Chapter 11. weight and the normal force
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weight and the normal force
Is it really possible to lose weight instantly?!
Here’s the deal. The machine has a platform at the top with
some scales on it. When you stand on the scales, they read
the same number of kilograms as they usually would in your
bathroom. No surprises there.
Here, the reading on
the scales is the same
number of kilograms
as it would usually be.
But then the platform you’re standing on suddenly moves
downwards - and the reading on the scales becomes lower.
Numbers don’t lie - so if the reading’s gone down then you
must have lost weight. Right?
You’re lighter here
than you were before.
Just before you reach the bottom of
the machine, the scales are switched
off to protect them from the impact
with the cushioned landing area.
There must be a trick involved
somewhere... but what is it? The scales
don’t look fake and read the same
number of kilograms as usual when
they’re not on the machine.
Set of
scales.
Platform moves
down when you
step on it.
Reading on scales
taken here.
Here, the reading on the
scales is lower than it was
at the top of the machine!
Scales switched
off here.
Spring-cushioned
landing area.
So maybe it’s something to do with what
the machine does and how the scales
produce a measurement.
How do you think the machine works?
(How do scales actually produce a measurement?)
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scales and springs
Scales work by compressing
or stretching a spring
Some scales work by compressing a spring. If
you put pieces of fruit on top that are all more
or less the same size, the spring will compress
by the same amount each time you add another
piece. The change in length of the spring is
converted into a reading in kilograms.
Change in
spring length.
Original
spring
length.
New spring
length.
Change in spring
length makes the
dial go round and
point to a number
of kilograms.
Another type of scales works by stretching a spring.
This is exactly the same principle as compressing a
spring, except that you hang an object from the spring
rather than putting it on top. Again, a change in length is
converted into a reading in kilograms.
A spring will
always compress/
stretch by the same
amount for the
same load, to give a
consistent reading.
A marker at the end of
the spring points to a scale
showing the current weight.
This loop’s attached
to the end of the
spring, and can be
pulled by something.
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This end is firmly
attached to something.
Spring is in here.
Inner bit with loop is
separate from outer
bit with scale on it.
weight and the normal force
I’m just trying to get my head around this.
You stand on the scales - and they read the
same as they usually would. Then the scales move
downwards in the machine - and the number of
kilograms they read goes down too.
Jim: Yeah, I’m struggling as well. I don’t see how the person’s lost
weight. It’s not like they were wearing a rucksack full of boulders
that they suddenly took off, or anything.
Joe: Maybe it’s something to do with how the scales make their
measurements. Scales don’t measure the number of kilograms
directly - scales measure the change in length of a spring.
Frank: Hmmm. You mean if I put the
scales against the wall and pushed them
with my hand, they’d register a number of
kilograms. Yeah, I can see that.
If you know HOW
your measuring
devices work, you
can trouble-shoot
your experiments
when unexpected
things happen.
Hand pushes scales
with this force.
Jim: That’s weird. Kilograms are units of mass, right? Mass is
the amount of ‘stuff ’ something’s made from. But if you push the
scales sideways like that, the reading depends on the force that you
push with, not on the amount of stuff your hand’s made from.
Joe: I guess that’s because the scales don’t really measure kilograms
directly - they measure the change in the length of the spring.
And that must depend on the force that the spring’s pushed with.
Frank: If I’m standing on the scales, I’m kinda pushing down on
the spring inside them because of gravity. I guess that because of
gravity, a certain number of kilograms must produce a certain force
- and a certain change in length of the spring. So the scales always
assume that you’re standing on them when making a measurement.
Joe: So if you use the scales differently from how the manufacturer
intended - by pushing them against the wall, or perhaps by making
them move down like the WeightBotchers machine does - then you
get a flakey reading.
Frank: That sounds plausible - but personally I’d like to get my
head around how force (which seems to do with pushing) and mass
(which is to do with the ‘stuff ’ you’re made of) are connected ...
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weight is a force
Mass is a measurement of “stuff”
Mass is an indication of how much ‘stuff ’ something is
made from, and is measured in kilograms. Mass is a
scalar, as ‘stuff ’ can’t have a direction - it’s just what’s there.
