Analysis: Pure twist – the couple

Here’s something quite elegant. What if there are two equal and opposite forces, F, separated by a distance, a? The forces cancel out, and they leave a pure twisting effect, a couple. The moment due to a couple is a times F, whatever point you choose when you calculate it. Sometimes we call it a pure couple, uncontaminated by any net force. You could try taking moments about different points. Pause the video and calculate the total moment of the two forces at, one, a point on the line of action of each force, and two, a point halfway between them.

You should get a times F, clockwise each time. Theory says that the effect of a pure couple on the equilibrium of a rigid object is independent of where it is applied. So you just add it in when you apply sum of the moments equals 0. We’ll do that later. Now, let’s look at the experiment on twist. The linkage applies a pure twist to the larger cardboard shape. We’ll follow the forces in the linkage and see how it works. Notice that the two small links are equalisers. They equalise the forces on each end of a link. Take moments about the centre of each one if you are not sure.

Starting with the forces on the lower right-hand link, can you see that the string on the right end pulls down the right end of the link above it? This generates an equal downward force on the left end of the top link. By Sir Isaac’s Third Law, action and reaction, the string at this end pulls up on the large cardboard shape. Now, we’ll analyse our experiment formally. We started with just the large shape loaded with 20 washers on each hook. This made it go lower, but without any rotation. Next, we loaded the bottom small link with 10 washers. This put a twist on the large shape that caused it to rotate clockwise.

Then we transferred five washers from the hook on the right of the large shape to the 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’ll apply free-body diagrams to the analysis. We’ll use the free-body diagram on the left. It’s quite subtle, so don’t worry if it doesn’t make sense straight away. Applying vertical equilibrium to the large shape gives us this expression.

The linkage applies two equal and opposite forces of magnitude W3 upon 2, so they cancel and needn’t appear in the equation for vertical equilibrium. So, we get this simplified equation. Now we apply rotational equilibrium. We can choose any point. I’ll take the point where the force FB is applied. So we take sum of the moments about B equals 0 positive clockwise, and we get this equation. The large cardboard shape rotated until the change in the forces in the transducers balanced out the twist imposed by the linkage. Let’s find the new values of FA and FB. Remember, the loads were 20 washers on each hook and 10 washers on the lower link.

From our analysis of vertical equilibrium, we get FA plus FB equals 20 plus 20 in washer units. From rotational equilibrium, 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 five 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 W3 on 2. So, with our rearranged loads, we get FA equals 15 plus 5, or 20 washers, and FB equals 20 washers, also.

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 the couple from the linkage. So what have we learnt 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 we swap the correct amount, we will bring the cardboard shape back to the horizontal again. The main thing you need to remember is that 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 are set.