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

Doc Brown's Physics Revision Notes

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

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.

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.

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)

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!

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 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).

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 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!)

Notes:

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

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

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

(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.

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, effectively a force meter or newtonmeter.

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.

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 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

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, 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 1: 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?

APPENDIX 2: Forces and circular motion

Velocity is a vector quantity, it has both size (the speed) and direction.

If either the speed or direction changes, you have a change in velocity - you have an acceleration!

With this in mind, imagine whirling a conker around on the end of a piece of string (right of diagram below).

What velocity are we dealing with? What force are we dealing with?

circular motion - velocity & centripetal force

To keep a body moving in a circle there must be a force directing it towards the centre.

This is called the centripetal force and produces the continuous change in direction of circular motion.

Even though the speed may be constant, the object is constantly accelerating because the direction is constantly changing via the circular path - i.e. the velocity is constantly changing (purple arrows, on the diagram).

For an object to be accelerated, it must be subjected to a force that can act on it - Newton's 1st law of motion.

Here the resultant centripetal force is acting towards the centre, so always directing the object to 'fall' towards the centre of motion (blue arrows on the diagram).

But the object is already moving, so the force causes it to change direction.

SO, the actual circular path of motion is determined by the resultant centripetal force (black arrows and circle) and the circling object keeps accelerating towards what it is orbiting.

The centripetal force stops the object from going off at a tangent in a straight line.

When you swing something round on the end of a string, the tension in the string is the centripetal force.

You yourself feel this force of tension as the 'pull' in the string.

If you could use a fast action camera to monitor the motion and the string broke, you would observe the object would fly off at the precise tangent to the circular path and in a straight line of constant velocity - the result resultant of Newton's 1st law!

Since gravity and air friction act on the object, you do have to keep on 'inputting' kinetic energy to keep it swinging round.

The centripetal force will vary with the mass of the object, the speed of the object and the radius of the path the object takes.

The same arguments on circular motion apply to the movements of planets around a sun, a moon around a planet and a satellite orbiting a planet. The orbits are usually elliptical, rarely a perfect circle, but the physics is the same.

In these cases, it is the force of gravitational attraction that provides the centripetal force and it acts at right angles to the direction of motion.

You should also realise that they are moving through empty space (vacuum), so there are no forces of friction to slow the object down.

This is why the planets keep going around the Sun and the moon keeps going around the Earth.

When satellites are put into orbit they are given just the right amount of horizontal velocity so that the resultant centripetal force of gravity keeps the satellite in its a circular orbit.

You can vary this horizontal velocity to position satellites at different distances above the Earth's surface.

Forces revision notes index

FORCES 1. What are contact forces & non-contact forces?, scalar & vector quantities, free body force diagrams

FORCES 4. Elasticity and energy stored in a spring

FORCES 5. Turning forces and moments - from spanners to wheelbarrows and equilibrium situations

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