WEBVTT
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This is College Physics
Answers with Shaun Dychko.
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We're going to re-do this question that
we've done before in an example
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but with the angle of between the arm
and the bicep going to be 120 degrees.
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So the hand is down here and the
book is on the hand down here.
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Okay.
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So, the torques that are
going counter-clockwise
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of which there is only one due to the bicep
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has to equal the total
torque going clockwise.
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We're assuming the pivot
is at the elbow here
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and so the force exerted on the elbow
has no term in our torque formula
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because its lever arm is zero.
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We're multiplying each of these forces
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by the perpendicular component
of this distance to the elbow.
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So, this angle *theta* which
is between the horizontal
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and the arm is going to be 120 minus
90, and that gives 30 degrees.
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So the horizontal component
of each of these lengths here
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is going to be the hypotenuse which is
the length, multiplied by cosine of thirty.
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If we consider this triangle
for the bicep first of all,
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it's this triangle here where this is *r b*
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and this is *r b perpendicular* that we want
to know, perpendicular to the force.
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This angle here is 30,
that's a right triangle,
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and so we multiply the hypotenuse
by cosine of 30 to get the adjacent.
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So we substitute that into
each of these terms here
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and we have *F b* times *r b* cos *theta*
equals force of weight on the arm,
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multiplied by the distance from the center
of mass of the arm to the elbow,
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times cos *theta*, plus force
of weight on the book,
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times the distance from the book to
the elbow, multiplied by cos *theta*.
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But this is cos* theta* is
a factor on every term
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and so we can divide both sides
by cos* theta* and it disappears.
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Then also divide both sides by *r b* and
you solve for the force due to the bicep.
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So then we plug in numbers.
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We have two and a half kilograms
times 9.81 newtons per kilogram,
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this is the force of the weight of the arm,
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multiplied by its 16 centimeter
distance to the elbow.
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I can use centimeters here because
since we have centimeters on top
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and centimeters on the bottom,
those units are going to cancel.
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It doesn't matter what the units are so long
as they are the same on top and bottom.
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You could convert them
to meters if you prefer.
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Then add to this the four
kilogram mass of the book
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times 9.81 times 38 centimeters
from the hand to the elbow,
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divided by four centimeters
lever arm of the bicep,
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or you know, not quite
the lever arm technically
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but it's the distance from where the
bicep attaches to the arm bone,
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the radius or the ulna.
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Anyway this works out to 471 newtons
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which is the same as we had
in the example by the way.
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The angle didn't make any
difference because it canceled.