Hooke's Law Introduction - Force of a Spring

Summary

The video demonstrates Hooke's Law, which explains the linear relationship between the force applied to a spring and its displacement from the equilibrium position. The presenter uses a force sensor and meter stick to show how force increases as the spring is compressed. The data is plotted, illustrating a negative slope, and the spring constant is calculated. The video clarifies concepts like the spring constant, restoring force, and the significance of the negative sign in the equation. The law holds until the spring reaches its elastic limit, beyond which it no longer obeys Hooke's Law.

Takeaways

• 🔬 Robert Hooke was a 17th-century English scientist known for coining the term 'cell' and formulating Hooke's Law.
• ⚖️ Hooke's Law states that the force of a spring is proportional to its displacement from equilibrium, represented as F = -kx.
• 📏 The spring constant (k) represents the force required to compress or stretch a spring by one meter, measured in Newtons per meter (N/m).
• 📉 A graph of spring force vs. displacement shows a linear relationship, with a negative slope indicating the force is opposite to displacement.
• 🧲 The spring force is a restoring force, meaning it always acts to return the spring to its equilibrium (rest) position.
• ⚠️ Hooke's Law only applies within a spring's elastic limit—beyond that, the spring is permanently deformed, and the law no longer holds.
• 🧮 The y-intercept of the best-fit line for force vs. displacement is zero, and the slope corresponds to the negative of the spring constant.
• ↔️ The negative sign in Hooke’s Law shows that the spring force and displacement are in opposite directions.
• 🎯 The AP Physics 1 equation sheet removes the negative sign in Hooke's Law by using absolute values for force and displacement.
• 🛠️ Hooke's Law is useful for calculating both the force required to compress a spring and predicting the displacement for a given force.

Q & A

Who was Robert Hooke and what is his contribution to science?

Robert Hooke was an English scientist who lived in the 17th century. He is credited with being the first to use the term 'cell' to describe the basic units of life he observed under a microscope.

What is Hooke's Law and how is it demonstrated in the video?

Hooke's Law states that the force required to compress or expand a spring is linearly proportional to the displacement of the spring from its equilibrium position. In the video, this is demonstrated by compressing a spring with a force sensor and measuring the force as the displacement increases.

What is the significance of the slope in the spring force vs. displacement graph?

The slope of the line in the spring force vs. displacement graph represents the negative spring constant (k). The magnitude of this slope gives the value of the spring constant, which tells us how much force is required to compress or extend the spring per meter.

Why is there a negative sign in Hooke's Law?

The negative sign in Hooke's Law indicates that the direction of the spring force is opposite to the direction of the displacement from the equilibrium position. For example, if the spring is compressed to the left, the force pushes to the right.

What is the spring constant and how is it measured?

The spring constant (k) is a measure of how much force is required to compress or expand a spring per unit of displacement. It is calculated as the slope of the force vs. displacement graph, and its units are typically Newtons per meter (N/m).

What is meant by the equilibrium or rest position of a spring?

The equilibrium or rest position of a spring is the position where the spring is neither compressed nor stretched, and no external force is applied. Displacement is measured from this position when the spring is compressed or expanded.

What is a restoring force, and how does it relate to Hooke's Law?

A restoring force is a force that acts to bring an object back to its equilibrium position. In the case of a spring, the spring force acts as a restoring force, always directed towards returning the spring to its rest position.

What happens if a spring exceeds its elastic limit?

If a spring is stretched or compressed beyond its elastic limit, it becomes permanently deformed and will not return to its original shape. Hooke's Law no longer applies once the spring reaches this point.

How are the units of the spring constant derived?

The spring constant is derived from Hooke's Law, which states that force equals the spring constant times displacement. Using standard units, force is measured in Newtons, and displacement is in meters, so the spring constant is in Newtons per meter (N/m).

How can we use the spring constant to predict force or displacement?

Once the spring constant (k) is known, we can use Hooke's Law to predict the force for a given displacement or the displacement for a given force. For example, if the displacement is known, the spring force can be calculated by multiplying the displacement by the spring constant.

Outlines

00:00

🔬 Introduction to Robert Hooke and Hooke's Law

The video begins with an introduction to Robert Hooke, a 17th-century English scientist known for his research in various fields. He is notably credited with coining the term 'cell' to describe the basic unit of life. The video then introduces the law Hooke experimentally proved, now known as Hooke's Law, and starts a demonstration using a horizontally oriented spring and a force sensor to explain how spring force changes with displacement.

05:00

📏 Measuring Spring Force and Displacement

This section demonstrates measuring the force of the spring as it is compressed using a meter stick and a force sensor. The presenter notes the linear relationship between the displacement of the spring and the force measured, leading to the display of data on a graph. The slope of the line is discussed, indicating a direct relationship between spring force and displacement, illustrating Hooke's Law, which states that force equals the spring constant multiplied by displacement.

⚙️ Defining Key Terms in Hooke's Law

Key terms from Hooke's Law are explained. 'F sub s' represents the force caused by the spring, 'k' is the spring constant (the slope of the force-displacement line), and 'x' is the displacement from the equilibrium or rest position. The equilibrium position is where the spring is at rest before a force is applied. The standard unit for the spring constant is Newtons per meter, but other units can also be used.

