Mathematics: A chain lying on the ground is 10 m long and its mass is 60 kg. How much work (in J) is required to raise one end of the chain to a height of 4 m?

Step-by-step explanation:I’m not sure but 20 suare feet...

A chain lying on the ground is 10 m long and its mass

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gg68814

13.04.2022

Mathematics

Approximately $4.70 \times 10^{2}\; \rm J$ , assuming that $g = 9.81\; \rm m \cdot s^{-2}$ .

Step-by-step explanation:

When the force is constant, moving by $\Delta x$ in the direction of the force would do a work of $F \cdot \Delta x$ .

However, in this example, as an increasing portion of the chain gets lifted up, the force lifting the chain would increase in size. Hence, apply the integral form of this equation.

Let F(x) denote the force required to lift the chain when the height above the ground is meters. At that moment, exactly (x / 10) of the chain would be above the ground. The mass of that much of the chain would be $(x / 10)\!$ the mass of the whole chain.

$\begin{aligned}F(x) &= m(x) \cdot g \\ &= 60 \cdot \frac{x}{10} \cdot g = 6\, g\, x\end{aligned}$ .

Integrate with respect to :

$\begin{aligned}\int \limits_{0}^{4} F(x) \cdot {\rm d} x &= \int \limits_{0}^{4} 6\, g\, x \cdot {\rm d} x \\ &= \left[(3\, g)\, x^{2}\right]_{x=0}^{x=4}\\ &= 48\, g \approx 4.7 \times 10^{2} \end{aligned}$ .

In other words, it would take approximately $4.7 \times 10^{2}\; \rm J$ of energy to raise one end of the chain to that height of $4\; \rm m$ .

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A chain lying on the ground is 10 m long and its mass is 60 kg. How much work (in J) is required to raise one end of the chain to a height of 4 m?

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