#### To determine

**To find:**

Work required to raise one end of the chain to a height of 6 m

#### Answer

W=1411.2 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) Given:**

Length of chain 10 m

Mass of chain 80 kg

**3) Definition 4:**

W=limn→∞∑t=1nfxi* ∆x=∫abfx dx

**4) Calculation:**

Chain lying on the ground is 10m long

Out of the 10m long wire, one end of the chain is raised to a height of 6m, 4 m of the chain lying along the ground.

Total length of the chain if one end of the chain is raised to a height of 6m is 6m

Total mass of chain 80 kg

Total length of chain 10 m

Mass per unit length of chain is (80kg)/(10m)=8 kg/m

The force acting on the chain is F=mg

Where m is mass of chain and g is acceleration due to gravity

Force per unit length acting on the chain is F=(8 kg/m)(9.8 m/s2)=78.4N/m

The work done acting on the chain is W=F·d

Where F is force and d is distance

To find the work required to raise one end of the chain to a height of 6 m

Use an argument similar to the one that leads to Definition 4.

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

The part of the chain xi* m from the lifted end is raised (6-xi*)m if 0≤xi*≤6 m, and is lifted 0 m if xi*>6 m.

Therefore, the work needed is

W=limn→∞∑i=1n6-xi*·78.4 ∆x=∫066-x78.4dx=78.46x-12x206=78.418=1411.2 J

**Conclusion:**

The work required to raise one end of the chain to a height of 6 m is 1411.2 J