A man of mass 94 kilograms ascended
a plane of length 90 meters, which was inclined at an angle of 30 degrees to the
horizontal. Determine the work done by his
weight to the nearest joule. Take 𝑔 to equal 9.8 meters per
Okay, so let’s say that this is our
plane at 30 degrees to the horizontal. And there’s a man of mass 94
kilograms climbing up the plane. We’ll call that mass 𝑚. And we’re told that the length of
the plane is 90 meters, which we’ll call 𝑙. We want to calculate the work done
by this man’s weight to the nearest joule.
We can say then that this man’s
weight, his mass times the acceleration due to gravity, a downward-acting force,
actually does work as this man climbs up the incline. That work is equal to the force
involved, that is, the man’s weight force, multiplied by the vertical distance
through which the man moves. That is, the distance we’ll use in
our calculation won’t be the length 𝑙. But rather it will be this vertical
distance 𝑑. That’s because this distance
represents the only component of the length 𝑙, we could call it, that is parallel
or antiparallel to the force involved.
Here’s where we can start then. We know that the force for which we
want to calculate work done is the man’s weight force, 𝑚 times 𝑔. Going a step further, since this
30-degree slope is part of a right triangle, we can say that the distance 𝑑 is
equal to 𝑙 times the sin of 30 degrees. This is so because the sine of this
angle is equal to the ratio of 𝑑 to 𝑙. Rearranging that relationship, we
get that 𝑑 equals 𝑙 sin 30.
Before we plug in the given values
for 𝑚, 𝑔, and 𝑙, it’s important to set up a sign convention. Note that as this man climbs the
incline, he moves upward vertically. This is in the opposite direction
to his weight force, which acts down. To make sure that we combine
quantities correctly, let’s say that one of these directions is positive, making the
other negative. If we say that the upward direction
is positive, then that means the value of 𝑙 times the sin of 30 degrees is
positive. But then that means that 𝑚 times
𝑔, which is directed downward, must be negative. The point here is that the force
involved, the man’s weight force, is acting in an opposite direction to the
displacement 𝑑. The net effect of this is that the
work done by this weight force will be negative.
Now that we’ve figured out the
signs, we’re ready to substitute in for 𝑚, 𝑔, and 𝑙. The man’s mass is 94 kilograms, 𝑔
is 9.8 meters per second squared, and 𝑙 is 90 meters. Entering this expression on our
calculator, we get a result, to the nearest joule, of negative 41,454 joules. This, to the nearest joule, is the
work done by this man’s weight as he ascends this plane.