Work, Energy, and Hydrostatic Force

Definition In SI units force is measured in newtons, distance in meters and work in joules, where

Pressure is force per unit area, measured in pascals, $1 \ \text{Pa} = 1 \ \text{N}/\text{m}^2$. In US customary units force is measured in pounds, distance in feet and work in foot-pounds. A mass $m$ near the surface of the earth has weight $mg$ with $g \approx 9.8 \ \text{m}/\text{s}^2$.

Definition Suppose an object moves along the $x$ axis from $x = a$ to $x = b$, with $a < b$, under a force $F(x)$ acting along the axis, where $F$ is continuous on $[a, b]$. The work done by the force is

This comes from cutting $[a, b]$ into $n$ pieces of length $\Delta x = (b - a)/n$, choosing a point $x_i^*$ in the $i$th piece, treating the force as the constant $F(x_i^*)$ on that piece, and taking the limit of the Riemann sums

Definition Hooke’s law says that, within the elastic limit of the spring, the force needed to hold a spring stretched or compressed $x$ units beyond its natural length is

where $k > 0$ is the spring constant. Here $x = 0$ is the natural length, $x > 0$ means stretched and $x < 0$ means compressed. The work needed to stretch the spring from $x = a$ to $x = b$ is

Definition To find the work needed to lift a body of liquid, cut it into thin horizontal layers. If a layer at height $y$ with thickness $\Delta y$ has volume $A(y) \, \Delta y$, the liquid has constant density $\rho$, and the layer must be lifted a distance $D(y)$, then the work for that layer is about $\rho g A(y) D(y) \, \Delta y$. When the liquid fills heights $a \le y \le b$ and $A$ and $D$ are continuous, the total work is

The same idea works for cables and chains, where each small piece is lifted its own distance.

Definition An object of mass $m$ moving with velocity $v$ has kinetic energy

Suppose it moves along the $x$ axis with position $x(t)$, twice differentiable, and the net force on it depends only on position, given by a continuous function $F(x)$. Newton’s second law $F(x(t)) = m x''(t)$ and the substitution $x = x(t)$ give the work energy theorem

A differentiable function $U$ with $U'(x) = -F(x)$ on an interval is a potential energy for $F$ there, and then for $a$ and $b$ in that interval the work done by $F$ from $a$ to $b$ is $U(a) - U(b)$. For gravity near the earth’s surface, $U = mgh$ at height $h$.

Definition In a fluid at rest with constant density $\rho$, the pressure caused by the fluid at depth $d$ below the surface is

This leaves out the air pressure on the surface, which acts on both sides of most walls and so usually cancels. For water $\rho g \approx 1000 \cdot 9.8 = 9800 \ \text{N}/\text{m}^3$.

Definition At any point in a fluid at rest, the pressure is the same in every direction. Together with $P = \rho g d$ this means the pressure depends only on depth. A flat horizontal plate of area $A$ at depth $d$ therefore feels the force

Definition Let a flat plate stand vertically in a fluid of constant density $\rho$. Measure depth $x$ downward from the surface, and suppose the plate occupies the depths $a \le x \le b$ with width $w(x)$ at depth $x$, where $w$ is continuous on $[a, b]$. The total force of the fluid on one side of the plate is