### Video Transcript

You are in a room in a basement
with a smooth concrete floor and a nice rug. The rug is three meters wide and
four meters long. You have to push a very heavy box
from one corner of the rug to its opposite corner. The magnitude of friction between
the box and the rug is 55 newtons, but the magnitude of friction between the box and
the concrete floor is only 40 newtons. Will you do more work against
friction going around the floor or across the rug? How much extra work would it
take?

This is an exercise involving
non-conservative forces. And to start out, letβs draw a
diagram. In this situation, we have a very
heavy crate on the corner of a three-by-four-meter rug. We want to move the crate to the
opposite corner. And weβre told that if we move the
crate straight across the rug, the force of friction is 55 newtons. While if we choose instead to move
the crate on the smooth concrete floor around the rug, the overall force of friction
is 40 newtons. We want to figure out which of
these two paths will take more work.

Recalling that work is equal to
force times distance, we can write that the work required to move the crate across
the rug equals πΉ sub π, given as 55 newtons, times the distance across the rug the
crate moves. Since the rug is a rectangle, that
distance π sub π is equal to the square root of three meters squared plus four
meters squared, or five meters. Plugging in these values and
calculating π sub π, itβs equal to 275 joules.

Next, we wanna calculate the work
done if we move the crate across the floor instead of the rug. This is equal to πΉ sub π, which
is 40 newtons, times π sub π, which is the distance the crate would move. This distance, skirting the edge of
the rug, is equal to four plus three, or seven meters. Plugging in for these two values,
when we calculate π sub π, we find itβs 280 joules. So, comparing the work done on the
rug to the work done on the floor, we see that more work is required to move the
crate across the floor by five joules.