Video Transcript
A lamina in the shape of a square
has a side length of 20 centimeters and a weight of 15 newtons, which is acting at
the point of intersection of its diagonals. The lamina is suspended from a
horizontal pin near its vertex 𝐴 such that its plane is vertical. And a couple is acting on the
lamina that makes it in equilibrium in a position where the line segment 𝐴𝐶 is
inclined to the vertical at an angle of 30 degrees. Determine the magnitude of the
moment of the couple.
Let′s begin with a diagram of the
scenario. We have the square lamina of side
length 20 centimeters and a weight of 15 newtons with a center of mass at the
intersection of the diagonals, that is, the geometric center of the square. The lamina is suspended from one of
its vertices, 𝐴, such that the angle between the line segment 𝐴𝐶, the line that
passes through the suspension point 𝐴 and the center of mass, is angled at 30
degrees to the vertical.
Since the lamina′s center of mass
is not directly below the pivot point 𝐴, the weight exerts a moment on the lamina
about the point 𝐴. The lamina is kept in equilibrium
by a couple, which must therefore exert a moment equal and opposite to the moment of
the weight. We cannot know the orientation and
position of the forces that produce this couple. So we cannot place them on the
diagram. But we do not need to.
To calculate the magnitude of the
moment of the couple, we just need to calculate the magnitude of the moment of the
weight of the lamina about the point 𝐴, since the couple is equal and opposite to
it. Recall that the moment 𝑀 of a
force acting from a point 𝑝 about a pivot point 𝐴 is given by the magnitude of the
force 𝐹 multiplied by the perpendicular distance 𝑑 between the pivot point and the
line of action of the force.
In this case, the perpendicular
distance 𝑑 between the line of action of the weight of the lamina and the pivot
point will be given by the opposite side length of this right triangle, enclosed by
the vertical line, the point 𝐴, and the center of the lamina, since the weight of
the lamina acts parallel to the vertical. The hypotenuse of this triangle is
half the length of the diagonal, which is equal to the side length, 20 centimeters,
multiplied by the square root of two. So the hypotenuse has length 20
root two over two, which simplifies to 10 root two centimeters.
The distance 𝑑 is therefore given
by 10 root two times the sin of 30 degrees. sin of 30 degrees is equal to
one-half, so 𝑑 is equal to five root two. The magnitude of the moment of the
weight is therefore equal to the magnitude of the weight, 15 newtons, multiplied by
the perpendicular distance between the line of action of the weight and the pivot
point five root two. This comes to 75 root two
newton-centimeters, which is equal and opposite to the couple acting on the
lamina. Therefore, the magnitude of the
moment of the couple is also equal to 75 root two newton-centimeters.
