Skip to main content
added 797 characters in body
Source Link
Agnius Vasiliauskas
  • 14.9k
  • 2
  • 22
  • 58

When force varies, you need to integrate force over displacement to get the total work done: $$ W=\int _{a}^{b}\mathbf {F(s)} \cdot d\mathbf {s}\tag 1 $$

For example. Let's say we need to find a work done, when we compress a spring from $0$ to a final position $\ell$.

Spring force needed to compress to $x$, by Hooke's law, is $$ F(x)=kx \tag 2$$

Now substitute (2) into (1) : $$ \begin{align} W &= \int_0^\ell F(x)dx \\ &= \int_0^\ell kx~dx \\ &=\frac {k~\ell^{~2}}{2} \end{align} \tag 3 $$

After integration, notion of a force at a specific point $F(x)$ in spring, is lost. We are left only with total displacement $\ell$, hence final formula of total work done doesn't care about specific application of force at particular $x \in [0,\ell]$ path point (or at particular time moment). This temporal/spatial information is lost. 

You can say to some sense, that varied force is averaged over the entire path by integration procedure. Actually, if you'll equate (3) to a work done by an average force:

$$ \frac {k~\ell^{~2}}{2} = \overline F ~\ell \tag 4,$$

you can extrapolate what is this average force in spring compression $ 0 \to \ell$ :

$$ \overline F = k~ \frac {\ell}{2} \tag 5 .$$

So, for getting total work done, you can either:

  • integrate varied force over displacement or time, if over time then you need in eq (1) to change integration variables by : $W=\int _{a}^{b} {F(s)} \cdot ds = \int _{t_1}^{t_2} {F(t)} \cdot \frac {ds}{dt} dt = \int _{t_1}^{t_2} {F(t)} \cdot v(t) dt$

  • or you just calculate average/effective force $\overline F$ over path and use it as in normal work done scenario.

Anyway, after integration or using an average force, temporal information like $F(t)=ma(t)$ is lost.

When force varies, you need to integrate force over displacement to get the total work done: $$ W=\int _{a}^{b}\mathbf {F(s)} \cdot d\mathbf {s}\tag 1 $$

For example. Let's say we need to find a work done, when we compress a spring from $0$ to a final position $\ell$.

Spring force needed to compress to $x$, by Hooke's law, is $$ F(x)=kx \tag 2$$

Now substitute (2) into (1) : $$ \begin{align} W &= \int_0^\ell F(x)dx \\ &= \int_0^\ell kx~dx \\ &=\frac {k~\ell^{~2}}{2} \end{align} \tag 3 $$

After integration, notion of a force at a specific point $F(x)$ in spring, is lost. We are left only with total displacement $\ell$, hence final formula of total work done doesn't care about specific application of force at particular $x \in [0,\ell]$ path point (or at particular time moment). This temporal/spatial information is lost. You can say to some sense, that varied force is averaged over the entire path by integration procedure.

When force varies, you need to integrate force over displacement to get the total work done: $$ W=\int _{a}^{b}\mathbf {F(s)} \cdot d\mathbf {s}\tag 1 $$

For example. Let's say we need to find a work done, when we compress a spring from $0$ to a final position $\ell$.

Spring force needed to compress to $x$, by Hooke's law, is $$ F(x)=kx \tag 2$$

Now substitute (2) into (1) : $$ \begin{align} W &= \int_0^\ell F(x)dx \\ &= \int_0^\ell kx~dx \\ &=\frac {k~\ell^{~2}}{2} \end{align} \tag 3 $$

After integration, notion of a force at a specific point $F(x)$ in spring, is lost. We are left only with total displacement $\ell$, hence final formula of total work done doesn't care about specific application of force at particular $x \in [0,\ell]$ path point (or at particular time moment). This temporal/spatial information is lost. 

You can say to some sense, that varied force is averaged over the entire path by integration procedure. Actually, if you'll equate (3) to a work done by an average force:

$$ \frac {k~\ell^{~2}}{2} = \overline F ~\ell \tag 4,$$

you can extrapolate what is this average force in spring compression $ 0 \to \ell$ :

$$ \overline F = k~ \frac {\ell}{2} \tag 5 .$$

So, for getting total work done, you can either:

  • integrate varied force over displacement or time, if over time then you need in eq (1) to change integration variables by : $W=\int _{a}^{b} {F(s)} \cdot ds = \int _{t_1}^{t_2} {F(t)} \cdot \frac {ds}{dt} dt = \int _{t_1}^{t_2} {F(t)} \cdot v(t) dt$

  • or you just calculate average/effective force $\overline F$ over path and use it as in normal work done scenario.

Anyway, after integration or using an average force, temporal information like $F(t)=ma(t)$ is lost.

Source Link
Agnius Vasiliauskas
  • 14.9k
  • 2
  • 22
  • 58

When force varies, you need to integrate force over displacement to get the total work done: $$ W=\int _{a}^{b}\mathbf {F(s)} \cdot d\mathbf {s}\tag 1 $$

For example. Let's say we need to find a work done, when we compress a spring from $0$ to a final position $\ell$.

Spring force needed to compress to $x$, by Hooke's law, is $$ F(x)=kx \tag 2$$

Now substitute (2) into (1) : $$ \begin{align} W &= \int_0^\ell F(x)dx \\ &= \int_0^\ell kx~dx \\ &=\frac {k~\ell^{~2}}{2} \end{align} \tag 3 $$

After integration, notion of a force at a specific point $F(x)$ in spring, is lost. We are left only with total displacement $\ell$, hence final formula of total work done doesn't care about specific application of force at particular $x \in [0,\ell]$ path point (or at particular time moment). This temporal/spatial information is lost. You can say to some sense, that varied force is averaged over the entire path by integration procedure.