WEBVTT
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This is College Physics Answers
with Shaun Dychko.
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This sandwich board is held
together by this chain
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and it has a total mass of 8.00 kilograms
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and our job is to figure out what is
the tension force on this chain.
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So we are going to figure out
the normal force on the feet
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of this sandwich board and that
can be found by saying that
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2 times the normal force on each foot
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equals the total weight downwards
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and when we look at it that way,
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we don't have to worry about this hinge.
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Now the force exerted on
a side by the hinge,
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there might be some angle to
this hinge force—
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we don't really know—
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but we can ignore any vertical component
to the hinge force because that's internal
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to both sides and it's not something
we need to be concerned with
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when we talk about the normal force
and gravity.
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Okay!
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So the normal force is half the weight,
in other words.
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Now we can also talk about
the horizontal forces—
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this is the first condition of equilibrium
that we are talking about—
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and the horizontal forces have to
balance out to total zero
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and so the x-component of this hinge force
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which is the hinge force times
*cosine* of *Θ*
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has to equal the tension force to the left.
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And thirdly, we need a third equation since
we have a hinge force that we don't know;
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a tension force that we don't know;
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and an angle that we don't know
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so we need a third equation to
deal with three unknowns.
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So we have the second condition of
equilibrium which is that
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the total clockwise torques have to equal
the total counter-clockwise torques.
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So the counter-clockwise torques are due
to the tension force in the chain
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and it has a distance *r* subscript *T*,
this is the lever arm
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for the tension force of 1.30 meters
total height minus this
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0.50 meters from the hinge is
where the chain's attached
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so that's 0.80 meters is the lever arm for
the chain taking the pivot to be
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the foot of one side.
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And then we also have the weight exerting
a counter-clockwise torque
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and it has a lever arm *r W* for weight
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of one-quarter this total width because
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we are told that this sandwich board is
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has a mass distribution that is uniform
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and so this hinge must be at
the half-way point
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between the two feet
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and then this center of gravity has to be
half-way point
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between the middle and the foot
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which makes it one-quarter of the way
between the feet.
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So we have 1.10 meters—total distance
between the feet—divided by 4
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that's 0.275 meters is this *r W*.
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So we multiply that by the weight to get
the additional counter-clockwise torque
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and then substituting *mg* over 2 for
this *F W* because this is
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half the total weight
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and that has to equal the torque that's
clockwise due to the hinge
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and so we have the lever arm of
the hinge is *r H*,
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which is the height of the sandwich board,
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and multiplied by the hinge force times
*cos Θ* because we want to
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get the component of the hinge force
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that's perpendicular to this lever arm.
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And then we substitute in *F T* in place of
the *F H cos Θ* because our
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'total force horizontally equaling zero'
condition for equilibrium led us to that
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and so we can replace *F H cos Θ* with *F T*.
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And then we are gonna subtract
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*F Tr T* from both sides
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and then factor out the *F T* and we have
*F T* times *r H* minus *r T*—
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of course, I have also switched
the sides around too—
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and so all this equals *mgr W* over 2
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and then divide both sides by
*r H* minus *r T*.
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And so we have then that the tension
force in the chain
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is *mg* lever arm of the weight
divided by 2 times
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lever arm of the hinge minus
the lever arm of the chain.
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So that's 8.00 kilograms times
9.80 newtons per kilogram
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times 0.275 meters divided by 2 times
1.30 meters minus 0.80 meters
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which is 21.6 newtons.
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And then in part (b), we want to find
the hinge force
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and we can say that *F H sin Θ* plus
the normal force equals
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the weight but we are...
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you know, this *Θ* is going to be zero
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because the weight is half the total weight
of the sandwich board, *mg* over 2,
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the normal force is *mg* over 2 as well and
that's what we figured out here
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and so that leaves nothing left over
for this *F H sin Θ*
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it has to equal zero and of course,
*F H* is not zero which means
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*sin Θ* is zero and that's true only when
you have the angle being 0 degrees.
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Okay!
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So then we can say that the hinge force
times *cos Θ*
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which is the horizontal component of
the hinge force to the right
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has to equal the tension force to the left
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and then divide both sides by *cos Θ* but
we have already established that
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this *Θ* is 0 and the *cos* of which is 1
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and so the hinge force then is going to
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have the same magnitude as
the tension force in the chain
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which is 21.6 newtons and
that's horizontally
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and that will be to the right on
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the right hand side of the sandwich board
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and it will be to the left on the left hand
side of the sandwich board.