Figure 8-41 a applies to the spring in a cork gun (Fig. 8-41 b ); it shows the spring force as a

Figure 8-41 a applies to the spring in a cork gun (Fig. 8-41...

Key Concepts

Kinetic Energy

Kinetic energy represents the energy of motion and is calculated as one-half the mass of the object multiplied by the square of its velocity. In problems involving energy conversion from a deformed spring to a moving object, this concept helps in deducing the speed of the object based on the amount of energy transferred from the spring’s potential energy.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is deformed from its equilibrium position. In many cases, this energy is given by a quadratic relationship to the displacement (as in Hooke's law), but more complex force-displacement relations can also arise. Grasping this concept is key for determining how much energy is available to convert into kinetic energy when the spring is released.

Work Done by a Variable Force

When the force acting on an object is not constant but varies with displacement, the work done by this force is determined by calculating the area under the force versus displacement curve. This concept is crucial when dealing with systems such as springs, where the force changes as the spring is stretched or compressed, requiring integration or geometric interpretation to determine the total work performed.

Conservation of Energy

This principle states that in a closed system, energy is neither created nor destroyed but is instead converted from one form to another. In mechanics problems like the one described, the potential energy stored in a spring is converted into the kinetic energy of the propelled object. Understanding this concept is vital to analyzing systems where forces do work, ensuring that the total energy before and after an event remains constant, aside from any non-conservative work.

Force versus Displacement Graph

This graph illustrates how the force exerted by a component, like a spring, changes with its displacement from equilibrium. The area under this curve corresponds to the work done by the force. Understanding the graph is essential as it visualizes the relationship between force and displacement, allowing for effective calculation of energy transfer in systems where forces are not constant.

Transcript

00:02 Okay, everyone.

00:04 So now we're looking at question 28 from chapter 8, and this is a spring in a cork gun question.

00:11 And so there is a graph right now.

00:14 So let me draw the graph for you guys.

00:16 I'm not going to draw the exact coordinates.

00:19 Actually, you know what? i'm going to do that.

00:21 I'll draw the exact coordinates as shown.

00:25 2, 4, negative 0 .2, negative 0 .4, negative 0 .4, negative 2, negative 4.

00:34 I just want to speed this up.

00:36 And so your y -axis is labeled force, and force is in newtons, whereas your x -axis is labeled basically displacement, and that's in centimeters.

00:47 So remember, we're going to have to convert that later on, too.

00:50 So the question wants to know, since figure, here 8 applies to the spring in a cork gun.

00:58 It shows the spring force as a function of the stretch or compression of the spring.

01:02 The spring is compressed by 5 .5 centimeters.

01:07 Let's keep that in mind.

01:08 Spring compressed, compressed 5 .5 centimeters.

01:15 Compressed equals 5 .5 centimeters.

01:19 Let's actually highlight this, okay? and now we know, what else do we know? we also know that it's used to propel a 3 .8 gram cork from the gun.

01:32 Okay.

01:33 So cork mass is equal to 3 .8 grams.

01:38 So it appears that we actually have a lot of like converting to do in this problem because none of the units are really matching up.

01:45 And so a wants to know what is the speed of the cork if it is released as the spring passes through, it's relaxed.

01:53 Okay, once again, we've done this type of problem before, right? this is just a conservation of energy problem.

02:04 So let's keep in mind that the conservation of energy principle says that the sum of your beginning energies is equal to the sum of your final energies.

02:12 So which energies do we have right now? well, beginning we have the potential energy of the spring, right? so we actually have pesp.

02:22 And then what do we have at the end? well, we have kinetic energy because this cork is being like shot out.

02:29 So, k .e.

02:32 Yeah, kinetic energy.

02:33 Fine.

02:34 Okay.

02:36 So all you have to do is literally write this out.

02:38 What is the potential energy of a spring? well, we know this.

02:41 Okay.

02:41 Potential of the energy of the spring is equal to one half k delta x squared.

02:48 And kinetic energy, everyone knows, 1 half m v final squared.

02:53 And what do you do from here? you literally solve for v final.

02:57 And we notice something interesting here.

02:59 The one halves cancel out because they both appear the same.

03:04 They both appear on both sides of the equation.

03:07 So now let's solve, let's solve for v final.

03:12 So k delta x squared all over m is equal to v final squared.

03:19 So v final is apparently equal to the square root of k delta x squared all over m why once again? because all of the potential energy that the spring contains is now being converted to kinetic energy...

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