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Physics of Hearing

- Define conservative force, potential energy, and mechanical energy.
- Explain the potential energy of a spring in terms of its compression when Hooke’s law applies.
- Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy.

A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.

We can define a potential energy (PE) for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added.

$\frac{kx}{2}\\$

. Thus the work done in stretching or compressing the spring is $W_\text{s}=Fd=\left(\frac{kx}{2}\right)x=\frac{1}{2}kx^2\\$

. Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of $\frac{1}{2}kx^2\\$

. We therefore define the $\text{PE}_{\text{s}}=\frac{1}{2}kx^2\\$

,
where The equation

$\text{PE}_{\text{s}}=\frac{1}{2}kx^2\\$

has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of $\text{PE}_{\text{s}}=\frac{1}{2}kx^2\\$

, where $W_{\text{net}}=\frac{1}{2}mv^2-\frac{1}{2}mv_0^2=\Delta\text{KE}\\$

.If only conservative forces act, then

Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is,

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

$\text{(conservative forces only),}\begin{cases}\text{KE}+\text{PE}=\text{constant}\\\text{or}\\\text{KE}_{\text{i}}+\text{PE}_{\text{i}}=\text{KE}_{\text{f}}+\text{PE}_{\text{f}}\end{cases}\\$

where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the - How fast is the car going before it starts up the slope?
- How fast is it going at the top of the slope?

KE_{i }+ PE_{i} = KE_{f} + PE_{f}

or

$\frac{1}{2}{mv_{\text{i}}}^2+mgh_{\text{i}}+\frac{1}{2}{kx_{\text{i}}}^2=\frac{1}{2}{mv_{\text{f}}}^2+mgh_{\text{f}}+\frac{1}{2}{kx_{\text{f}}}^2\\$

,
where $\frac{1}{2}{kx_{\text{i}}}^2=\frac{1}{2}{mv_{\text{f}}}^2\\$

.
In other words, the initial potential energy in the spring is converted completely to kinetic energy in the absence of friction. Solving for the final speed and entering known values yields$\begin{array}{lll}v_{\text{f}}&=&\sqrt{\frac{k}{m}x_{\text{i}}}\\\text{ }&=&\sqrt{\frac{250.0\text{ N/m}}{0.100\text{ kg}}}\left(0.0400\text{ m}\right)\\\text{ }&=&2.00\text{ m/s}\end{array}\\$

$\frac{1}{2}{kx_{\text{i}}}^2=\frac{1}{2}{mv_{\text{f}}}^2+mgh_{\text{f}}\\$

.
This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy and partly to kinetic energy. The final speed at the top of the slope will be less than at the bottom. Solving for $\begin{array}{lll}v_{\text{f}}&=&\sqrt{\frac{kx_{\text{i}}^2}{m}-2gh_{\text{f}}}\\\text{ }&=&\sqrt{\left(\frac{250.0\text{ N/m}}{0.100\text{ kg}}\right)\left(0.0400\text{ m}\right)^2-2\left(9.80\text{ m/s}^2\right)\left(0.180\text{ m}\right)}\\\text{ }&=&0.687\text{ m/s}\end{array}\\$

- A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken.
- We can define potential energy (PE) for any conservative force, just as we defined PE
_{g}for the gravitational force. - The potential energy of a spring is ${\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}\\$, where
*k*is the spring’s force constant and*x*is the displacement from its undeformed position. - Mechanical energy is defined to be KE + PE for a conservative force.
- When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,

$\begin{cases}\text{KE}+\text{PE}=\text{constant}\\\text{or}\\\text{KE}_{\text{i}}+\text{PE}_{\text{i}}=\text{KE}_{\text{f}}+\text{PE}_{\text{f}}\end{cases}\\$

where i and f denote initial and final values. This is known as the conservation of mechanical energy.

- What is a conservative force?
- The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer dives from it, starting just before the swimmer steps on the board until just after his feet leave it.
- Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act?
- What is the relationship of potential energy to conservative force?

- A 5.00 × 10
^{5}-kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant*k*of the spring? - A pogo stick has a spring with a force constant of 2.50 × 10
^{4}N/m, which can be compressed 12.0 cm. To what maximum height can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40.0 kg? Explicitly show how you follow the steps in the Problem-Solving Strategies for Energy.

$\frac{1}{2}{\text{kx}}^{2}\\$

where