Even though the scales indicate otherwise, the person on the
WeightBotchers machine always has the same mass - it’s not
like they took off a rucksack or had a haircut halfway down
and lost a whole lot of matter.
MASS is how much
“stuff” something’s
made of. It’s a scalar,
because ‘stuff’ doesn’t
have a direction.
Weight is a force
Applying a force in this
direction compresses
the spring and makes
the dial go round.
If you put the scales against the wall, you can exert
a horizontal force on them by pushing them with
your hand and compressing the spring.
Force is a vector because it has direction - the
direction that you’re pushing the spring in.
Although the scales give a reading in kilograms, they actually make
measurements based on the change in length of the spring. So if
you put fruit on the scales and the spring’s length changes, there must
be a force involved.
The force vector
of the fruit’s
weight points in
this direction.
The change in length comes about because the spring has to
counteract the fruit’s weight, which is there because the fruit is in the
earth’s gravitational field. The fruit’s weight is the force exerted on it
by the earth’s gravitational pull. You can draw the fruit’s weight as a
force vector arrow pointing down, towards the center of the earth.
WEIGHT is the FORCE you
experience as a result of being
in a gravitational field.
As weight is a force,
this is a force vector.
On Earth, your weight vector points
down, towards the center of the earth.
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weight and the normal force
But people say things like “I weigh 60
kilograms” all the time. How can you say
that mass and weight are different?
Mass and weight are different!
In everyday speech, people use the words “mass”
and “weight” like they’re the same thing. But in
physics we need use these words more carefully.
Mass. Stuff. Scalar.
Weight. Force. Vector.
If you go to the moon, your mass is the same
number of kilograms as it is on earth, as you’re
still made from the same amount of ‘stuff ’.
But weight is the force you experience as a result
of being in a gravitational field. And as the
moon’s gravitational field is smaller than the
earth’s, your weight is less on the moon than it is
on Earth even though your mass is still the same.
Your mass is the same on
the earth and the moon.
So the scales measure the force it takes
to compress a spring, then convert the force
that causes a certain change in spring length into
kilograms? It sounds like the relationship between mass
and weight is really important here.
Earth.
The way that the scales convert a
force into a reading in kg is crucial.
If you stand on the scales on the moon, the
scales will read the wrong number of kilograms
- even though your mass hasn’t changed. This is
because the scales assume you’re on earth when
they convert the change in length (as a result of
an applied force) into a reading in kilograms.
If you can work out the relationship
between force and mass that the scales use
to do this conversion, you’ll be able to debunk
the WeightBotchers machine.
Weight
on Earth
Force vector
arrow.
Moon.
Weight
on Moon
Your weight is six
times larger on
the earth than
on the moon.
Have you seen an equation
that involves both force and
mass somewhere before?
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force and mass
The relationship between force and mass involves momentum
In chapter 10, you figured out that when you apply the same force
for the same amount of time to any object, you always give it
the same change in momentum. As long as there are no other
forces acting on the object, you can write this as an equation:
F t =
p
Force applied ...
... for a period of time
... gives a change in momentum.
But momentum is mass × velocity. So you can substitute in mv
every time you see a p and rewrite this equation as:
F t =
(mv)
Momentum, p = mv
m1
Large mass
Small velocity.
Same change
in momentum.
F
Same force
applied for same
amount of time.
v1
F
p 1 = m1v 1
Small mass
m2
v2
p 2 = m2v 2
Large velocity.
This equation works
if F is the only force
acting on the object.
F t =
Here, we’ve called the
elephant object 1, so it has
mass m1 and velocity v1. Using
numbers in subscripts is a
common way of distinguishing
between objects in physics.
Using numbers in subscripts
makes equations more general. We
could still write down the same
equation, p1 = m1v1 if we swapped
the elephant for a duck, whereas
pe = meve would be confusing.
