PhysicsLesson 12 of 18About 40 minutes

Conservation of mechanical energy

A swinging pendulum keeps trading height for speed and speed for height. Kinetic and potential energy change all the time, but their total does not. That one idea solves problems that would be very hard with forces alone.

By the end of this lesson you can

explain what it means for a quantity to be conserved;
calculate mechanical energy and follow it through a motion;
solve problems with conservation of mechanical energy;
explain what friction does to mechanical energy.

Conserved quantities

A quantity is conserved if its total amount stays the same, even though it may move around or change form. The number of marbles in a closed box is conserved however much you shake it. The money in your pocket is not: you can spend it.

Energy is conserved. It is never created or destroyed, only transferred from one object to another or converted from one form to another. This lesson looks at the most useful case: energy moving between kinetic and potential forms.

Mechanical energy

Definition

Mechanical energy is the sum of an object's kinetic energy and all its forms of potential energy:

$$ME = KE + \sum PE$$

Other forms, such as thermal energy, sound, light and chemical energy, are called nonmechanical energy.

Energy in a swinging pendulum

Interactive

$KE$

$PE_g$

$ME$

Release angle

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

Release the pendulum with friction off. Watch the three bars. Where is the kinetic energy largest? Where is the potential energy largest?
What happens to the green total bar during the swing?
Turn friction on and release again. What happens to the total now, and where does the missing energy go?

Here is the same idea in numbers, for a 0.200 kg apple dropped from 1.80 m (zero level at the floor, no air resistance):

Height (m)	$PE_g$ (J)	$KE$ (J)	$ME$ (J)
1.80	3.53	0	3.53
1.20	2.35	1.18	3.53
0.60	1.18	2.35	3.53
0 (just before landing)	0	3.53	3.53

Every joule of potential energy lost becomes a joule of kinetic energy gained.

Conservation of mechanical energy

When friction and other non-conservative forces can be ignored, mechanical energy is conserved:

$$ME_i = ME_f$$ $$\tfrac12 mv_i^2 + mgh_i = \tfrac12 mv_f^2 + mgh_f$$

When a spring stores energy, include its $\tfrac12 kx^2$ on both sides.

Two features make this equation powerful:

The path does not matter. Only the starting and ending heights and speeds appear. A slide can be straight, curved or bumpy.
The mass often cancels. If only gravity does work, every term contains $m$, so you can divide it out.

Worked example 1

A child starts from rest at the top of a frictionless slide 2.50 m high. How fast is she moving at the bottom?

Choose the zero level at the bottom of the slide. Then $h_i = 2.50\ \text{m}$, $h_f = 0$, $v_i = 0$.
Write the conservation equation. $0 + mgh_i = \tfrac12 mv_f^2 + 0$
Cancel $m$ and solve. $$v_f = \sqrt{2gh_i} = \sqrt{2(9.81)(2.50)} = 7.00\ \text{m/s}$$
$v_f = 7.00\ \text{m/s}$, whatever her mass.

Two frictionless slides of the same height

Interactive

straight ramp

steep curve

Scroll sideways to see the whole demo, or turn your phone sideways.

Try this

Predict first: which ball wins the race to the bottom, and which arrives with more speed?
Release both and compare the times and the final speeds.
Explain why the final speeds match even though the accelerations along the way are very different.

Worked example 2

A ball is thrown from a balcony 8.0 m above the ground with a speed of 10.0 m/s. Ignoring air resistance, how fast is it moving when it hits the ground? Does the direction of the throw matter?

Zero level at the ground. $v_i = 10.0\ \text{m/s}$, $h_i = 8.0\ \text{m}$, $h_f = 0$.
Conservation, with $m$ cancelled. $\tfrac12 v_i^2 + gh_i = \tfrac12 v_f^2$, so $v_f = \sqrt{v_i^2 + 2gh_i}$.
Substitute. $$v_f = \sqrt{(10.0)^2 + 2(9.81)(8.0)} = \sqrt{257} = 16.0\ \text{m/s}$$
Direction. Kinetic energy depends only on speed, not direction. Thrown up, down or sideways at 10.0 m/s, the ball lands at the same speed. (It lands at different places and times, which energy does not tell you.)
$v_f = 16.0\ \text{m/s}$ in every case.

