#### To determine

**To find:**

Work done in lifting the lower end of the chain to the ceiling

#### Answer

W=62.5 ft-lb

#### 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) Given:**

i) Weight of the chain 25 lb

ii) Length of the chain 10 ft

**3) Calculation:**

Let’s place the origin at the top of the ceiling and x- axis pointing downward as in the figure.Divide the cable 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*

Total Length of chain 10 ft and total weight of chain 25 lb

The length of chain per foot is 25lb/10ft=2.5 lb/ft

The weight of the ith part is (2.5 lb/ft)(∆xft)=2.5 ∆x lb

Divide the length of chain 10 ft into equal parts 0≤5≤10

The upper part of the chain is 0≤xi*≤5 and lower part is 5≤xi*≤10

To find the work done in lifting the lower end of the chain to the ceiling so that it is level with the upper end.

The lower part of the chain is 5≤xi*≤10

Distance of point xi* in the interval 5≤xi*≤10 is xi*-5ft

When lifting the point xi* will travel 2(xi*-5) ft

Therefore,

The lower part of the chain ( for 5≤xi*≤10 ) has to be lifted 2(xi*-5) ft,

The work needed to lift for the ith subinterval of the chain is 2(xi*-5)(2.5∆x)

Where 2(xi*-5) is the distance the chain in ith interval travels

2.5∆x is the force (due to weight) required in lifting the ith part of chain

The total work needed is

W=limn→∞∑i=1n2xi*-52.5∆x

=∫5102x-52.5dx

=5∫510x-5dx

=512x2-5x510

=550-50-252-25

=5252

=62.5 ft-lb

**Conclusion:**

The work done in lifting the lower end of the chain to the ceiling is 62.5 ft-lb