A bucket full of sand has a mass
of 25 kilograms, including
the bucket and the sand.
The bucket is to be lifted
to the top of a building
15 meters tall by a rope
of negligible weight.
15 meters tall by a rope
of negligible weight.
However the bucket has a hole
in it and leaks 0.2 kilograms
of sand each meter it is lifted.
We're going to find the
work done lifting the bucket
to the top of the building.
So looking at our notes
below if an object moves
in a straight line by a
continuous force, big F of x,
then the work done by
the force from x equals a
to x equals b is given by this definigral.
So the first is defined
big F of x force function.
Notice in this problem
we're given information
about the mass not the force.
So to find big F of x we'll
have to use the formula,
force equals mass times acceleration,
where because the mass is in kilograms,
the acceleration due to
gravity is 9.8 meters
per second squared.
So using force equals
mass times acceleration.
Let's work on returning big F of x.
Big F of x is equal to...
Now we'll have to write
an expression for the mass
because the mass starts at 25 kilograms
then leaks 0.2 kilograms each meter.
So the expression for the mass would be
25 minus 0.2x,
25 minus 0.2x,
25 minus 0.2x,
rex with the number of meters,
the bucket and the sand is lifted
and then times the
acceleration due to gravity,
which is 9.8 meters per second squared.
Let's go ahead and distribute big F of x
is equal to 9.8 times 25 is equal to 245
is equal to 9.8 times 25 is equal to 245
minus 9.8 times 0.2x is equal to 1.96x.
minus 9.8 times 0.2x is equal to 1.96x.
And because the bucket
full of sand is lifted
to the top of 15 meter tall building,
we know x be greater
than or equal to zero,
unless you're equal to 15.
So we'll use this interval as
the interval of integration,
which means the work,
done lifting the bucket
from the top of the building
is equal to the integral
from zero to 15 of big F of x,
from zero to 15 of big F of x,
which we now know is 245 minus 1.96x
which we now know is 245 minus 1.96x
integrated with respect to x.
Let's go and evaluate
this on the next slide.
Integrating with respect to x,
we would have 245x
we would have 245x
minus 1.96 times x squared divided by 2.
minus 1.96 times x squared divided by 2.
Well 1.96 divided by 2 is equal to 0.98.
So if minus 0.98x squared,
So if minus 0.98x squared,
and then get limited
integration from 0 to 15.
Now I'll need to find big
F of b minus big F of a.
So next is 15, we have 245 times 15
So next is 15, we have 245 times 15
minus 0.98 times 15 squared.
minus 0.98 times 15 squared.
Of course when x is zero
both terms are zero.
So simplifying, 245 times
15 is equal to 3,675
So simplifying, 245 times
15 is equal to 3,675
minus 0.98 times 15
squared is equal to 220.5.
minus 0.98 times 15
squared is equal to 220.5.
Subtracting, we get 3,454.5
Subtracting, we get 3,454.5
Subtracting, we get 3,454.5
and the units would be joules.
So going back to our first
slide just for a moment.
So going back to our first slide,
the amount of work done lifting the bucket
to the top of the building
again is 3,454.5 joules.
to the top of the building
again is 3,454.5 joules.
I hope you found this helpful.