Worked example 3

A toy launcher has a spring with $k = 250\ \text{N/m}$. It is compressed 10.0 cm and fires a 50.0 g ball straight up. How high above the launch point does the ball rise? Ignore the small rise of the ball while the spring expands.

Start and end. At the start the ball is at rest and all the energy is elastic. At the top the ball is momentarily at rest and all the energy is gravitational.
Conservation. $\tfrac12 kx^2 = mgh$
Solve. $$h = \frac{\tfrac12 (250)(0.100)^2}{(0.0500)(9.81)} = \frac{1.25\ \text{J}}{0.491\ \text{N}} = 2.55\ \text{m}$$
$h = 2.55\ \text{m}$

When friction acts

Think first

Slide your palms back and forth against each other as fast as you can for ten seconds. What do you notice, and what became of the energy of that motion?

Compare your answer

Your hands get warm. Friction turned the kinetic energy of your hands into thermal energy.

Friction does negative work, so mechanical energy decreases. The lost mechanical energy has not vanished: it has become thermal energy, sound, and sometimes damage to the surfaces. Total energy is still conserved, even though mechanical energy is not. You can use the difference to work out how much energy friction removed.

Worked example 4

A 20.0 kg sled starts from rest at the top of a snowy hill 5.00 m high and reaches the bottom at 8.00 m/s. How much mechanical energy was turned into other forms by friction?

Mechanical energy at the top. $ME_i = mgh = (20.0)(9.81)(5.00) = 981\ \text{J}$
Mechanical energy at the bottom. $ME_f = \tfrac12 mv^2 = \tfrac12 (20.0)(8.00)^2 = 640\ \text{J}$
The difference is the energy lost to friction. $981 - 640 = 341\ \text{J}$
About 341 J became thermal energy in the sled runners and the snow.

Check yourself

Question 1

A diver steps off a platform 10.0 m above the water. How fast is she moving when she reaches the water?

Show solution

$v = \sqrt{2gh} = \sqrt{2(9.81)(10.0)} = 14.0\ \text{m/s}$

Question 2

A pendulum bob moves at 2.20 m/s at the bottom of its swing. From what height above the bottom was it released from rest?

Show solution

$h = \dfrac{v^2}{2g} = \dfrac{(2.20)^2}{2(9.81)} = 0.247\ \text{m}$

Question 3

A roller-coaster car starts from rest at the top of a 30.0 m hill and coasts to the top of a second hill 12.0 m high. Ignoring friction, how fast is it moving at the top of the second hill?

Show solution

It has dropped $30.0 - 12.0 = 18.0\ \text{m}$: $v = \sqrt{2(9.81)(18.0)} = 18.8\ \text{m/s}$.

Question 4

A basketball player jumps straight up, leaving the floor at 3.00 m/s. How high does her center of mass rise?

Show solution

$\tfrac12 mv^2 = mgh$, so $h = \dfrac{(3.00)^2}{2(9.81)} = 0.459\ \text{m}$.

Question 5

A toy cannon fires a 20.0 g marble horizontally using a spring with $k = 120\ \text{N/m}$ compressed by 4.00 cm. How fast does the marble leave the cannon?

Show solution

$\tfrac12 kx^2 = \tfrac12 mv^2$, so $v = x\sqrt{\dfrac{k}{m}} = 0.0400\sqrt{\dfrac{120}{0.0200}} = 3.10\ \text{m/s}$.

Question 6

You hold a bowling ball hanging from the ceiling on a long rope against the tip of your nose, then let go (without pushing). Ignoring air resistance, what happens when it swings back?

It hits your nose, because it speeds up on the way back
It stops just short of your nose, because it cannot rise higher than where it started
It hits your nose if the ball is heavy enough
It depends on the length of the rope

Show solution

With no energy added, the ball's mechanical energy cannot exceed what it had at release, so it can rise back only to its starting height, exactly where your nose is. With a little air resistance it stops slightly short. (Do not push it!)

Summary

A conserved quantity keeps the same total while it changes form. Energy is always conserved.
Mechanical energy: $ME = KE + \sum PE$.
Without friction, $\tfrac12 mv_i^2 + mgh_i = \tfrac12 mv_f^2 + mgh_f$ (plus $\tfrac12 kx^2$ for springs). The path does not matter, and the mass often cancels.
Friction turns mechanical energy into thermal energy; the difference in $ME$ tells you how much.

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