Theory

Theory 1

Work is a measure of energy expended to achieve some effect. According to physics:

$$\text{Work}=\text{Force}\times \text{Distance}$$ $$W={\int }_a^{b}F(x)dx\text{OR}W={\int }_a^{b}xdF$$

To compute the work performed against gravity while lifting some matter, decompose the matter into horizontal layers at height $y$ and thickness $dy$. Each layer is lifted some distance. The weight of the layer gives the force applied.

The work performed on each single layer is summed by an integral to determine the total work performed to lift all the layers:

Work performed

$$\text{Work to lift a layer}=\text{height raised}\times g\times \text{density}\times A(x)\times dx$$ $$\text{Total work}={\int }_a^b\rho gh(x)A(x)dx$$

• $A(x)=\text{area of layer}$
• $h(x)=\text{height layer is lifted}$
• $\rho =\text{mass density}=1000\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ for water
• $g=9.8\mathrm{m}/{\mathrm{s}}^2=\text{constant of gravitational acceleration}$