The figure shows two different ways of combining a pair of...

The figure shows two different ways of combining a pair of identical springs, each with spring constant k. We refer to the top setup as parallel and the bottom one as a series arrangement. Instruction (a) For the parallel arrangement, 1. Draw a free body diagram and draw all the forces acting on the connector piece on the left. 2. write down the net force equation based on the free body diagram. 3. Using the fact the F = - k x, determine the equivalent spring constant of the whole setup. Explain whether the combined spring constant should be interpreted as being stiffer or less stiff. (b) For the series arrangement, 1.Draw free body diagrams and draw all the forces acting on the connector piece on the left and the joint point of the two springs. 2. write down the net force equations based on the free body diagram. 3. Using the fact the F = - k x, determine the equivalent spring constant of the whole setup. Explain whether the combined spring constant should be interpreted as being stiffer or less stiff.

Transcript

00:01 So in this question we are given two situations where two springs of equal spring constants are in the first case attached to a bar, both attached separately to the bar and the other end is attached to a wall and we are pulling with a force f.

00:19 And in the second case they are attached to each other and their extreme ends of the combination are attached to bar and the wall and the bar is being pulled by force f.

00:29 So this is a parallel spring arrangement.

00:33 And this is a series spring arrangement.

00:37 So we are supposed to find out the draw the free body diagram for both cases and write the equations for the two cases and find out the equivalent spring constant if this were replaced by a single spring.

00:54 And same here if this was replaced by a single spring.

00:57 What would the k -e -b? so let's start with the parallel arrangement first.

01:03 So just to make our analysis a little bit more general, we we will assume that the springs have spring constants k1 and k2.

01:15 And finally we'll put k1 equal to k2 equal to k and then find out the value of the equivalent spring constant.

01:19 But to begin with, we'll do this.

01:21 And so if you draw the free body diagram of this par, you have force f on one side.

01:29 And let's say that there is a displacement x for the system.

01:35 And in which case this x is displacement for spring one and spring two because both of them are moving with the same distance because they are attached separate to the same bar.

01:44 That means that the force on this side to the spring would be k1x and k2x.

01:52 So here the force balance is very easy.

01:54 So for the parallel case, f is equal to minus k1x plus k2x.

02:05 And we are taking this minus sign because we are signifying that this force will always be opposite to the direction of the force from the spring.

02:15 So from here it's very easy to see that we can combine this and say that force is equal to minus k1 plus k2 times x and if a general equation for a spring is that f is equal to k minus k equivalent times x.

02:33 From here we can find out that k equivalent is basically k1 plus k2 in this case or in the special case when k1 is equal to k2 is equal to k, we find that k equivalent in the case of parallel is 2k.

02:48 So the equation becomes f is equal to minus 2k x.

02:53 So this is your part a for parallel case.

02:58 And so this is simple because this is two different forces acting on the bar.

03:03 So it's easy to combine them in a free water diagram.

03:06 Now let's look at the other case.

03:09 Let me draw it all the way down here.

03:11 So this is the bar.

03:13 This has force f.

03:14 Now the important thing to note here is that there is a total displacement of the system...

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