Main content

## Physics library

# More on Normal force (shoe on floor)

David explains how to determine the normal force for a variety of scenarios (extra forces, diagonal forces, acceleration) involving a shoe on the floor. Created by David SantoPietro.

## Video transcript

- [Voiceover] Check out
this fine looking sneaker right here, we're gonna
use this shoe to illustrate some more challenging
normal force problems and we're gonna take this
as an opportunity to discuss a lot of the misconceptions
that people have about the normal force. So one misconception is that people forget normal force is a contact force. You only have a normal
force when two surfaces are in contact, so when
the shoe's in contact with the floor, there will be
a normal force on the shoe and a normal force on the floor. Or if the shoe were in
contact with the wall, there would be a normal force on the wall and a normal force on the shoe. But if the shoe were just
falling through the air, here's what happens for a lot of people. Let's say the shoe's just falling, and you got a question
and the question said, draw the forces that
are exerted on the shoe while it's falling through the air. People get so used to having normal forces that they make a mistake, they
do this, they say alright, let me draw it over here. They say there's a gravitational
force and that's just fine. There will be a gravitational
force, there's always gravity, the Earth's always pulling down, and it pulls down with an amount mg. But they're so used to
having normal forces, I mean normal forces pop up
in so many different questions it's almost just like a reflex, people just automatically put,
whoop, there's a normal force there's gotta be a normal force, right? There's always a normal force? But there isn't always a
normal force, if the shoe is not in contact with the
surface, you don't have a normal force, it's not until this
shoe makes it to the ground or touches another surface
that you'll have that normal force, so if we
stick this shoe right here and we let it rest on the ground, now you'll have a normal
force and that normal force will point up, and this
is what people wanna say and it's true when the
surfaces are in contact but if they're not in contact, you don't have a normal force. And then here's another misconception, people think the normal
force is always equal to mg, because again, it's equal to mg in so many different
scenarios that people just wanna say well it's always
equal to mg, and again, it's just like a reaction. People see normal force, they
just automatically replace it with mg, and that'll be
true in the simple case, but I'll show you coming up
how that's not gonna be true and what you do if it's not true. So for instance if we wanted
to find what's the normal force if this shoe has a mass m, so let's assume that
the shoe has a mass m, what would the normal force be? We can use Newton's second law, we can always use Newton's second law, so we'll say that acceleration
equals the net force divided by the mass, and in this case, since these are vertical forces, I'm gonna consider the acceleration in the vertical direction
and the net force in the vertical direction. And so what is the acceleration
for the shoe vertically if it's just sitting here in
a room, sitting on the ground, at rest and not changing it's motion, not changing it's velocity,
the acceleration's just gonna be zero, so the vertical acceleration, acceleration, excuse me, should be zero. For the net force, I've
got an upward normal force, so I'm gonna make that positive, if fn represents the
magnitude of the normal force, this would be positive Fn, I'm
just gonna put positive here even though I don't really
need it but to show you that it's upward, we're gonna
consider upward to be positive and then I've got this
downward gravitational force, and if mg represents the size
of the gravitational force, I'm gonna put a negative
here to represent that that gravitational force is
down, and then I divide by the mass of the shoe and
if I do this I get that these two forces, this net
force divided by the mass has to be zero, according
to Newton's second law. But I can multiply both sides
by the mass and if I do that, the left hand side is still zero, and I'll get that this is
equal to the normal force minus mg, so I'll have
normal force minus mg, and if I finally solve
for the normal force, I'll get that the normal
force is gonna equal mg, and a lot of people are like
yeah I already knew that, duh. Normal force is always equal
to mg, but it's only equal to mg in this case because
those were the only two forces, look at the assumptions we made. Only two forces were
the normal force and the gravitational force and we
assumed that the acceleration was zero, if you relax
any of those requirements, normal force is no longer
going to be equal to mg. And it was on a horizontal surface, if you relax that requirement,
again, there's no reason to think this has to
be in the Y direction, you could have normal
forces in the X direction. So let's slowly, one point at a time, try to relax some of these
requirements and see what that does to the normal force. In other words, what if we
just added another force? What if we let the shoe
sit here on the ground and I push down on it? So I'm pushing down on this
shoe, I'm gonna say I'm pushing down with a force, I'll
just call it F1, so a force of magnitude, F1, and
it's pointing downward, how would that change this now? So this is, we're stepping it
up, this is gonna be a little harder, what do we do? Well the acceleration is still
zero, let's say it's still just sitting there, so we
don't have to do anything with the left hand side that's still zero, multiplying by m still makes that zero, but now up here in this force up here, I'm gonna have another
