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Chapter 11. weight and the normal force

# Chapter 11. weight and the normal force

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438   Chapter 11

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.

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?)

you are here 4   439

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

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.

440   Chapter 11

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

devices work, you

can trouble-shoot

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 ...

you are here 4   441

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.

442   Chapter 11

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?

you are here 4   443

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.

444   Chapter 11

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

you are here 4   445

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.

446   Chapter 11

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).

you are here 4   447

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

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.

448   Chapter 11 