📊 Analyzing the Spring Constant and Data

Using the data from the experiment, the spring constant is calculated to be 640 Newtons per meter. This value allows predictions about the spring force based on different displacements. For example, a spring displaced by 0.019 meters results in a force of about 12 Newtons. Likewise, given a spring force, the corresponding displacement can be calculated, showing the practical application of Hooke's Law.

🛠️ Comparing Different Spring Constants

The video compares two springs with different spring constants. A larger spring constant requires more force to compress the spring to the same displacement compared to a spring with a smaller constant. This relationship is clearly shown through a graph. The negative sign in Hooke's Law is also explained, indicating that the direction of the spring force is opposite to the direction of displacement.

🔄 Understanding the Restoring Force and the Negative Sign

The negative sign in Hooke's Law shows that the spring force and displacement are in opposite directions. This is emphasized by comparing it to the acceleration of an object in freefall, where the acceleration due to gravity is positive but the object's y-direction acceleration is negative. Additionally, the spring force is described as a restoring force, meaning it always tries to bring the spring back to its equilibrium position.

📚 Conclusion: Elastic Limit and Applications of Hooke's Law

The video concludes by discussing the elastic limit, beyond which a spring becomes permanently deformed, and Hooke's Law no longer applies. It also mentions how the AP Physics equation sheet expresses Hooke’s Law without the negative sign, focusing on the magnitude of the force and displacement. The presenter prefers using the version with the negative sign to show the opposite directions of force and displacement.

Keywords

💡Hooke's Law

Hooke's Law states that the force required to compress or extend a spring is proportional to the displacement of the spring from its rest position. In the video, this law is demonstrated by compressing a spring and measuring how the force increases as the displacement increases. It is the main focus of the experiment shown.

💡Spring Constant (k)

The spring constant (k) is a value that represents the stiffness of a spring, measured in Newtons per meter (N/m). In the video, a spring with a spring constant of 640 N/m is used, meaning it takes 640 Newtons of force to compress or stretch the spring by one meter. This constant is a key part of Hooke's Law, as it determines how much force is needed for a given displacement.

💡Displacement (x)

Displacement in this context refers to the distance the spring is compressed or stretched from its equilibrium (or rest) position. It is represented by 'x' in Hooke’s Law. In the video, displacement is measured in millimeters, and as the spring is compressed by different amounts, the corresponding force is recorded.

💡Equilibrium Position

The equilibrium position is the point where the spring is at rest and no external force is applied to it. In the video, this is the position of the spring before it is compressed or stretched, and displacement from this point is measured to calculate the force applied to the spring.

💡Restoring Force

Restoring force is the force exerted by a spring that acts to return it to its equilibrium position. In the video, this force is demonstrated when the spring pushes back against the force sensor as it is compressed, always directed in the opposite direction of the displacement.

💡Elastic Limit

The elastic limit is the maximum amount a spring can be compressed or stretched before it is permanently deformed and can no longer return to its original shape. The video mentions that Hooke's Law is only valid until the spring reaches this point, beyond which the spring will not behave elastically.

💡Linear Relationship

A linear relationship is a relationship where one variable changes in direct proportion to another. In the video, the graph of spring force versus displacement shows a straight line, indicating a linear relationship between force and displacement, as predicted by Hooke's Law.

💡Negative Sign in Hooke's Law

The negative sign in Hooke’s Law indicates that the force exerted by the spring is in the opposite direction to the displacement. In the video, the spring is compressed to the left, so the force pushes back to the right. This is an important aspect of understanding the directional nature of forces in physics.

💡Force Sensor

A force sensor is a device used to measure the amount of force applied to an object. In the video, the force sensor is used to measure the force exerted by the spring as it is compressed, providing data points that are used to create a graph and validate Hooke's Law.

💡Meter Stick

A meter stick is a tool used to measure distance. In the video, it is used to measure the displacement of the spring in millimeters, helping to calculate how much the spring has been compressed or stretched in relation to the force applied.

Highlights

Robert Hooke was the first to use the term 'cell' to describe basic units of life observed in microscopes.

Hooke's law describes the force required to compress or expand a spring as linearly proportional to its displacement from rest.

A force sensor is used to measure the spring's force as it is compressed, with readings taken every two millimeters.

The spring force increases as the spring is displaced, shown as a linear relationship on a graph.

The slope of the best-fit line in the spring force vs. displacement graph is negative, indicating opposite directions of force and displacement.

The spring constant (k) is defined as the magnitude of the slope of the force vs. displacement graph.

The spring constant units are typically Newtons per meter, though other units like Newtons per centimeter or dynes per millimeter can also be used.

The spring constant of the demonstrated spring is calculated to be 640 Newtons per meter.

Hooke's law can be used to predict the force based on displacement, or the displacement based on force.

The spring constant determines how much force is required to compress or expand the spring by a given distance.

A larger spring constant means more force is required to achieve the same displacement compared to a spring with a smaller constant.

The negative sign in Hooke's law indicates that the force direction is opposite to the displacement direction.

Despite the negative slope in the graph, the spring constant is always a positive value.

The spring force is called a 'restoring force' because it works to bring the spring back to its equilibrium position.

Hooke's law applies only within the elastic limit of the spring—beyond that, the spring is permanently deformed, and the law no longer holds.

The AP Physics 1 equation sheet shows Hooke's law without the negative sign, focusing on the magnitude of force and displacement.

Transcripts

- 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.