LECTURE 28: Spring force and potential energy

Transcription

1 Lectures Page 1 LECTURE 28: Spring force and potential energy Select LEARNING OBJECTIVES: i. ii. iii. Introduce the definition of spring potential energy. Understand the shape of a spring potential energy vs displacement graph. Demonstrate the ability to extract forces from potential energy vs displacement graphs. TEXTBOOK CHAPTERS: Giancoli (Physics Principles with Applications 7 th ) :: 6-4 Knight (College Physics : A strategic approach 3 rd ) :: 10.4 BoxSand :: Energy ( Potential Energy ) WARM UP: What is the change in gravitational potential energy of the box in the figure below. Also, what is the work done by gravity on the box? In this lecture, we will see that the work done by a spring force is a conservative force. We have already encountered one other conservative force, the force of gravity. Recall that if a conservative force is internal to our system, we can define a potential energy for the work instead of calculating the work directly. Thus we will define the spring potential energy in this lecture. The force an ideal spring exerts on an object connected to its end is This force is often referred to as Hooke's Law. Why the negative? It tells us that a spring is a "restoring" force,

2 that is to say that it always tries to push or pull an object back to some equilibrium location. The below figures might help visualize this. Note some of the important features of the above figure showing the spring force. First, when an object is at the equilibrium position, there is no spring force. Next, notice that the spring force has the same form -kδx for both compression and stretching, where Δx is the x-displacement from the equilibrium location. Also notice that I used a different coordinate system for the stretching case, this is just to highlight that the choise of coordinate system does not matter. Notice how the spring force must include a negative sign in front of it because it always points in the opposite direction of the x-displacement. The "k" that shows up in the force of spring equation is known as the spring constant. This is a material property. Can you figure out the units of this spring constant? I should also point out that since the choice of coordinate system does not matter, some textbooks and online sources place the coordinate system as the equilibrium location. This makes Δx = xf - xeq = xf - 0 = xf x, thus the spring force has the form -kx. I will use the original Δx form to help illustrate that it is the displacement from the equilibrium location. We need to point out one more very important feature of the spring force, it is a function of position. The spring force is not a constant value; we need to be careful when using it. Approximations (Ideal spring) The above discussion about springs is valid for an ideal spring as compared to a real spring. What makes an ideal spring ideal? For one, an ideal spring always obeys Hooke's law (it can be described the spring force in the above equations for all stretch and compression lengths). Attach a spring you find in your house to the ceiling, add mass to the bottom end, and measure the Δx the spring stretches for each mass added. If you then plot a force vs displacemnt graph, you will notice it only obeys Hooke's law for a range of forces, then begins to diverge from the ideal behavior. Thus, if we are using Hooke's law to analyze a spring, we should keep in mind that at some larger stretch length Hooke's law will no longer give us a percise result. Ideal springs are also assumed to be much less massive than the objects attached to their unfixed end; we often call this scenario a massless spring. This approximation lets us ignore the kinetic energy of the spring as it stretches and compresses since it would be a negligible contribution to the massive object attached to its free end. Lectures Page 2

3 Lectures Page 3 Before we define the spring potential energy, we should first go through and calculate the work via the hard way. Remember our definition of work is only valid for a constant force. The spring force is not a constant force so we must seek a different approach to finding the work done by it. Recall if we plot a force vs position graph the area under the curve will return the work done by that force. EXAMPLE: Consider an ideal spring attached to a wall at one side with a box of mass m pushed up against the free end as shown in the figure below. The box is initially compressing the spring some value Δx. As the spring pushes the box across the table, the force from the spring on the box is shown in the graph below. Consider a system of just the box, what is the work done on the box by the spring as it moves from its compressed position to the equilibrium location? As it turns out, no matter the path the box takes, the work from the spring will always be the same value for a given initial and final location. For example, consider the same example problem above, but now the spring continues through the equilibrium location, and stretches to some x 2 location, then travels back to the equilibrium location. We calculate the total work that the spring does on the box as it goes from 0 to x eq to x 2 and then back to x eq the same way as above, construct the force vs displacement graph and sum up the work from each segment.

4 Lectures Page 4 Notice how the work done by the spring is the same as going from 0 to x eq as it is going from 0 to x eq to x 2 to x eq. This is analogous to the work from the force of gravity. It is only dependent on the initial and final locations; The spring force is a conservative force, and the work due to the spring force is conservative work. If the spring force is external to your system, all you need to do is use 1/2 k Δx 2 for the work due to the spring force. Now let's look at a system that includes the spring within our system as shown below. Since the spring force is now internal, we define a spring potential energy. So the work kinetic energy equation for the above system would simplify the following way What did we do? The kinetic energy of the system is the sum of the kinetic energy of the box and the

5 Lectures Page 5 spring, but the spring is an ideal spring so it's mass is much smaller than the box and the kinetic energy is also much smaller, basically zero. The spring is an internal force which is also conservative, so the work internal is the negative change in spring potential energy. Finally, the external work is due to gravity (since Earth is not in our system), and the normal force (since the table is not in our system), but both external forces are perpendicular to any displacement the box undergoes. PRACTICE: Consider the graph below showing the magnitude of a spring force of an ideal spring vs displacement from equilibrium. What is the spring constant of this spring? PRACTICE: Plot the spring potential energy vs displacement from equilibrium on the graph below.

6 Lectures Page 6 PRACTICE: Consider an ideal spring that is initially compress so that the spring potential energy is 5 J. If the spring constant is 15 kg/s 2, what is the change in spring potential energy if the spring is then stretched 50 cm past the equilibrium location. Questions for discussion: (1) (2) (3) What role does the negative sign play in Hooke's law? If the change in spring potential energy is negative, do we know if the spring was initially compressed and left to stretch out, or if the spring was initially stretched and left to compress? A spring with a spring constant of "k" is replaced with a different spring with a spring constant larger than the initial spring. How does this new spring change the dynamics of the system?