force, I'm gonna have F1, that points down so in my force diagram I'd have another force
that points down, F1, that means I'd have to
subtract it when I find the net vertical force, I'd have F1, this would be a negative F1 right here, and when I solve for Fn
I'd add mg to both sides to cancel it and then
I'd add F1 to both sides to cancel this F1, this negative F1, and I'd get mg plus F1. So I get the normal
force is gonna be bigger, bigger by an amount F2
and that makes sense, if you push down on a, oh no F2, wow, F1, sorry about that, it's
gonna be bigger by an amount F1, so if I push down with an
extra 10 Newtons of force, there's more pressure, right? That makes sense, the
pressure between the ground and the shoe is gonna be
greater, you're squashing these two surfaces together
with greater force, so the ground's gotta
push up to keep the shoe out of the surface, that's
what this normal force does, it exerts a force to keep the
object out of the surface, to keep the object from
penetrating that surface, so if I push down on an
object, into a surface, that normal force increases
and it increases by the amount you're pushing down, so that makes sense. If you had an upward
force, let's say you had an upward force, someone's
pulling up on the shoe while you push down, you're
fighting over the shoe, you're wrestling over it with
somebody because they just, they love the shoe, they
recognize the beauty of the shoe, well crafted shoe, so if
there's an F2 pointing up, we now have another force in our diagram, that force would point
up, we would call this F2, over here how would this change? Again, still acceleration to zero, but I'd have an upward
force now so I'd have to add F2 vertically because
that's another force, I'd have a plus F2 right here, and then over here when I solve for this, I add mg to both sides,
I add F1 to both sides, and I have to subtract F2 from both sides so now I'd have Fn is mg plus F1 minus F2, this also makes sense,
if you pull up on a shoe, you're relieving some of the
pressure between the shoe and the other surface,
the shoe and the floor. So if I pull up with 20
Newtons I'm gonna reduce the normal force by 20
Newtons because I'm relieving some of that pressure between
the shoe and the floor. Let's make it even harder. Let's make this thing scary,
sometimes you get really crazy problems and you don't
know what to do, let's say, we have another force, let's
say this force is gonna be a diagonal force, so
we're gonna pull this way. That was not a well drawn
force, let me draw it like this. So we got a force this way at an angle. Now we're talking, this is F3. F3, at an angle of, we'll
call it Fi, so the angle from this horizontal line here is Fi. Now what do we do? So I've got this crooked
angle in here, now this F3 is gonna be pointing this way,
so I'll add another force to my force diagram, and I
can figure out how to include this into my vertical force
version of Newton's second law, I can't include the entire
F3 force, here's a mistake people make, they wanna
just add F3 or subtract F3, but I can't do that, this
is the vertical form of Newton's second law, this is
only applying to the vertical direction, the Y direction,
but F3 is pointing both vertically and horizontally. So I have to only include
the vertical part of F3 in this formula so what I
have to do is say that alright F3 is gonna have a vertical component, that vertical component,
I'll call it F3 Y, for F3 in the vertical direction. And it's also gonna have
a horizontal component. I'll just call that F3 X for
F3 in the horizontal direction. So if I wanna solve for F3 Y,
I'll just use the definition of sine and I know to use
sine because this side is the opposite to this angle, I know sine relates opposite side, so
I'm gonna write this as sine of phi is gonna equal
the opposite side is F3 Y so F3 in the Y direction
divided by F3 total, the total magnitude of F3,
and if I solve this for F3 Y, I get F3 in the Y direction
is gonna equal F3 times sine of Fi, now I can include this
in my force, my net force, because this points upward,
so since it points up, this vertical component's
gonna add a plus, F3 sin theta over here,
or sorry not theta, it's gonna be F3 sin Fi, and
I'll have a plus F3 sin fi right here, and when we
subtract this F3 sin phi from the other side to get it over to here, we're gonna get minus F3
sine phi and that makes sense because if we pull up, we
know we're reducing some of the pressure, we'll
reduce the normal force, so this component, look at
this component points up, it's reducing some of the pressure on the bottom of the shoe,
the normal force goes down, and so we subtract the F3 sin fi. And one more way to step this
problem up to the next level would be to say that this
room isn't really just a room, maybe it's an elevator, and this elevator is accelerating upward with some acceleration
A zero, in that case, nothing would change on
this right hand side. Sometimes people think
if there's acceleration, there's gonna be some new force,
but if these are the forces those are the forces, the only
thing that changes over here if this was in an elevator
that's accelerating up, is that instead of zero,
you'd replace this with a zero or whatever the
acceleration is, that's it, that's the only change, you
could still solve for fn the same way, when you
multiply by m it wouldn't be zero on the left anymore,
you'd have a m a zero, and then a plus m a zero when you solve for the normal force. Alright, I think we've
pretty much exhausted this example of a shoe on the floor, it's probably harder than
any example you'll see but now you know how to
handle any type of force you might meet or acceleration.