 Hey everybody, welcome to Tudor Terrific. Today, I'm going to do a physics problem for you involving circular motion and this problem involves Tarzan over here and he is swinging on a rope over a ravine. The rope is 8.5 meters long, his mass is 98 kilograms, and we're told he could pull on that rope to stay on the rope with a maximum force of 2,000 newtons. The problem asks us at the bottom of his trajectory here, this partial circular trajectory, what is the maximum speed he could tolerate being able to pull with that force? Now let's determine that by looking at a zoomed-in picture of where he's contacting the rope. He is in essence pulling down on the rope and to hold his weight up and the rope, go into Newton's third law, will pull with an equal opposite force of tension back up on him as long as he can maintain a good grip. Now that tension force is acting on him and he is pulling down on the rope. So this is not part of his free-body diagram, but this is. So the tension force points straight up. Now he will be swinging along here, and so we have to understand what force is causing him to move in this trajectory. Let's look exactly at the bottom of the trajectory at his free-body diagram. Here he is, he's a dot. Gravity points down, which is, you know, kind of the problem which causes this to be a strenuous activity to hold onto this rope, and upwards is a force of tension. Now in order for him to swing in this direction and not fall straight down, the tension force has to be greater than the force of gravity. And so we realize now that the net force is upwards. Not just upwards, throughout his entire trajectory, the net force points in the direction of the rope approximately. And so this net force is the centripetal force required for this motion. Now this is the equation we're going to use to find the velocity, but we had to figure out what FC was. It is the net of these two forces. So we can write force of tension minus force of gravity equals MV squared over R. Okay, so we need an equation for this. Well, I know the tension force is 2,000 Newtons, because it's equal and opposite to the pull force that he generates over there. So I can put 2,000 Newtons in for that, and then I can subtract his gravity. So his gravity will be mass times 9.8. So 98 kilograms times 9.8 meters per second squared. So if we compute this really fast, we will get the following. A net force of 1039.6 Newtons. I'm going to keep all the digits right now, so I'm not double rounding. So this is equal to MV squared over R. And now our job is to solve for the velocity. Well, that will involve moving the radius of rotation to the other side. It will involve dividing by the mass and then square rooting. So V equals 1039.6 square root, by the way, times the radius of our circular path. Remember the whole reason we could do this is because this is somewhat circular. It's a portion of a circle. This is a 8.5 meter radius of that rotation, the length of the rope, and divided by his mass, 98 kilograms. So let's compute this now. When I compute that, I get 9.4957. So that, if we look at the initial conditions here, we have only really one sig fig for that and two sig figs for these two numbers, the mass and the radius. So what I'll do is I will allow two significant figures here, just so I don't have to deal with just one. Assume this is a little more accurate than just one sig fig, and I get 9.5 meters per second. So that's the maximum speed he can tolerate, assuming that we are maxing out our centripetal force. And this is quite a fast speed for someone swinging. If you think about that, that's 9.5 meters he's moving every second. If he moves faster than that, this V will go up requiring a larger centripetal force. He would have to generate a larger tension force in order to meet the centripetal force required for that motion at that velocity. So that's why it says max V that he could tolerate with 2,000 Newton. Alright guys, thank you so much for watching that video. This is Falconator signing out.