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Analysis: Adding forces Part 2

You've got the concepts for adding vectors; now it's time to apply them.
In the previous video, we learned about vector addition. In this video, we’ll use it to analyse our experimental data for two forces acting at a point. We’ll find out how the two forces acting at a point share the load. You’ll need this for your design task. We’ll deal in numbers now from our experiment. They are available in the data PDF. If you did the experiment yourself, you can analyse your data, too. To join in, you’ll need paper and pencil and a scientific calculator. A scientific calculator is one that has trigonometrical functions on it. You might have your old scientific calculator, for example, from when you were at school or when someone you know was.
If not, you’ve probably got one on your mobile phone. Find the basic calculator, then turn the phone sideways and see what happens. With many phones, a scientific calculator appears. Here’s our setup. We measured the forces in the transducers and recorded the supported weight. We’ll calculate the combined effect of the two force transducers. By equilibrium, this should be the same as the weight of the pan. We’ll see how close it is. To begin the analysis, draw an FBD for the object shown. The object is the weight pan, string, and a little section of each transducer. Pause the video and draw it for yourself. Here is what it should look like.
The resultant of the two transducer forces must balance the supported weight. To describe the force vectors, we’ll need to know their magnitude and direction.
For the next bit, you’ll need to use inverse trigonometrical functions on your calculator. Whereas the cosine function, for example, tells you the cosine of an angle, the inverse cosine function tells you the angle if you know the cosine. A trigonometrical ratio is, for example, t is the cosine of theta, or t equals cos theta. An inverse trigonometrical ratio is, for example, theta is the angle whose cosine is t, or theta equals cos to the minus 1 t. As an example, to access the inverse cosine on my phone, you need to touch second and use the cosine minus 1 key. Let’s sort out the directions first. You can do some of this yourself.
Pause the video and use the dimensions given, which were measured in an actual experiment, to work out the angles shown in the diagram. Tip– think cosine.
Did you get the answer? If you didn’t get it, here is the trick. From basic trigonometry, cosine alpha equals h over the distance AB. So angle alpha is found by cos to the minus 1 h over AB, giving alpha equals 18.9 degrees. And similarly, cosine beta equals h over distance CB. Angle beta is found by cos to the minus 1 h over CB, giving b equals 33.1 degrees. Now you have found the angles, you can find the resultant of the two forces from the force transducers. You can vector add using the cosine and sine rules, but the scheme that follows is probably easier. One, split the force from each force transducer into horizontal and vertical rectangular components.
Two, add the vertical components to get the vertical component of the resultant.
Three, add the horizontal components to get the horizontal component of the resultant. You will need a sine convention. For example, to the right is positive to the left is negative.
Four, vector add the two components to get the resultant magnitude and angle.
Pause the video and find the magnitude of the resultant and its angle to the vertical, and we’ll use positive is a clockwise angle.
Here are the answers. The vertical component of AB is 35.0 washers. The horizontal component of AB is minus 11.97 washers– minus because it is to the left. Of the vertical component of BC is 18.4 washers. The horizontal component of BC is 12.0 washers. It’s plus because it is to the right. The vertical component of the resultant is 53.4 washers. The horizontal component of the resultant is 0.033 washers. That’s to the right. So we have the resultant is 53.4 washers and the angle is 0.04 degrees. That’s clockwise. Here’s how to get those answers. The vertical component of the force in AB is F AB times cos alpha.
The horizontal component of the force in AB is minus F AB sine alpha– minus because it is to the left. Substituting the data gives 35.0 washers and minus 11.97 washers, respectively. Similarly, the vertical component of the force in CB is F CB cos beta. The horizontal component of the force in CB is F CB sine beta. Plus this time, because it is to the right. Substituting the data gives 18.4 washers and 12.0 washers, respectively.
Now add the two vertical components– that’s 35.0 and 18.4– to get the required vertical component of the resultant, which is 53.4 washers. And add the two horizontal components– minus 11.97 and 12.00– to get the required horizontal component of the resultant, noting the signs. You should get 0.03 washers. It’s plus, so it’s to the right. Here’s how to get the answer for the resultant force and the angle. The resultant is obtained using Pythagoras’s theorem. And you should get F equals 53.4 washers. And the angle is obtained from trigonometry. You should get phi equals 0.04 degrees to the right of vertical. Because the weight goes straight down, the resultant force to balance it must go straight up.
In other words, when you vector add the two loads in the force transducers, the vertical component of the resultant should equal the load in the load pan, and the horizontal component should be zero. Let’s compare this with our measurements. The weight in the load in the experiments was 50 added washers plus the weight of the pan, which was two washers. The total in washers was 52. How do the numbers from calculation compare with this? Some discrepancy is expected. The percentage discrepancy in the vertical force is given by e1 equals 53.4 minus 52 divided by 52 times 100 to make it a percentage. And that gives the discrepancy in the vertical force e1 equals 2.8%.
The discrepancy in the horizontal component as a percent of the vertical force is 0.03 over 52 times 100, which is e2 equals 0.06%. What do you think? Report your opinions on the error using the discussion to answer these questions. Now would be a good time to do some tutorials before you start the design task.

You’ve got the concepts for adding vectors; now it’s time to apply them.

Data from the experiment are given in the Downloads section below. You can use your own data too.

This video will show you how to analyse these data. You will see how well theory and practice agree.

Talking points

  • What do you think about the agreement between our simple experiment and the analysis?
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