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A 56 -kg student runs at $5.0 \mathrm{m} / \mathrm{s},$ grabs a hanging rope, and swings out over a lake (Fig. $45 ) .$ He releases the rope when his velocity is zero, $(a)$ What isthe angle $\theta$ when he releases the rope? (b) What is the tension in the rope just before he releases it? ( c) What is the maximum tension in the rope?

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A) $29^{\circ}$B) $480 \mathrm{N}$C) $690 \mathrm{N}$

Physics 101 Mechanics

Chapter 8

Conservation of Energy

Work

Kinetic Energy

Potential Energy

Energy Conservation

Moment, Impulse, and Collisions

Cornell University

Rutgers, The State University of New Jersey

Hope College

University of Sheffield

Lectures

04:05

In physics, a conservative force is a force that is path-independent, meaning that the total work done along any path in the field is the same. In other words, the work is independent of the path taken. The only force considered in classical physics to be conservative is gravitation.

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

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So we have a person here who were swinging and there's a rope hanging here and he runs for the velocity off five meters per second, catches hold of the rope and be of water here. And then finally he swings to this location and makes an angle off Terror and his velocity at this point become zero and the rope link this 10 meters. So we need to find out the angle. You know, when he releases the roof. So we can assume that at this point wise equals to zero. And V one is five meters for a second, and Vito's equals to zero. And we just need to find out the height this height, which is by two. So we applied the law of conservation of energy are these two locations one locations to so well right You won equals e too have and even squared plus m g buy one should be equals to have m b two squired less mg vie to. So we quit this equation half m v one square because this is zero So therefore his heart and we went square on the left equals mg by two because the velocity become zero dislocation. So therefore we get by two is equals. To be one squared over to G is basically equals to this Etch this vertical height. So now in this location here, if you find the co sign of this angle, so co sign of this angle should be equals two. No, this is EJ than this distance is l minus h in this right angle triangle call sign of Peter. Is this divided about the hyper genius? So they're focusing on a feature Is L minus X divided by hill, which is actually equals to one minus edge by L, which is equals to one minus V one squared by two g l So because in this equation we have edges equals two If you can see here in this equation here, so action by L So edge by l which is this is equals to be one squared by two g. So if he quit this questions we have cost hater. So tater is equals to cause in verse one minus V one squired by to geo. So you're plugging in this equation. We'll get 29 degrees asked the value of terror. The B part is that they asking? What attention the room just before he releases it. So for the B part off the equation here. So this angle is Keita and the tension is FT. And we have the four sect ing as mg. So we have this angle s Ted off so we can find out the net force along the radius, which is equals to the tension minus mg cost ada. So at this point at this point than it force which is equals to zero. So therefore the tension force is equals two mg course I inte which is equals two 56 times. Sorry. 56 times 9.8 Dames cost 29 and that should give you 480 Newton's. Now, at this point, the velocity zero so f t minus m D'Costa zero. And it is actually calls to Embry squired Bauer the centripetal force. But since there is no velocity at this point, the centripetal force must be equals to zero hands Net forces, you know, therefore I put it equals zero. So therefore attention. He's going omg cost later on, we can capture attention. As for 80 mutants. All right, what is the maximum tension in the rope. So rewrite the Newton's law as net force is equals. Two f t minus mg Constanta is equals two m v squired by our so therefore FT is equals two m v squared by our place mg cost header. You know, ask this angle increases Sorry decreases their tension increases and at the speed increases the tension increases. Now both effects are the greatest at the bottom of the swing, so that is where the tension will be at its maximum. So therefore we have f d mex is equals. Two mg causes zero and the bottom of the spring plus M v squired by our where the tension is going to be the maximum. So we just simply get 56 times five squared, divided by 10 and that should be equals 26 90 Newton's as the maximum tension at the lowest point of the rope

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