Video Transcript
A cube sits on a slab of wood which
is mounted at an angle of 15 degrees to the horizontal, as shown in the figure. The cube weighs 40 newtons and has
a side length of five centimeters. Calculate the pressure on the
wooden slab due to the weight of the cube.
In this question, we have a cube
sitting on a slab of wood. And we want to calculate the
pressure on the wooden slab due to the weight of the cube. We can recall that the pressure 𝑝
on an area 𝐴 is given by the equation 𝑝 equals 𝐹 over 𝐴, where 𝐹 is the force
acting perpendicularly to the area.
We are told in the question that
the cube weighs 40 newtons. This is the weight, which we will
label as 𝑊, and this is a force that acts vertically downwards. However, to calculate the pressure
on the wooden slab, we need the component of force that acts perpendicular to the
face of the cube that is resting on the wooden slab.
So let’s resolve the weight into
two independent components: one that acts parallel to the wooden slab, which we will
label as 𝐹 sub 𝑥, and the other component that acts perpendicular to the slope,
which we will label as 𝐹 sub 𝑦. We can now use the definition of
cosine to find that cos 15 degrees equals 𝐹 sub 𝑦 over 𝑊. If we multiply both sides of this
expression by 𝑊, then we find that the force 𝐹 sub 𝑦 is equal to 𝑊 multiplied by
cos 15 degrees.
We now need to calculate the area
of the face of the cube that is resting on the wooden slab. We are told in the question that
the cube has a side length of five centimeters, which is equal to 0.05 meters. The face of the cube is a
square. So the area 𝐴 will be equal to
0.05 meters multiplied by 0.05 meters, which is equal to 2.5 times 10 to the power
of negative three meters squared.
Now that we have values for the
force acting perpendicular to the wooden slab and the area of the face resting on
the wooden slab, we can now calculate the pressure on the wooden slab. Substituting the values into our
equation for pressure, we find that the pressure 𝑝 is equal to 40 newtons
multiplied by cos 15 degrees over 2.5 times 10 to the power of negative three meters
squared. Completing this calculation, we
find that pressure is equal to 15455 newtons per meter squared to the nearest whole
number. And this is our final answer.
The pressure on the wooden slab due
to the weight of the cube is equal to 15455 newtons per meter squared.
