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(III) A skier of mass $m$ starts from rest at the top of a solidsphere of radius $r$ and slides down its frictionless surface.(a) At what angle $\theta$ (Fig. 36) will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle?FIGURE 36 Problem 28

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A) $48.2^{\circ}$B) $\cos \theta_{\mathrm{crit}}=\frac{v_{\mathrm{crit}}^{2}}{r g}$ greater

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

University of Washington

University of Winnipeg

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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.

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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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Okay, so the setup is as follows. You have the skier. Um um I'm just showing this one section of the of the sphere skier is on a slow Is Adam Ingles data from the top? Um, and so there's ah, force of gravity acting downwards of normal forced upwards. So, um, if you want to resolve forces, um, the radio force Ah, if you'll remember, call is M v squared over r ah. And that will be equal to the net force here. So that force here is, um, mg co sign data minus, uh, fn are normal for us. Okay, um and so has this. Um, So for the critical velocity are the critical angle. Fada, You want point at which offend is zero, but skier is just touching the surface on. There's no normal force yet, so I found a zero would imply that v squared. It's called a secret for critical velocity. The square recruit squared R is she co sign, uh, state a critical angle. Um, and so co sign of the critical angle is ah, the critical velocity squared over gr and we'll call that equation one. Then we use energy conservation. Okay, so he initially will be cool that your final, um so v want initially there's no velocity of you wanted zero. Uh, you find the relationship between Why, um huh. And are, as with little chicken arms tree wise. Just negative. That's increasing downwards. So why is positive Downwards? Why is negative? Uh, are wise. Increasing downwards is negative Are times one minus co sign data. Okay, this is from simple trigonometry on. So why one is negative Are times one minus co sign zero. What? Its coastline. Zero. It's one. So why one is zero? So initially you have no kinetic energy and no potential energy. Okay, so zero because final kinetic energy 1/2 and be one square. Uh, plus final potential energy, which is em Jew times. Why, no 1/2 of you to square plus mg times y two. And why two is again One minus. Co sign Saito s. So we have that Weise view two squared is so the ends Cancel on the seaview to square. Therefore, is, uh, two g r times one minus co sign data. Okay. Um, therefore, we can substitute this in, uh, equation one we have is co sign of critical angle is equal to just this To JI are times one minus co Sign of the critical angle, Uh, over flips over. And she are. And so the gr is cancel. We're left with co sign. Data critical is equal to two minus to co sign Saito. Critical. Therefore, the critical angle. Our co sign data critical, is to over three. So take the Arco sign of two or three and you get that. The critical angle is 48 0.2 degrees, all right? And finally, and part B. You want to know what happens when fiction force that when fiction forces added you have that zero equals 1/2 on V once? Quite right. The kinetic energy replaced the potential energy. Um minus. Okay, Uh, plus, um, work done by fiction. Okay. And so, whatever this v two is, uh, well, um, but there will be a subtraction in the view to term here, um, by friction work done due to friction. And so he too, will go down. Be too. Decreases be to decreases. As a result. Uh, therefore, you can see from equation one that when beauty goes down, co sign data crit critical angle increased goes up and there directly proportional. Therefore, your critical angle Ah is larger in that case, and that's it.

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