#### To determine

**To find:**

The amount of work donewhen the bucket finishes draining just as it reaches the 12 m level.

#### Answer

**W≈3857 J**

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Definition:**

W=limn→∞∑t=1nfxi* ∆x=∫abfx dx

**3) Calculation:**

Let’s place the rope at the top of the height and hang it with the bucket; x- axis is pointing downward

Divide the height of the rope into small parts with length ∆x.

If xi* is a point in the ith such interval, then all points in the interval are lifted by approximately the same amount, xi*

Suppose the part xi* of the rope has just reached level 12 m. Then (12-xi*)m is hanging downwards. The force acting on xi* is due to gravitational force. The total mass below xi* contributes towards the force acting on xi*. Thus the force acting on xi* is due to the sum mass of rope hanging below xi*, mass of water remaining and mass of bucket.

Mass of water goes from 36kg to 0 as bucket moves from height 0 to 12. The rate of leaking is constant. So for every meter 36/12 = 3kg water is leaked out. So when the xi* part of rope has just reached top the mass of water remaining is 36-3 xi* = 3(12-xi*) kg. Let’s compute the total mass at xi*.

Total mass per meter is the sum of mass of rope, mass of bucket, and mass of water.

First mass of rope

By the formula,

Mass=linear density*length

(0.8 kg/m)(12-xi*m)

=0.812-xi*

=9.6-0.8xi*kg

Now,

Mass of water

3kg/m12-xi*m

=36-3xi*kg

And mass of the bucket is 10kg

Therefore,

Total mass=9.6-0.8xi*kg+36-3xi*kg+10kg

=9.6-0.8xi*+36-3xi*+10kg

=55.6-3.8xi*kg

By formula, force is given by,

f=mg

Where, m is total mass and g=9.8 m/s2 acceleration due to gravity

f=9.855.6-3.8xi*N

By formula,

The total work done is

W=limn→∞∑i=1n9.855.6-3.8xi* ∆x

Where, f=9.855.6-3.8xi* and distance=d=∆x

=∫0129.855.6-3.8xdx

=9.855.6x-1.9x2012

=9.8393.6

≈3857 J

**Conclusion:**

The amount of work in pulling the system up to 12 m level is

≈3857 J