gravitational field so the spring action scale can be read in kg. Bathroom scales, or any other scales, would give a very false reading on the moon!

You can measure weight using a calibrated spring balance force meter (newtonmeter).

Along side the spring is a scale calibrated in newtons, the unit of force.

Some simple example calculations:

Q1 What is the weight of 70 kg adult on (a) the Earth, (b) the moon.

Using the gravitational field constants quoted above

(a) W = m x gearth = 70 x 9.8 = 686 N

(b) W = m x gmoon = 70 x 1.6 = 112 N

Quite a difference!


Q2 An astronaut on Mars found an object of mass 5.50 kg gave a reading on an electronic balance meter of 20.41 N. Calculate the strength of gravity on the surface of Mars.

W = mg, so gmars = W/m = 20.41/5.50 = 3.71 N/kg

Note the value of the gravitational field strength is more than the moon (smaller mass) and not as large as on Earth (bigger mass). This is ignoring their different sizes and densities, its just a surface gravity comparison.


Q3 The force of gravity on the dwarf (minor) planet Pluto is 0.71 N/kg. What would be the mass of an object on Pluto that would experience a weight of 10 N?

W = mg, so m = W/gpluto = 10/0.71 = 14.08 kg


Other aspects of weight and gravity phenomena


If an object is 'weightless' it is apparently not being subject to any gravitational field force.

The most obvious example is an object out in deep space well away from any star or planet.


Why does a feather and a hammer fall at the same velocity in vacuum?

Experiment A: When you drop from a few metres height a heavy object like a hammer and a light object like a feather at the same time, your experience will tell you to expect the hammer, quite rapidly, to hit the floor first and the feather to follow on far more slowly.

Experiment B: If you repeat the experiment e.g. with a small but heavy weight and a small feather at one end of an enclosed large glass sealed tube from which all the air is pumped out, you see a very different result. Both objects fall at the same rate.

The reason for the result you see in experiment B is because all objects, whatever their mass, experience the same accelerating force due to the gravitational force field (of the Earth). The acceleration is actually ~10m/s2.

In experiment A the feather is much lighter has a much greater surface area to mass ratio and the friction effect (drag) of it passing through the air is much greater than what the hammer experiences. So the descent of the feather is slowed down. If the air is removed, there is no drag effect on either object and they accelerate to Earth at the same rate. You would see exactly the same effect on our moon, which has virtually no atmosphere and the first men on the moon did a similar B experiment.


What is the centre of mass of an object? How can be determine it by experiment?

For some calculations, and, of great importance to structural engineers, you may need to know where the centre of mass (sometimes called the centre of gravity).

The centre of mass is a single point in the object through which the whole weight of an object is considered to act.

For regular shaped objects of uniform density its quite easy to figure it out. e.g. the centre of mass of a cube will be at the centre, equidistant from the 8 vertices.

For a rectangular block, the centre of mass point is defined by the co-ordinates H/2, B/2 and L/2. The same argument applies if H = B = L for a cube.

For a sphere of uniform density, the centre of mass will be at its dead centre.

For an irregular shaped object like yourself its a bit more tricky!

 However, if it is a 'flat' object like a sheet of thick cardboard, wood or metal you can do quite a simple experiment to determine the centre of mass (centre of gravity).

 In the school/college laboratory it quite easy. You pin the object at various points, ensuring it can hang freely under its own weight and hang a weighted plumbline down from the same point. The pin holes should be as near to the edge of the irregular object as possible.

When the object is quite stationary you mark another point on the other side of the object so you can then join them both up to give a locating line (e.g. line A). You then repeat this, choosing another point further round the object giving line B and then C etc.

You should find that all the lines intersect at point G, the centre of mass.


The method works because when the object hangs freely, there is equal mass (weight) on either side of the plumbline and this is independent of the pin point.

You can determine the centre of mass of a teachers by pinning them up by the tips of their ears, fingers or toes!


APPENDIX weight, work and gravitational potential energy

Two types of calculations follow on from the 'weight and gravity' notes above.

You may encounter either of them before or after studying 'weight and gravity', but they are closely related and follow on from the notes above.

If you allow a weight to fall it can do work, because a raised weight is an energy store of gravitational potential energy (GPE).

 The general formula for work done (energy transferred) is:

work done in joules = acting resultant force in newtons x distance through which the force acts in metres

W (J) = F (N) x d (m)

You can then apply this equation to calculate the energy stored as GPE on raising a weight (mass x gravitational force) a given height. You therefore have also calculated the energy that can be released (ignoring friction) if the weight is allowed to fall.

The force (F) involved will be the weight of material raised or lowered

In general an object or material possesses gravitational potential energy by virtue of its higher position and can then fall or flow down to release the GPE e.g. winding up the weights on a clock, water stored behind a dam that can flow down through a turbine generator. Any object falling or material flowing downwards is converting GPE into kinetic energy and any object raised in height gains GPE.

Since gravitational energy is a form of stored energy, it does nothing until it is released and converted into another form of energy.

The amount of gravitational potential energy gained by an object raised above ground level can be calculated using the equation:

GPE = mass gravitational field strength height, Egpe = m g h

gravitational potential energy, Egpe, in joules, J

mass, m, in kilograms, kg

gravitational field strength, g, in newtons per kilogram, N/kg

height, h, in metres, m

Note: (i) In any calculation the value of the gravitational field strength (g) will be given.)

(ii) In the equation you should realise that the m x g = weight, the first two parts of the right-hand side of the equation. This is effectively the force that moves through the height the object is raised or lowered. This means the GPE equation is just a variation of the general equation for work done or energy transferred from one energy store to another.

GPE = mgh is another form of W = Fd, so I hope you can see the connection?



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