A 3 meter chain with linear mass density rho(x) = 2x(4 − x) kg/m lies on the ground, where x = 0 is the top of the chain. Calculate the work required to lift the chain from its top end so that its bottom is 2 meters above the ground.

Step-by-step explanation:

Step-by-step explanation:

Given that,

Length of chain

L = 3m

Linear mass density is

ρ(x) = 2x(4 — x) kg/m lies on the ground

When x = 0, is top of the chain

Work done to lift the chain from top end so that the bottom is 2m above the ground.

Considered the segment of the chain of length ∆x that will be lifted in the positive y direction (+j)m from the foot.

The work needed to lift this segment is given as

Work = mass density × ∆x × gravity

W = ρ × ∆x × g

g is acting downward = 9.8j

Summing over all segment of the chain and passing to the limit as ∆x→0.

Therefore, the total work done needed to full extend the chain is

W = ∫ ρ × ∆x × g x = 0 to 3

Since g is constant

∆x = xdx

Then,

W = g ∫ 2x(4—x)x dx. x= 0 to x = 3

W = 9.81 ∫ (8x² — 2x³)x dx

W = 9.81 ( 8x³/3 — 2x⁴/4)

W=9.81(8x³/3— ½x⁴) from x=0 to x=3

W = 9.81[8(3)³/3 — ½(3⁴)] — 0

W = 9.81 × (72 —40.5)

W = 9.81 × 31.5

Work done= 309.015 J

Lifting the entire chain requires to light the weight

Weight = ∫ρgdx

Weight = g ∫ρ dx. From x=0 to x=3

Weight = g ∫ 2x(4-x) dx

Weight = 9.81 ∫(8x-2x²)dx

Weight = 9.81 [ 8x²/2 - 2x³/3]

Weight = 9.81[4x²-⅔x³] x=0 to x=3

Weight = 9.81[4(3²) — ⅔(3³)]

Weight = 9.81(36—18)

Weight = 9.81 × 18

Weight = 176.58N

Now, this weight is lifted to a height of 2m, then using potential energy formula, we have

P.E = Work = mgh = Weight ×height

Work = W×h = 176.58 × 2

Work = 353.16 J

Then, total workdone is

W = 353.16 + 309.015

W = 662.18 J

The required Work done required to lift the chain from top so that it's bottom is 2m from the ground is 662.18J

the answer will be the last one

step by step solution :

step   1   :

1.1     4 = 22

(4)3 = (22)3 = 26

equation at the end of step   1   :

  ((23) • 26)2

step   2   :

equation at the end of step   2   :

  (23 • 26)2

step   3   :

final result :

  218

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