WEBVTT

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This is Giancoli Answers
with Mr. Dychko.

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We need to find out
what the mass of this box must be

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given that an applied force is

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at an angle of 28 degrees
above the horizontal,

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and the coefficient
of static friction his 0.4.

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The coefficient of kinetic friction
is an irrelevant piece of information

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because we're told
that the box is not moving

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when there's a force of 23 newtons,

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so, that means it's static friction,

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that's involved in non kinetic friction.

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So, the way I've drawn it here is
to show all the forces involved.

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Upwards we have
the static friction force

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and then we also have some
component of the applied force

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that's also in the y direction upwards.

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And those two,
this component here

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and this friction force,
both being up,

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have to equal the total forces down

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which is only one,
this gravity downwards.

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Since it's not moving, the total up
has to equal the total down

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and because it's not moving

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we also know the total to the left
has to equal total to the right.

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So, there is a component
of the applied force in the x direction

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and that's going
to equal the normal force

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which is to the left.

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So, we can say
it with algebra now,

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we can say that the component

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of the applied force in the y direction
plus the friction upwards,

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that total has
to equal gravity down.

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And then we can create expressions
for each of these terms

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and then rewrite this equation
with some substitutions.

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So, we'll say that the component

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of the applied force
in the y direction is

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the applied force, <i>Fa</i>,
times sine theta

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because this is the opposite leg
of this applied force triangle.

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And then we'll also say
that the friction force is

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coefficient of static friction
times the normal force.

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But the normal force equals
the x component of the applied force

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since there's
no movement horizontally,

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so, that means,
or no acceleration horizontally

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and that means that this
has to equal this,

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and so, we can make
that substitution

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and the x component
of the applied force

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is the applied force
times cosine theta

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because it's the adjacent leg
of this triangle.

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And then finally we'll say that
force of gravity is <i>mg</i>.

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So, we'll substitute
for each of these things.

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So, <i>Fa y</i> is <i>Fa</i> sine theta.

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And force of friction is
<i>muS</i> times <i>Fa</i> cos theta.

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And then force
of gravity is <i>mg</i>.

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So, we have
this equation now here.

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And we'll solve this
for the mass that the box must have

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by dividing both sides by <i>g</i>
and switching the sides around.

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Then we have the mass
must be 23 newtons applied force

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times sine 28
plus 0.4 times cos 28,

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all divided
by 9.8 newtons per kilogram.

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And that gives 1.9 kilograms
must be the box's mass.