Newton's second law | Physics | Khan Academy

- [Instructor] Today in the gym,

when my wife was doing dumbbell curls,

I started wondering, see,
she's putting a force

on that dumbbell upwards, right?

But does that force stay a constant

as she moves the dumbbell up or not?

Does it change? And if it does
change, how does it change?

Does it increase, does it
decrease? What happens to it?

Guess what.

We can answer this question by the end

of this video using Newton's second law.

So let's start with a simpler example.

We have a ice hockey ground over here,

and there's a puck moving
on top of it at some speed.

If there are no frictional
forces acting on this,

if you assume that, then the
forces acting on this puck

would be balanced because in
the horizontal, you can see

that there are no forces
because we're ignoring friction.

And in the vertical,
the gravitational force,

which is pulling down on
it is completely balanced

by the force that the
ground is pushing up on it,

the normal force, they balance it out.

And so since there are no
unbalanced forces acting

on this puck from Newton's first law,

we know that this thing will
continue its state of rest,

or in this particular case,
the state of uniform motion.

So it'll continue to move
with that same velocity.

But now comes the question,

what if there was an
unbalanced force acting on it?

What happens because of that?

Well, let's find out.

For that, let's just whack
it with a hockey stick.

No.
(instructor laughing)

So if I whack it to the
right, let's say in this case,

I will now put an unbalanced
force to the right.

What will happen? Well,
we can probably guess it.

That puck's velocity will now be higher.

It'll just get blasted off over there.

So its velocity will increase.

In other words, it will accelerate.

Ooh, this means when there's
an unbalanced force acting

on a an object, in other words,

if there is a non-zero net
force acting on an object,

which is the same thing as
saying an unbalanced force,

but whenever this net
force acts on an object,

what does it do?

It accelerates our puck.

The puck undergoes, or the
object undergoes an acceleration.

This is the essence of
Newton's second law.

Now all we gotta do is
analyze the situation

even more carefully

and see if we can concretize
this relationship.

So let's do that.

The first question we could have is yeah,

so a net force causes an acceleration,

but how long does that acceleration last?

Well, let's see.

When the stick hits the puck,

that's when it starts accelerating,

which means as long as
the stick is in contact

with the puck, as long as
it's in contact with it,

like right now here, it's during that time

there will be acceleration.

But what happens once it loses contact?

Once it loses contact, again,
net force goes to zero.

And now coming back to Newton's first law,

it'll continue moving with
that same increased velocity.

This means the acceleration only happened

during this time when the hockey stick

was in contact with it.

In other words, the
acceleration lasts as long

as the net force lasts.

Okay, next, let's think
about what would happen

if the net force was higher?

For that let's imagine
we whacked it harder.

What's gonna happen now?

Or you can imagine it'll get
blasted off even more faster,

even faster, right?

Which means it'll have a higher velocity

when it loses contact.

Ooh, that means there'll
be bigger acceleration.

So if the net force is larger,

it means you'll have
a larger acceleration.

If the net force is smaller,
you get a smaller acceleration.

In other words, we see
a direct relationship

between acceleration and the net force.

All right, what else can we deduce?

Hey, let's think about the direction.

What is the direction of the acceleration?

Well, in this case, the
net force is to the right,

and our puck's velocity

is also increasing towards the right.

So in this case, the
acceleration is to the right.

So in this case, if the
net force is to the right,

the acceleration is to the right.

What would happen if the
net force was to the left?

So let's imagine we whack that puck now

to the left, what would happen?

Well, we can again imagine the puck

would now get blasted off to the left.

But let's look at it carefully.

Since the puck is already moving
to the right, if we push it

to the left, now we're gonna slow it down.

The puck will come to a stop first.

It'll happen very quickly
that we won't even see it.

But it has to happen before
going to the left, right?

Which means when you go from here to here,

notice even though the puck
is moving to the right,

it is slowing down, which
means the acceleration

is to the left.

So when the net force is to the left,

we're seeing an
acceleration is to the left.

After that, its velocity
might increase to the left,

which means again, the
acceleration is to the left.

Ooh.

So if the net force is to the left,

the acceleration is to the left.

If the net force is to the right,

the acceleration is to the right.

So the acceleration will
be in the same direction

as that of the net force.

Okay, is there anything else
that affects our acceleration?

Well, let's see.

If you come back over here,

what if you use the same bat,

whacked it with the same force,

but instead of a puck,

let's say there was a bowling ball

moving with the same velocity.

What would happen now?

(instructor laughing)

I'm pretty sure you can
feel it in your bones now.

It would be much harder
to stop that bowling ball

and make it turn backwards, right?

I mean, the same thing will happen.

You will slow it down, but
it'll be much, much harder.

It'll take a much longer
time to slow it down,

even though you're putting
the same amount of force.

So the net force has stayed the same.

But what has happened to our acceleration?

Since the velocity changed
over a much longer time,

the acceleration became smaller.

Hey, why did the
acceleration became smaller?

