# 7.4 Conservative forces and potential energy  (Page 5/10)

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Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in [link] . Note also that we do not consider details of the path taken—only the starting and ending points are important (as long as the path is not impossible). This assumption is usually a tremendous simplification, because the path may be complicated and forces may vary along the way.

## Phet explorations: energy skate park

Learn about conservation of energy with a skater dude! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space!

## Test prep for ap courses

Two 4.0 kg masses are connected to each other by a spring with a force constant of 25 N/m and a rest length of 1.0 m. If the spring has been compressed to 0.80 m in length and the masses are traveling toward each other at 0.50 m/s (each), what is the total energy in the system?

1. 1.0 J
2. 1.5 J
3. 9.0 J
4. 8.0 J

A spring with a force constant of 5000 N/m and a rest length of 3.0 m is used in a catapult. When compressed to 1.0 m, it is used to launch a 50 kg rock. However, there is an error in the release mechanism, so the rock gets launched almost straight up. How high does it go, and how fast is it going when it hits the ground?

20 m high, 20 m/s.

What information do you need to calculate the kinetic energy and potential energy of a spring? Potential energy due to gravity? How many objects do you need information about for each of these cases?

You are loading a toy dart gun, which has two settings, the more powerful with the spring compressed twice as far as the lower setting. If it takes 5.0 J of work to compress the dart gun to the lower setting, how much work does it take for the higher setting?

1. 20 J
2. 10 J
3. 2.5 J
4. 40 J

(a)

Describe a system you use daily with internal potential energy.

Old-fashioned pendulum clocks are powered by masses that need to be wound back to the top of the clock about once a week to counteract energy lost due to friction and to the chimes. One particular clock has three masses: 4.0 kg, 4.0 kg, and 6.0 kg. They can drop 1.3 meters. How much energy does the clock use in a week?

1. 51 J
2. 76 J
3. 127 J
4. 178 J

(d)

A water tower stores not only water, but (at least part of) the energy to move the water. How much? Make reasonable estimates for how much water is in the tower, and other quantities you need.

Old-fashioned pocket watches needed to be wound daily so they wouldn’t run down and lose time, due to the friction in the internal components. This required a large number of turns of the winding key, but not much force per turn, and it was possible to overwind and break the watch. How was the energy stored?

1. A small mass raised a long distance
2. A large mass raised a short distance
3. A weak spring deformed a long way
4. A strong spring deformed a short way

(c)

Some of the very first clocks invented in China were powered by water. Describe how you think this was done.

## Section summary

• 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 $\left(\text{PE}\right)$ for any conservative force, just as we defined ${\text{PE}}_{g}$ 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 $\text{KE}+\text{PE}$ for a conservative force.
• When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,

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

## Conceptual questions

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?

## Problems&Exercises

A $5\text{.}\text{00}×{\text{10}}^{5}\text{-kg}$ 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?

$\text{7.81}×{\text{10}}^{5}\phantom{\rule{0.20em}{0ex}}\text{N/m}$

A pogo stick has a spring with a force constant of $2\text{.}\text{50}×{\text{10}}^{4}\phantom{\rule{0.20em}{0ex}}\text{N/m}$ , 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 [link] .

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
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