Circular and Satellite Motion Legacy Problem #9 Guided Solution

A good problem solver reads the problem carefully and develops a mental picture of what's going on. Oftentimes, a diagram is drawn to represent that situation. A good problem solver identifies known and unknown quantities, and oftentimes even records that information right on the diagram. A good problem solver plots out a strategy to get from the known to the unknown quantities. This is most often dependent upon a good understanding of what quantity is related to what quantity. Here we have a problem about a bucket that's tied to a string and whirled in a vertical circle. We're asked some questions about the bucket at the top of the circle and at the bottom of the circle. In my solution, what I've done is I've divided the page into two halves, one side being the top of the circle and the other side being the position of the bucket at the bottom of the circle. For the left side of my page, what I've done is drawn a section at the top of that circle, and I've put a dot at the 12 o'clock position. I've labeled for that dot that v equal 3.9 meters per second and r equal 1.3 meters. I wish to find the A, the F net, and the tension force. To calculate the A, what I've done is taken the speed of 3.9 meters per second, and divided it by the radius of 1.3. I get about 11.7 meters per second squared, and I understand this acceleration to be towards the center of the circle. Now if I look at my diagram, towards the center would be down. The net force is just m times A, and I'm told the mass of the bucket and the water within is 1.8 kilograms. So 11.7 times 1.8 gives me 21.06 newtons. Once more, I know the direction of this net force is towards the center of the circle, which is down. Now I drew a free body diagram for the bucket. I'm thinking of an upside down bucket above the center of the circle. Gravity, as always, is down, and its value can be calculated as m times g. That comes out to be 17.64 newtons. But the other force is tension, the force of the rope. So you have to ask yourself, which way will the rope pull on the bucket when the bucket is at the top? And the answer is, it will pull it down. So the tension force is also down. Now I figured that the net force is 21.06 newtons. That means that the two individual forces must add up as vectors to this 21.06 newtons down. So if the gravity force is 17.64 newtons, the tension force must make up the difference between these two forces. 21.06 newtons minus 17.64 newtons gives 3.42 newtons as the tension force. Now when we look at the other side of the page, the bottom of the loop, I did much the same thing. I began by drawing myself a picture at the bottom section of a circle. I put a dot at the so-called 6 o'clock position. And then I recorded the speed at the bottom, 6.4 meters per second. And of course the radius is still 1.3 meters and the mass 1.8. I used the speed and the radius to calculate the acceleration. I got 31.5 meters per second per second. Direction is important. I asked, which direction is this? Well, I thought it's got to be towards the center of the circle. Which way is the center of the circle? Well, looking at my diagram, I figured the center of the circle is up. So the 31.5 meters per second per second acceleration is an upward acceleration. And when I multiply it by a mass, I get an upward net force of 56.71 newtons. Now I drew a free body diagram. As always, gravity is down. And its value is still 17.64 newtons. The tension force has got to be up. The center of the circle, the hand that's holding the string, is right there in the center of the circle, pulling up on the string and up on the bucket. And so the tension force is up and the gravity force is down. And together the two forces have to add up to 56.71 newtons up. That means that my tension force must be much, much greater than my gravity force. It has to be so much greater that when you add the negative gravity to the positive tension, you end up getting 56.71 newtons. I can find this tension force if I add the 17.64 newtons to the F net. And in doing so, I get 74.35 newtons.