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Analysis: Pure Twist – the couple

We started to look at pure twist in Week 3. Now that you have understood more about forces on rigid objects we can look at couples in more detail.
SPEAKER: We’ve already explained the effect of a couple on a rigid body. You know that no matter where it is applied, it has the same effect. Here we will see how it all works now you can use equilibrium to find forces on rigid objects. Remember, if there are two equal and opposite forces F acting on a rigid object, and they are separated by a distance a, then the forces cancel out and they leave a pure twisting effect– a couple. Earlier, we found that the moment due to a couple is a times F, whatever point you choose when you calculate it. Now we will revisit the experiment on pure twist.
We started by putting 20 washers on each of the two hooks on the large cardboard shape. When we added 10 washers to the lower right link, the large cardboard shape rotated but the height of its centre didn’t change. Then we transferred five washers from the hook on the right of the large shape to a hook on the left. Hey, presto! The large shape became horizontal again. We measured the length of the transducers and they were the same as when we started, confirming that there was no change in the net force on the shape. Now we will apply free body diagrams to the analysis. We will use the free body diagram on the left.
Applying vertical equilibrium to the large shape gives us this expression. The linkage applies two equal and opposite forces of magnitude W3 divided by 2. So they cancel and needn’t appear in the equation for vertical equilibrium. So we can use this simplified equation. Now we apply rotational equilibrium. We can choose any point on the cardboard shape. I’ll take the point where the force, Fb, is applied. You take sum of the moments about b equals nought positive clockwise. And we get this equation. The large cardboard shape rotated until the change in the forces on the transducers balanced out a twist imposed by the linkage. Let’s explore the values of forces Fa and Fb.
Remember, the load started off as 20 washers on each hook and 10 washers was on the lower link. From our analysis of vertical equilibrium, we get Fa plus Fb equals 20 plus 20, in washer units. From rotational equity, then we get Fa equals 20 plus 10 divided by 2. That gives us Fa equals 25 washers, and, by substitution, Fb is 15 washers. When we transferred 5 washers from W1 to W2, the main link became horizontal again. Let’s see why. If we change W2 to 25 And W1 to 15, we still get Fa plus Fb equals 40. And we still get Fa equals W1 plus W2 divided by 2.
So with our rearranged loads, we get Fa equals 15 plus 5, which makes 20 washers, and Fb equals 20 washers, as well. The forces in the transducers have returned to their starting values. So the extension must be the same as it was before. But loads W1 and W2 are changed. We transferred washers to do this, and they create a twist that balances out a couple from the linkage. So what have we learned from this? Our linkage applies a pure torque to the large shape of W3 times C divided by 2 clockwise.
This does not affect the sum of the forces in the vertical direction, but the pure torque causes the shape to rotate until the changed forces, Fa and Fb, balance out the torque. If we move load from W1 to W2, we won’t change the sum of forces in the vertical direction, but we will change the equation for rotational equilibrium. If what we swap is correct, we will bring the cardboard shape back to the horizontal again. We have looked at this in great detail. Just in case you’ve lost the main points along the way, here they are again. Two equal, opposite forces, F, a distance, a, apart generate a couple. That’s a pure twist of F times a.
The external effect of a couple on a rigid object is the same wherever it is applied. Now learn some more conventional interactions in the next video and you will be set.

We started to look at pure twist in Week 3. Now that you have understood more about forces on rigid objects we can look at couples in more detail.

Talking points

  • In what ways have you understood more about couples now that you have an understanding of forces on rigid objects?
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