- Good morning.
Robert Hooke was an English scientist who lived in the 17th century.
He performed experiments and did research on a number of subjects.
He is credited with being the first to use the term cell
to describe the basic units of life he observed in microscopes.
Today, we are going to demonstrate a law he experimentally proved which is now called "Hooke's law"
Flipping Physics
Let's start by looking at this horizontally oriented spring.
On the left is a fixed wooden block to which I have attached the spring
in order to keep the left end of the spring stationary.
On the right end of the spring, I have attached a small wooden block
so that I can use a force sensor to compress the spring.
Here we go, using a force sensor to compress the spring.
Notice as I slowly compress the spring,
the force measured by the force sensor slowly increases in magnitude.
Now, I can add a meter stick to measure the distance the spring is compressed from its initial position.
And you can see that when the spring is displaced to the left by two millimeters,
the spring force is 0.9 newtons.
When the spring is displaced to the left by four millimeters, the spring force is 2.4 newtons.
When the spring is displaced to the left by six millimeters the spring force is 3.7 newtons.
And we can continue taking all those measurements every two millimeters until we get to 22 millimeters.
All of those measured spring forces are now shown in the data table.
And now, we can display all of the data on a graph of the spring force as a function of spring displacement.
- That is totally a linear relationship.
- Yeah, and the slope of the line is negative.
- And the y-intercept is definitely zero.
- You are all correct.
Which means we can add a line of best fit of the form y equals m x plus b, where b, the y-intercept, equals zero.
But notice, we have the force of the spring on the y-axis and the displacement of the spring on the x-axis.
So the equation of the best-fit line for this data is the spring force
equals the slope of the best-fit line times the displacement of the spring.
We have just illustrated "Hooke's law."
Hooke's law is the force of the spring equals negative k times x, and let's now define both k and x.
F sub s is the force caused by the spring.
Lowercase k is called the "spring constant."
Notice the spring constant is the magnitude of the slope of the line
created by the spring force as a function of displacement graph.
I've also seen the spring constant called a "force constant" or a "proportionality constant";
however, I will simply call lowercase k the "spring constant."
x is the displacement from what is called the equilibrium position or rest position of the spring.
Unfortunately, I pretty much see it called both equilibrium and rest position equally,
so I will refer to it as both the equilibrium and rest position.
The equilibrium position for a spring is where the spring is at rest before a force is applied to the spring.
For our spring, you can see it is where the spring is at rest before we push on it with the force sensor.
As we compress the spring, the displacement from rest position is x the measurement we took for our data.
Notice x, the displacement from equilibrium position,
is negative in our data because the spring is displaced to the left from rest position.
Bobby, please work out standard units for the spring constant.
- Sure.
Hooke's law states the spring force equals the negative of the spring constant
times displacement from rest position.
We could solve for the spring constant and plug in standard units for force,
and displacement from rest position.
And we get Newtons per meter for the units for the spring constant.
- So the units for the spring constant are Joules?
- No, Joules are Newtons times meters. The spring constant is in Newtons divided by meters.
- Oh, yeah, thanks!
- Realize the units for the spring constant could also be Newtons per centimeter, dynes per millimeter,
or even pounds per furlong, however,
Newtons per meter are the most common units for the spring constant.
Going back to the data we collected on the spring,
the slope of the best-fit line is negative 640 Newtons per meter.
That means the spring constant of our spring is 640 Newtons per meter.
- That means we can use the spring constant and Hooke's law
to predict what the spring force would be at specifics displacements.
For example, if the spring were displaced 0.019 meters to the left of equilibrium position,
the spring force would be about 12 Newtons.
- Or if we know the value of the spring force, we can predict how large the displacement would need to be.
For example, if the spring force is 11 Newtons,
the spring needs to be displaced about 0.017 meters to the left of rest position.
- That is correct. Very nice!
Realize the spring constant is just how much force it requires to compress or expand a spring per meter.
We can use a second spring to illustrate what that means.
Our original spring is on your right and has a larger spring constant than the spring on the left.
A larger spring constant requires a larger force
to compress the spring to the same displacement from rest position.
However, a smaller spring constant requires a smaller force
to compress the spring to the same displacement from equilibrium position.
You can see this clearly illustrated in the graph.
- Mr. P? - Yes, Bo?
- You have not talked about the vector symbol in Hooke's law.
- I don't think he needs to.
Force and displacement are both, by definition, vectors. That is all the vector symbol means. Right?
- Okay, sure.
- What I'm curious about is the negative in the equation. What does the negative mean?
- The negative in Hooke's law indicates that the direction of the spring force is
opposite the direction of the displacement of the spring from equilibrium position.
In our example, that means the force caused by the spring is to the right
because the displacement of the spring from rest position is to the left.
If we were to pull the spring to the right, that would mean the force caused by the spring would be to the left
because the displacement of the spring from equilibrium position is to the right.
Again, the negative simply indicates the direction of the spring force is
opposite the direction of the displacement of the spring from rest position.
- Mr. P?
- Yes, Bobby?
- I'm still confused by the negative. Does that mean the spring constant is negative?
- Ah, okay, thank you Bobby. So, no that does not mean the spring constant is negative.
In fact, the spring constant is always positive.
Because the force of the spring and its displacement are in opposite directions,
the slope of the best-fit line is always negative.
The slope of the best-fit line equals negative k;
therefore, because the slope of the best-fit line is always negative,
the value of the spring constant is always positive.
-Oh, I get it.
This is just like the acceleration in the y-direction of an object in freefall
equals the negative of the acceleration due to gravity.
The acceleration due to gravity is positive,
but the acceleration in the y-direction of the object in freefall is negative.
Just like the spring constant is positive, but the slope of the best-fit line is negative.
- Good job, Bobby.
- Very nice.
- Thanks! - I'm impressed.
- Another thing to realize here is that the spring force is also called a restoring force.
A restoring force is a force that tends to bring an object back to the equilibrium position.
Notice how the force of the spring is always directed towards the rest position.
That is what makes the spring force a restoring force.
The basic idea of Hooke's law is that it shows that the force required to compress or expand a spring
is linearly proportional to the distance the spring is compressed or expanded.
That is what the equation for Hooke's law means.
I should also point out that Hooke's law is only applicable on a spring
until the spring reaches its elastic limit.
The elastic limit being the maximum displacement before the spring will be permanently deformed.
In other words, once the spring reaches beyond the elastic limit,
the spring will no longer return to its original shape, and Hooke's law no longer holds true.
Lastly, the AP Physics 1 equation sheet gives Hooke's law as the absolute value of the spring force equals
the spring constant times the absolute value of the displacement of the spring from equilibrium position.
This equation removes the negative sign
and the negative sign shows that the spring force in the displacement from rest position are
opposite in direction.
I consider that useful information, which is why I use the version of Hooke's law which includes the negative sign.
Thank you very much for learning with me today.
I enjoyed learning with you.