8.8

The previous video set the scene for choosing cables and shackles for suspending a loudspeaker. This video will guide you as you calculate the forces that the cables and shackles must sustain. Once you have calculated them, you will select suitable cables and shackles from lists. If you prefer, you can just watch the video, but it’s probably better to be active. Here are the data you will need. They are also available in a PDF file. Some of these data are not required for the calculations, but can be useful in other ways. Here’s what you do. First, draw a free-body diagram of the speaker box from the front, assuming only the front cables are taking the load.

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Label the forces in the cables Fa and Fb, and label the weight w. Pause the video and try it.

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Now, check your free-body diagram against this list. Was your free-body diagram free? This is the most important requirement. Did your free-body diagram include all the forces? This is vital, too. Were your force arrows in the right place? In particular, was the head or tail of the weight arrow roughly in the center of the box? That is your estimate of the location of the center of gravity. Did your free-body diagram resemble the actual item? The dimensions won’t be used for calculations, but they do enable you to draw the FBD roughly to scale. Was your free-body diagram roughly to scale? Drawing an FBD way out of scale can sometimes lead to problems by confusing you.

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If you have any questions, pause and post them on the discussion. Now, to use the FBD. You will see in the next week that you could solve for the forces directly from this diagram, but here we will find the resultant of the two cable forces first. Notice that the two cable forces don’t meet in a point. We get around this by using the sliding vector property of a force on a rigid body. This diagram shows by dotted lines how you can slide the forces to the point where the two lines of action intersect, and then you add them there. Now you are ready to add the two forces in the cables to get their combined effect.

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We’ll use the method we developed in the video on adding forces. First, we found the horizontal and vertical components of the two cable forces. Then, we added the two vertical components and we added the two horizontal components, and then we combined them to find the resultant force. Now we can use this on our loudspeaker. Use the symbols Fa and Fb to write equations for the horizontal component, Fx, and the vertical component, Fy, of the resultant, F. You’ll need to use trigonometry as before, with angle Theta this time. Pause the video and try it.

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Here’s what you should get. Notice that axes define the x and y directions, and also which directions are positive. We have two unknowns, Fa and Fb. To solve them, we will need two equilibrium equations. As we saw in the analysis video, horizontal equilibrium tells us that the horizontal component of the resultant must be zero. That is, the resultant must go straight up. Use this to relate Fa and Fb. Pause the video and try it.

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Here’s the answer. Can you see that you can have determined this result from the symmetry in this case? You can now find the vertical component of the resultant in terms of Fa or Fb. Pause the video and try it.

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Here is the answer.

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Before we move on, consider the following questions. Did you understand the comment that you could have known that Fa equals Fb from symmetry?

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Is it obvious how the axes specify the positive sense for vertical and horizontal components?

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Did you get the answer straight away? What problems did you encounter?

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You might like to post your thoughts on the discussion. Now redraw your FBD, replacing the two cable forces with their resultant. Don’t forget to include the axes that define the sign conventions. Pause the video and try it.

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Here is the answer. It doesn’t matter what you specify for your sign convention. For example, you could have specified left is positive, but it is important that you follow it in your later working. This is where you apply vertical equilibrium. For vertical equilibrium, adding the vertical forces on the FBD must give zero– which in this case comes to the same thing as saying that the resultant of the two cable forces must balance the weight. Pause the video and write an equation for this, making sure that one, you keep the formant of forces on the left-hand side of the equation equals zero on the right-hand side, and two, you follow your sign convention.

438.7

Here is the answer. You’ll notice that the sign convention we specified showed up as positive, and that the working follows this convention by having the weight as negative. Whatever sign convention you choose, you should get the same value for Fa and Fb. You will need some very simple algebraic manipulation. With mass in kilograms and g in meters per second squared, the units for Fa and Fb will be Newtons. Here is the data for this problem. Using these data, pause the video and find the force in each cable.

491.2

Here are the answers we should get. If you can’t see how to get these values, pause and post a question on the discussion. Now we’ll decide on cables and shackles. The speaker manufacturer specifies a safety factor of 10 on the breaking load of the cable-shackle system, so the cables and shackles will need to sustain 10 times 832 Newtons, or 8,320 Newtons. Each link in the chain must be able to sustain this load. Taking all this into account, pause the video and select suitable items from this list. It is based on a brochure from a yachting supplies company. The list is also available in the downloadable PDF file.

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Here are the selections we made. The required breaking load as a safety factor of 10 is 8,320 Newtons. We selected cable five, which has a breaking load of 10,868 Newtons, and we selected shackle three, which has a breaking load of 12,358 Newtons. Did you agree with our choices? If not, find out why not. The discussion for this video might help. Congratulations. You have completed this item. You might like to keep your working as a record of your activity. Just a quick final point– would you be happy to stand under the speaker if it were suspended by your choice of cable and shackle? Post your response to this on the discussion.