Two masses hanging from a pulley Video Lecture | Physics Class 11 - NEET

let's solve some more of these systems
problems if you remember there's a hard
way to do this and an easy way to do
this the hard way is to solve Newton's
second law for each box individually and
then combine them and you get two
equations with two unknowns you try your
best to solve the algebra without losing
any signs but let's be honest it usually
goes wrong so the easy way to do this
the way to get the magnitude of the
acceleration of the objects in your
system that is to say if I want to know
the magnitude at which this five
kilogram box accelerates or that this
three kilogram box accelerates all I
need to do is take the net external
force that tries to make my system go
and then I divide by my total mass of my
system this is a quick way to get what
the magnitude of the acceleration is of
the objects in my system but it's good
to note it'll only work if the objects
in your system are required to move with
the same magnitude of acceleration and
in this case they are what I have here
is a five kilogram mass tied to a rope
and that rope passes over a pulley pulls
over and connects to this three kilogram
mass so that if this five kilogram mass
has some acceleration downward this
three kilogram mass has to be
accelerating upward at the same rate
otherwise this rope would break or snap
or stretch and we're assuming that that
doesn't happen so this rope is the
condition that requires the fact that
this rope doesn't break is what allows
us to say that the system is just a
single big total mass with external
forces exerted on it so how would we
solve this I just say that well what are
the external forces keep in mind
external forces or forces that are
exerted on the objects in our system
from objects outside of our system so
one external force would just be the
force of gravity on this five kilogram
mass so I'm going to have a force of
gravity this way and that force of
gravity is just going to be equal to
five kilograms times 9.8 meters per
second squared because that's how we
find the force of gravity should I make
it positive or negative
well this five kilogram is going to be
the one that's pulling downwards so if
the question is I hold these masses and
I let go what's the acceleration this
five kilogram mass is going to
accelerate downward
it's going to drive the system forward
that's the force making this system go
so I'm going to make that a positive
force and then I figure out are there
any other forces making this system go
no they're not you might say well what
about this tension over here isn't the
tension on this three kilogram mass
isn't that tension making this system go
not really because that's an internal
force exerted between the objects in our
system and internal forces are always
opposed by another internal force this
tension will be pulling the three
kilogram trying to make a move but it
opposes the motion with the five
kilogram mass and if we think of this
three plus five kilogram mass is a
single object these end up just
cancelling on our single object that
we're viewing is one big eight kilogram
mass so those are internal forces we
don't include them they're not a part of
this trick we have to figure out what
other forces will try to make this
system go or prevent it from moving
another force that tries to prevent it
from moving is the force of gravity on
the three kilogram mass or one force
that tries to prevent the system from
moving would be this force of gravity
how big is that that's three kilograms
times 9.8 meters per second squared and
that's trying to prevent the system from
moving this five kilogram mass is
accelerating downward and this force is
in the opposite direction of motion that
trips people out sometimes they're like
I don't understand they're both pointing
down shouldn't they have the same sign
they would when we're using Newton's
second law the way we usually use it but
when we're using this trick what we're
concerned with our forces in the
direction of motion this is an easy way
to figure it out forces in the direction
of motion we're going to call positive
and any forces opposite the direction of
motion we're going to call negative so
forces that propel the system forward
we'll just call that the positive
direction forces that resist the motion
we're just going to call that the
negative direction and since this is on
this side of the motion of the system
this system is everything in this system
is going this way the three kilogram
mass goes up the string over here goes
up the string up here goes to the right
the string right here goes down the five
kilogram mass goes down because all the
motion of the system is this way we
define that way as positive
this force of gravity on the three
kilogram mass is the opposite direction
it's opposing the motion of the system
it's preventing the system from
accelerating as fast as it would have
that's why we subtract it and now we
just divide by the total mass and the
total mass ajiz five plus three is going
to be eight kilograms and I get the
acceleration of my system so if I just
add this up I get two point four five
meters per second squared
so this is a really fast way to get what
the acceleration of our system is but
you have to be careful if the question
is what's the acceleration of the five
kilogram box well technically that
acceleration of the five kilogram box
would be negative two point four five
what we really found here since we were
just finding the magnitude was the size
of the acceleration since it's five
kilogram box is accelerating down and we
usually treat down as negative you won't
want to forget that negative in putting
in that answer the acceleration of the
three kilogram box however would be
positive two point four five meters per
second squared so when you're applying
this to an individual box you have to be
very careful and make sure you apply
that acceleration with the correct sign
for that particular box and if you
wanted to find the tension now now it's
easy to find the tension I can find this
tension right here if I wanted to if the
next step was find the tension in the
string connected to the boxes now I can
just use Newton's second law but the way
we always use it I'm done with the trick
the trick is just a way to get the
magnitude of the acceleration now that I
have that I'm done treating it as a
system or a single object I'll look at
this single five kilogram mass all alone
and I'll say that the acceleration of
the five kilogram mass which is Newton's
second law is going to equal the net
force on the five kilogram mass divided
by the mass of the five kilogram mass I
know the acceleration of the five
kilogram mass but if I'm going to treat
up as positive now I got to plug this
acceleration in with a negative sign so
negative two point four five meters per
second squared is going to equal the net
force on the five kilogram mass I've got
tension up you might be like wait we
said that was an internal force it was
an internal force and we didn't include
it up here but we're doing the old rules
now
normal second law in the vertical
direction so I use vertical forces and
if their upward I'm going to treat them
as positive and if their downward like
this five times 9.8 I'm going to treat
it as a negative because it points down
five times 9.8 m/s^2
and I divide by the five kilogram mask
because that's the box I'm analyzing I'm
not analyzing the whole system
I'm just analyzing the five kilogram box
now and I can solve and I can get my
tension the alternate way to do this
would be to say all right let's just
treat down as positive for this five
kilogram mass I then plug my
acceleration in is positive and I plug
my force of gravity and positive then my
tension would be negative I'd get the
same value out here I'm just solving for
the magnitude of the tension anyway so
if I solve this if I plug this into the
calculator and solve for tension I'm
going to get thirty six point seven five
Newton's which is less than the force of
gravity which it has to be because if
the tension was greater than the force
of gravity this five kilogram mass would
accelerate up we know that doesn't
happen the tensions got to be less than
the force of gravity so that this five
kilogram mass can accelerate downward so
that's a quick way to solve for the
magnitude of the acceleration of the
system by treating it as a single object
we're saying that if it's a single
object or thought of as a single object
which we can do because these are
required to have the same acceleration
or same magnitude of the acceleration
then if we're treating it like a single
object only external forces matter and
those external forces that make the
system go are going to accelerate the
system and those external forces that
resist the motion are trying to trying
to reduce the acceleration and we divide
by the total mass of the system that
we're treating as one object we get the
acceleration if that still seems like
mathematical witchcraft if you're not
sure about this whole idea I encourage
you to go back and watch the video we
solved one of these types of problems
the hard way and you see you really do
end up with the force that tries to make
the system go externally and the
external force that tries to stop it
divided by the total mass gives you the
acceleration essentially what we're
saying is that these internal forces
cancel if you're thinking of this system
as one single object because these are
applied internally and they're opposed
to each other one tries to make the
system go one tries to make the system
stop