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

and the consequences of gravitational fields - calculations and various phenomena explained, plus work done and GPE

 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?

Sub-index for this page

(a) Introduction to gravity

(b) What is mass? What is weight and how do you calculate it?

(c) Some simple examples of weight calculations

(d) Other aspects of weight and gravity phenomena including centre of mass

(e) Weight, work and gravitational potential energy

(f) Forces, gravity and circular motion

(a) 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 non-contact force exists wherever there is mass.

All objects have a gravitational field around them.

Therefore, there is always a force of attraction between all objects all the time.

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.

It was Isaac Newton who first realised that objects fell to Earth due to gravitational field attraction.

He not only recognised that gravity was a universal law of nature, but realised you could not only apply it to 'falling objects', but also to the motion of the planets around our Sun and our Moon orbiting the Earth.

BUT, remember, you are attracting the Earth at the same time as it attracts you. You don't notice this because your mass is so much smaller than the Earth. You only consciously experience yourself being attracted to the Earth.

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 the two attracted objects, d = distance from centres of gravity of the two objects

So you can relate this arithmetical proportionality equation to the two rules above)

 

You don't really notice gravitational attraction between objects around you because the gravitational fields are too weak. However, because the huge mass of the Earth and its very strong gravitational field, you definitely notice objects falling towards it. BUT, with gravity, we are always dealing with mutual attraction, so, what you don't notice is that your body is attracting the Earth towards it - think about it!

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(b) What are mass and weight - 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 the same as weight.

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!

BUT your weight might be anything from 400 N on the Earth's surface, to zero in outer space - all will be explained!

One consequence of gravity is that you experience weight, which is always acts as a downward force due to a gravitational field effect and is always an attractive force.

Note : Mass is a scalar measurement (just has size or magnitude).

Weight (or any force) is a vector measurement (it has both size and direction - always downwards for gravity).

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 the gravitational field strength (gravitational acceleration) 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)

OR you may need to think in terms of:

force in newtons = mass of object in kilograms x gravitational acceleration

This expression is equivalent to F = ma, the equation of Newton's 2nd Law of Motion.

The general equation is easily rearranged:

W = m x g,    m = W ÷ g,    g = W ÷ m

(learn to rearrange, its better than using the triangle)

Notes:

(i) There are three are variables, W, m and g.

(ii) Weight is proportional to mass for a give g.

(iii) The gravitational field strength constant (g) varies from planet to planet because the mass of the planets varies.

On a massive planet like Jupiter, the gravitational field strength is much greater than that on Earth.

On a smaller planet or our Moon, with far less mass, the gravitational field strength is much less.

Examples of surface gravitational field strength constants:

which can be units of weight in N/kg, but also units of acceleration in m/s²

Our Moon 1.62, planets: Mercury 3.70, Earth 9.81, Jupiter 24.79. To put these objects in perspective, an extremely dense neutron star may have a gravitational acceleration of 7 x 10¹² m/s² and its even greater for a black hole. kapow !!!!

(iv) g also varies with the distance you are from the centre of a large body like a planet e.g. it decreases the further up you are from the Earth's surface e.g. on the top of a mountain.

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!

In travelling from the Earth to the Moon, you would seem to have lost a lot of weight!

Unless of course, you take Moon calibrated weighing scales!

You can measure weight using a calibrated spring balance, effectively a force meter or Newton meter.

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

You can use a balance that is calibrated in g/kg and multiply by 9.8 to get the weight of the object on the Earth's surface.

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(c) Some simple example weight 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 g_earth = 70 x 9.8 = 686 N

(b) W = m x g_moon = 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 g_mars = 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 0.710 N/kg.

What would be the mass of an object on Pluto that would experience a weight of 10.0 N?

W = mg, so m = W/g_pluto = 10/0.71 = 14.1 kg (3 sf)

Q4 Imagine an astronaut in a space station experiencing 'weightlessness'.

Suppose the astronaut pushes against the wall with a force of 30 N and moves backwards with an acceleration of 0.40 m/s². What is the mass of the astronaut?

Here you have to think of the 'weight' equation as

force in newtons (N) = mass of object (kg) x acceleration (m/s²)

(of F = ma, if you have studied Newton's 2nd law of motion)

Substituting in the equation gives:

20 = mass x 0.40,  therefore mass of astronaut = 30 / 0.40 = 75 kg

Note that in this calculation, the acceleration does not have to be due to a gravitational field.

 

Q5

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(d) Other aspects of weight and gravity phenomena including centre of mass

Weightlessness

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 to fall quite rapidly and 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 and, more importantly, 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 with a hammer and feather.

In the old 'black and white' footage from the first moon landing by US astronauts, you could clearly make two important observations:

(i) On dropping, the hammer and feather hit the moon's surface at the same time - no air resistance.

(ii) The hammer, quite plainly, fell with much less acceleration than on Earth because of the much smaller gravitational field force on the moon, due to its much smaller mass.

On the Earth's surface, the gravitational acceleration is 9.8 m/s² and on the Moon it is only 1.6 m/s².

What is the centre of mass of an object? How can it be determined 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.

Its quite easy to envisage where it is for a regular shape e.g. a rectangular block - shown in profile in the diagram below.

The stability of a free standing object

A standing object becomes unstable when the vertical line through its centre of mass falls outside its base, which effectively acts as a base.

The weight of the object causes a turning effect about the pivotal base. The idea is illustrated by the diagram below of a regular shaped block, shown in profile, and tilted at various angles (but it could be a bus!).

1. The vertical line from the centre of mass passes right through the centre of the object's base.

The object is completely stable - no moment (turning force) is generated.

2. The vertical line from the centre of mass still passes through the base, but not its centre, and the object is unstable, so it will wobble a bit from side to side, and eventually settle down in an stable upright position as in 1.

The edge of the object touching the ground acts as a pivot point.

The weight of the object creates an anticlockwise moment (turning force) that makes the object fall back in an anticlockwise direction, but not sufficient to topple the object over.

3. The vertical line from the centre of mass passes outside of the object's base. The object won't even wobble, it is highly unstable and will just topple over on its side (to become stable!).

Again, the edge of the object touching the ground acts as a pivot point.

Again, the weight of the object creates a clockwise moment (turning force) that makes the object fall over in a clockwise direction, and sufficient to topple the object over on its side.

Tests on stability in terms of the centre of mass are important e.g. road vehicles like buses are safety tested to see the maximum angle allowed when tilted over without toppling over in an accident.

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 plumb line 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 plumb line 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!

See also Turning forces & moments - spanners to wheelbarrows, levers, gears & equilibrium situations

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(e) 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, E_gpe = m g h

gravitational potential energy, E_gpe, 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?

Also note that when an object /material falls, the GPE is converted into kinetic energy.

The gravitational potential energy store of the material decreases and the kinetic energy store of the material/object increases.

When a object stops falling, its maximum KE equals the GPE it had with respect to the height the object falls.

See Gravitational potential energy store calculations includes examples of using E_gpe = m g h = KE = 0.5 m v²

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