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A “swing” ride at a carnival consists of chairs that are swung in a circle by 15.0-m cables attached to a vertical rotating pole, as the draw- ing shows. Suppose the total mass of a chair and its occupant is 179 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.

3510 $\mathrm{N}$14.9 $\mathrm{m} / \mathrm{s}$

Physics 101 Mechanics

Chapter 5

Dynamics of Uniform Circular Motion

Newton's Laws of Motion

Applying Newton's Laws

University of Washington

Hope College

University of Sheffield

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

03:43

In physics, dynamics is the branch of physics concerned with the study of forces and their effect on matter, commonly in the context of motion. In everyday usage, "dynamics" usually refers to a set of laws that describe the motion of bodies under the action of a system of forces. The motion of a body is described by its position and its velocity as the time value varies. The science of dynamics can be subdivided into, Dynamics of a rigid body, which deals with the motion of a rigid body in the frame of reference where it is considered to be a rigid body. Dynamics of a continuum, which deals with the motion of a continuous system, in the frame of reference where the system is considered to be a continuum.

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here we've defined both the tension and the lost E for a seat on a swing ride. Now we'll start like we do with most of our problems by drawing a free body diagram. If the red dot here is the seat, we know that there's a force of gravity pointing straight down as well as a force of tension pointing in the direction of the string. Right now, the force of tension can be split into two components the white component right here and the X component over here. Now, using this information, we can solve for our attention. So as we can see in the free body diagram, the vertical component of tension is almost balance out the force of gravity. So t y equals and G. Now, when we look at the drawing, we can also see that our attention must be equal to the co sign of our angle data between our seats and the vertical. So t who sent data equals M G and T is mg over coast data. Now we know m we nog and were given our angle as well. So all this have to do here is plug in gonna do appear and we find that our answer is equal to 179 kilograms times 9.8 meters per second squared. You write that little bit neater with 79 kilograms times 9.8 meters per second, squared all over co sign of 60 when we plugged in a drawer calculator We get that the tension is equal to 3508.4 Newtons and if you want to round two the next 10th that would just be 3000 510 Newtons, you wanna double check? You can also look at your units are us when we solve the problem are kilograms meters per second squared which is equal to Newton's. So therefore our answer makes sense and we are in fact working in unit to force. And so 3510 you don't answer to part a Now the problem asked us to find the tension of the velocity. Excuse me that our chair is traveling in. So let's go back to our free body diagram as we can see here on the left the force that's pointing toward the center of our circular motion is the horizontal component of our tension. Therefore, that horizontal component must be responsible for the centripetal force. So t X equals M V squared over our now If the white component of tension is t sign data, then 40 coasted up. My bad then our horizontal component must be key sign of theater and that is gonna be equal to m v squared over R Now someone for V you find that V is equal to he signed data times are over m and now we realize that our is not actually given in the problem But we could determine it very easily using geometry so are right. Is this section that I'm dawning red right now? Um, but since we know our angle theta and we know that our high pop news is 15 meters, we can see very easily that are must be equal to 15 times the sign of data. Since the sound of data is equal to our over 15 by trigonometry laws. So we'll rearrange a rewrite. Rather, our equation for B, which we now find, is equal to t signed data squared times l the length or rope gonna plug in 15 for that in a minute, divided by m and now we're gonna plug in our quantities. So all this have to do is put the store calculator 300 5008 0.4 Newtons. That's attention times 15 are lengths, times sign of 60 squared, all over 179 kilograms. And I'll extend the radical just to make it clear that 179 is also under their move. Me solve, we get that this is equal to 14 point eight 48 and our units work out three meters per second. It makes sense. And so this is our final answer for part B.

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