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
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?
Subindex 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 noncontact 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
m_{1} x m_{2}/d^{2},
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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subindex
(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 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 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/s^{2}
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 10^{12} m/s^{2}
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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subindex
(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/s^{2}. 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/s^{2})
(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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subindex
(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/s^{2}.
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/s^{2} and on
the Moon it is only 1.6 m/s^{2}.
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 coordinates 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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subindex
(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
righthand 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 v^{2}
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subindex
(f) Forces,
gravity 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.
For more on motion and acceleration see Acceleration, velocitytime graph interpretation and calculations,
problem solving
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.
For more on motion and acceleration see Acceleration, velocitytime graph interpretation and calculations,
problem solving
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subindex
Forces revision notes index
FORCES 1. What are contact forces &
noncontact forces?, scalar & vector quantities, free body force diagrams
FORCES
2. Mass and the effect of gravity force on it  weight, (mention of work done and
GPE)
FORCES 3. Calculating resultant forces using vector
diagrams and work done
FORCES
4.
Elasticity and energy stored in a spring
FORCES 5. Turning forces and moments  from spanners
to wheelbarrows and equilibrium situations
FORCES 6. Pressure in liquid fluids and hydraulic
systems
FORCES 7. Pressure & upthrust in liquids, why do
objects float or sink in a fluid?, variation of atmospheric pressure with
height
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