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(Anzar Abbas) - Think about all the things with springs.
Locks, clocks, mattresses and sofas, pens,
devices that hold batteries, trampolines and toys.
Springs are so important,
they have their own sound effect.
(spring boinging)
In physics, there's a set of scientific laws
surrounding springs that involve elasticity,
torsion, and force.
All of those come into play with springs.
That's what British physicist, Robert Hooke,
figured out in the 1600's.
Hooke's Law helps explain elasticity.
It states that...
All springs, whether it's a rubber band or a Slinky,
have a natural position based on how they're made.
Some, like this Slinky, are naturally compressed.
So when you stretch them,
they want to spring back into the compressed shape.
A spring on a garage or a porch door
are naturally compressed.
Mattress springs and springs in some ballpoint pens
are compressed springs.
Whatever the spring's natural position is,
is called its equilibrium position,
where it's either not stretched or not compressed,
depending on the type of spring.
So here's the equation.
F sub s equals negative k times x.
F sub s is the spring force.
And it's measured in Joules.
Once it's stretched or compressed,
how much the spring moves or is displaced is x,
which is measured in meters.
The k is called the spring constant,
which is how the particular spring behaves.
The units for that are Newtons per meter.
The x is a negative value to show that
once the spring is displaced by force,
for it to spring back to its equilibrium state,
the displacement is opposite the direction of the initial force.
So, if you pull down on a spring,
the spring moves downward.
But when you let it go,
it springs back in an upward force,
due to the elasticity of the spring.
That's the negative kx.
This is why the spring force is also sometimes called
the restoring force.
Because it wants to restore the spring
back to the equilibrium position.
Whenever you do a problem with springs and elasticity,
it's always important to remember to note
the direction of this restoring force.
We're going to use Hooke's apparatus,
named after Robert Hooke,
to demonstrate springs and all they can do.
To see how much a spring stretches or compresses,
let's get some spring action by adding some weight.
Remember, weight is the mass times the acceleration
due to gravity, g.
With weight added to a vertical spring,
let's see how much it stretches or compresses.
Now, let's add a heavier weight, and do this several times.
Each time you add weight,
plot the spring action on a graph,
with force on the y-axis and displacements on the x-axis.
How far does the spring stretch or compress?
Check it out for yourself.
You'll find that it's likely to be a straight line.
The slope of this line is the spring's k constant.
You might also discover that not all springs
follow Hooke's Law perfectly.
... then the spring is considered Hookean.
In real life situations,
the ideal springs are called Hookean,
and non-ideal springs are called non-Hookean.
Rubber bands, for example, are non-Hookean.
What does that mean?
... that spring is considered to be non-Hookean.
If a spring is stretched too much,
it loses some of its springiness,
and becomes non-Hookean.
In that case, that straight line
on the force versus displacement graph
becomes a curve.
Now, what if we pulled on one of these weights,
and there was no friction to stop the springing action?
It would continue to bounce up and down forever.
That same thing is true of a horizontal spring.
This is called simple harmonic motion,
and we will explore this idea in a later unit.
Let's pull the spring away from its equilibrium position.
Now, we have spring potential energy,
since we know it will spring back
into its equilibrium position.
Here's the equation to figure out
the amount of potential energy in the spring.
PE sub s, that's the potential energy of the spring,
is equal to 1/2 k, the spring's constant,
times x squared.
x is the displacement we've pulled the spring
away from the equilibrium in meters.
And the potential energy is in Joules.
So we can relate the spring's potential energy
to gravitational potential energy.
From another segment,
you learned that gravitational potential energy
is due to the change in vertical displacement.
So, if the spring action is vertical,
you're dealing with both spring
and gravitational potential energy.
They're related to each other.
And here's how.
Remember the gravitational potential energy,
written PE sub G equals mgh,
that's mass times gravity times height,
which is just the force times distance.
When I drop an object...
work is done.
Gravitational potential energy is converted to kinetic energy.
In other words, the initial gravitational potential energy
equals the final kinetic energy.
The same applies to a spring's potential energy,
and the work it can do once the potential energy
is converted to kinetic energy.
A spring's potential energy can be converted
into kinetic energy.
And in an ideal world,
this would be done conservatively,
without energy transformed into heat by friction.
The equation would look much like the equation
you learned for the conservation of energy.
When you stretch a spring vertically,
the potential energies of gravity and of the spring
are included along with a kinetic energy of the spring.
So that the initial spring potential energy,
plus gravitational potential energy,
plus kinetic energy
equals the final spring potential energy,
plus gravitational potential energy,
plus kinetic energy.
For springs that are stretched or compressed horizontally,
the equation looks much the same.
But you can cancel out gravitational potential energy
on both sides,
since the spring isn't moving in a vertical direction.
Let's go back to the graph.
If you want to know what the PE is
when a spring is stretched or compressed by a certain amount,
pick a point, the displacement,
on the graph,
and calculate the area of the triangle that it makes.
Since the area of a triangle is 1/2 base times the height,
and that's the same as 1/2kx squared,
you get the same answer as the average of the potential energy,
which is 1/2 times the PE final minus the PE initial.
It's another way to figure out the spring's potential energy.
So now you know the relationship between
spring potential energy
and gravitational potential energy.
The key to understanding physics
is being able to relate the various forms of energy
to one another mathematically.
That's it for this segment of "Physics in Motion",
and we will see you next time.
- For more practice problems,
lab activities, and note-taking guides,
check out the "Physics in Motion" Toolkit.