(mv)
Momentum, p = mv
This is Newton’s Second Law. It shows that objects
with more mass have more inertia, or more resistance to
changing how they’re currently moving. If you apply the
same force for the same time to push two different objects,
the object with the larger mass is more ‘resistant to change’
and has a smaller change in velocity at the end.
The equation F t = (mv) gives you a relationship
between force and mass that you can use to work out
what’s going on with the WeightBotchers machine.
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This equation works for ANY object. We
don’t have a specific object in mind here,
so there are no subscripts on the ‘m’ or ‘v’.
weight and the normal force
Newton’s Second Law:
If you apply a force
NET
to any object for a
period of time, the
change in the object’s
momentum always has
the same value.
Fnet t =
(mv)
This equation works for any number
of forces acting on the object added
together to make the net force, Fnet.
But sometimes you push something
with a force and it stays still. Where’s
the change in momentum there?
It’s the net force that matters.
Two people pushing the mouse with equal
forces in opposite directions looks like this:
F
F
m
Fnet = 0
When you add together these force vectors
by lining them up ‘nose to tail’, the overall,
or net force you end up with is zero. And
the mouse doesn’t go anywhere, so its
momentum doesn’t change.
But if the left-to-right force became larger,
it would start to ‘overpower’ the right-to-left
force, and there’d be a net force to the right.
So the mouse would start moving to the
right - its momentum would change in the
direction of the net force.
a. After introducing a subscript to make it clear that it is the net force that causes the change in
momentum, the equation on the opposite page, Fnet t = (mv) can be rearranged to say Fnet =
Use this equation to work out the units of force.
(mv)
t
(mv)
. Do both m and v change with time while a force is applied?
t
(Assume that the situation is one where an elephant or mouse has been pushed with a net force.)
b. Your equation contains the term
c. Does your answer to part b give you any ideas about how you might simplify your equation Fnet =
Hint: What other equations
do you know where a variable
changes with time?
(mv)
t
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if mass is constant...
a. After introducing a subscript to make it clear that it is the net force that causes the change in
momentum, the equation on the opposite page, Fnet t = (mv) can be rearranged to say Fnet =
Use this equation to work out the units of force.
[m] = kg [t] = s
[v] = m/s
[F] = kg.m/s = kg.m/s2
s
(mv)
t
If you say this out loud, it’s:
“kilogram-meters per second squared”
(mv)
. Do both m and v change with time while a force is applied?
t
(Assume that the situation is one where an elephant or mouse has been pushed with a net force.)
b. Your equation contains the term
The velocity changes but the mass doesn’t change.
c. Does your answer to part b. give you any ideas about how you might simplify your equation Fnet =
You could turn it into F = m v as the mass is constant.
t
v
And t is the acceleration. So it could become F = ma.
(mv)
t
Don’t worry if you
didn’t spot this.
If the object’s mass is constant, Fnet = m a
Newton’s Second Law says that if you apply a net force to an object
for a period of time, then its momentum changes. So force is the
rate of change of the momentum of an object:
Fnet =
(mv)
t
Rate of change
of momentum
Typically, the mass of an object doesn’t change during the time
that the force is applied. This means that m is constant and only
v
v changes with time. And you already know that
is the rate of
t
change of velocity - in other words, the acceleration.
So you can rewrite Newton’s Second Law as:
Fnet = ma
This shows you that the units of force are kg.m/s2. However, as this
is a rather unwieldy unit to write out, physicists have come up with a
new unit, the Newton (N) where 1 N = 1 kg.m/s2.
So if you do a calculation to work out a force where the mass is in
kg and the acceleration is in m/s2, you’d write your answer as 10 N
instead of 10 kg.m/s2.
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The form of Newton’s
Second Law that
you’ll use the most is:
Fnet = ma
Net force
Mass
Acceleration
weight and the normal force
Q:
So why not just say “Fnet = ma” from the start? Why all this
stuff about momentum first?
A:
This book is about understanding physics. Rather than
nodding and accepting “Fnet = ma” with no reason for it, we went back
to what you discovered about momentum in chapter 10, when you
used a force to change the momentum of various objects. You’ve just
used what you already knew about momentum to work out this form
of Newton’s Second Law for yourself.