What changed?

From the puck to the bowling
ball, the mass changed,

the mass increased.

So this means mass also
affects the acceleration,

but how does it affect it?

Well, we saw that the
mass increased right now.

What did that do to the acceleration?

It decreased, and this
is kind of intuitive.

The bigger the mass, the
harder it is to accelerate,

meaning the smaller the acceleration,

which means acceleration
has an inverse relationship

with the mass.

So now, everything that we just
analyzed about acceleration,

its direction, its
dependency on the net force,

how it depends on the mass,

all of it can be put down in an equation.

And that equation is pretty
much right in front of us.

So the acceleration will
equal the net force divided

by the mass.

This is our Newton's second law.

And look, the equation
is saying the same thing.

Direct relationship between acceleration

and the net force, inverse
relationship between acceleration

and the mass and the arrowheads are saying

that acceleration and the
net force will always be

in the same direction.

Isn't it amazing that we can pack all

of that information in just
one beautiful equation?

And of course, you may
have seen this written

as f equals ma in some
sources, it's the same thing.

I like to write it this way

because acceleration
is caused by the force.

So once we decide the
force is and the mass,

then the acceleration gets fixed.

But anyways, what will happen
if the net force is zero?

What if we plug in over here zero?

Well, then the acceleration
also goes to zero.

What does this mean?

Well, this means we have all
the balance forces acting

on an object.

And if the acceleration is zero, it means

that the velocity stays a constant.

In other words, this
is Newton's first law,

which says an object
continues to stay at rest

or in uniform motion.

That is zero acceleration, right?

When there are no unbalanced
forces acting on it.

So notice, Newton's first
law is just a special case

of Newton's second law,

which means this equation is
encompassing both the second

and the first law as well.

And finally, speaking
about Newton's first law,

what we also noticed over
here is bigger the mass,

smaller the acceleration.

In other words, if the mass
is bigger, it is harder

to change its velocity.

It's much harder to do
that, which means objects

that have more mass have more inertia.

That's something again we
learned in Newton's first law.

Inertia is the property due

to which objects continue to stay at rest

or continue to stay in
uniform motion, isn't it?

It can fight acceleration,

and we can now see what
inertia depends on.

Inertia is the mass.

More the mass of an
object, more than inertia,

harder it is to accelerate.

Newton's second law could arguably be

the most important equation
of all of classical physics.

I say classical physics
because we now know

that if objects are moving
very close to speed of light,

then this breaks down, it doesn't work.

Now we'll have to resort

to Einstein's theory of relativity.

On the other extreme,

if we consider extremely tiny particles,

like subatomic particles like electrons,

protons, and neutrons,
well, even over there,

turns out Newton's laws don't work.

So even over there, it breaks down.

But as long as you don't
go to such extremes,

this equation will work for us.

So now let's see if we can apply this

to our original question.

When she just started moving the dumbbell,

dumbbell's velocity was increasing.

After that, let's say
there was a small phase

during which the velocity was constant,

and finally, when the
dumbbell is about to stop,

its velocity is decreasing.

So now the question is how
do we figure out what happens

to the force that she's
putting on the dumbbell?

Well, let's apply Newton's second law.

For that, let's first think
about the acceleration.

Well, over here, we are dealing
with increasing velocity.

Therefore, the acceleration is upwards

in the same direction as it's moving.

Then we have a constant velocity,

which means the acceleration is zero.

Finally, we have a decreasing velocity,

since the dumbbell is still
going up, decreasing velocity,

which means acceleration must be down

in the opposite direction.

Now, because we know the
direction of the acceleration,

we can figure out the
direction of the net force.

It has to be exactly
the same, upwards here,

zero here, downwards here.

We are applying Newton's second law,

the direction part over here.

Now, finally, this is the
direction of the net force.

We want to know what happens to the force

that we are putting on the dumbbell

or she's putting on the dumbbell actually.

How do we do that?

Well, let's look at all the
forces acting on the dumbbell.

Well, we know that there's
gravitational force acting

on the dumbbell all the time.

That force is a constant,

and therefore, our force is
in the opposite direction

of the gravitational force.

Now, in this case, when she's
just lifting the dumbbell,

if the net force needs to be upwards,

that means her force must be larger

than the gravitational force.

Only then her force will win out,

giving a net upward force, right?

Okay, what about over here?

We want the net force
to be zero over here.

How can that happen?

Ah, her force has to be exactly the same

as gravitational force

because only then the forces get balanced.

Finally, what happens over here?

Well, we want the net
force to be downwards,

which means we want gravity to win.

That means her force must be smaller

than the gravitational force.

Look, even though we did
simplify it a little bit,

I mean, I'm not really sure

that her dumbbell was moving
at a constant velocity,

but once we simplified a little
bit, we were able to analyze

what happened to the force

that she was putting on the dumbbell.

It went on decreasing as
the dumbbell moved upwards.

Isn't that incredible

how we used Newton's second law do that?

Amazing, isn't it?