Q:
Won’t the mass of an object always be constant? So you
can always use Fnet = ma?
A:
Sometimes, both the mass and velocity of an object can
change. For example, a rocket going into space carries a large mass
of fuel, which it continually burns. As time goes on, its velocity gets
larger, but its mass gets smaller as the fuel gets used up. So both the
mass and velocity change with time, which means that you’d need to
treat the (mv) part of the equation Fnet t = (mv) differently.
But you don’t need to worry about this too much, since it’s not the
part of the physics that we’ll cover in this book.
g is the
gravitational
field strength.
On earth,
g = 9.8 m/s2
Different physics courses use
slightly different values for g.
AP Physics uses 9.8 m/s2
Weight = mg
Q:
If an object’s mass stays the same, you can say Fnet = ma.
But if its mass changes, you have to say Fnet t = (mv)?
A:
Yes. The equation Fnet t = (mv) works for any object, whether
its mass is constant or not.
The equation Fnet = ma only works for an object whose mass is
constant.
Q:
A:
But how do I know which equation to use?
If you’re interested in the object’s velocity or momentum
rather than its acceleration, Fnet t = (mv) is the most useful form of
Newton’s Second Law.
If you’re interested in the object’s acceleration, then Fnet = ma its the
most useful form of Newton’s Second Law (as long as the mass of
the object is constant).
But we’re interested in weight! When I put an
apple on scales, its velocity doesn’t change and
it doesn’t accelerate, but it still has a weight!
Weight is the force that causes an
object to accelerate when it falls.
If you drop an apple, it accelerates at a rate of
9.8 m/s2. This is because the earth’s gravitational field
strength is 9.8 m/s2. You now know that for something
to accelerate, a net force must act on it.
The only force acting on the falling apple is its weight.
You can think of this as a gravitational force which
results from the stuff that the earth’s made of and the
stuff that the apple’s made of attracting each other.
Even when the apple isn’t falling, it’s still subject to the
same gravitational force, so it still has the same weight
- its mass × the gravitational field strength, or mg (we
use the letter g to represent the gravitational field
strength).
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support force
So where have we got
to now? Weight is a
force, right?
Jim: Right - and my weight is due to the “stuff ” I’m made of and
the “stuff ” the earth’s made of attracting each other. So we can
think of weight as being a gravitational force.
Joe: Yeah, your weight is the reason you accelerate towards
the ground at 9.8 m/s2 when there’s nothing to support you. And
Fnet = ma, so if I have a mass of 80.0 kg, my weight must be
80 × 9.8 = 784 N as that’s the gravitational force on me.
Frank: Yeah, and if you’re not accelerating, that force of your
weight’s still there, and is still 784 N, as weight = mg. I guess that
means that if my mass is constant, then the force of my weight is
constant whatever’s going on - my weight is still mg.
Practical point: Different
physics courses use slightly
different values for g. 2
AP Physics uses 9.8 m/s
- but generally expects you to
quote answers to 3 significant
digits even though this value
for g given in the AP table
of information only has 2
significant digits.
Jim: But the force that the WeightBotchers machine measures goes
down when the scales move downwards!
Jim: Yeah, that’s a puzzle. The scales can’t be measuring weight
directly, or else they would always have the same reading. So if the
scales don’t measure weight, what force do the scales measure?!
Joe: I think the key thing might be that the scales on the
WeightBotchers machine are accelerating towards the ground when
the reading changes.
Frank: But why would that change the reading?
Joe: I guess that the scales aren’t supporting you as much as they
were before they started to move.
If the scales (or
the earth) didn’t
provide a support
force, you’d just
keep on falling!
Jim: Yeah ... when you stand on the scales, the spring inside the
scales compresses until it provides enough force to support you - to
stop you moving down any further. And it’s the compression of the
spring that the scales measure.
Joe: Yeah, the scales measure the support force!
Frank: So if the scales aren’t totally supporting your weight, the
reading would be less?
Joe: I think that’s probably right.
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