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A solid uniform marble and a block of ice, each with thesame mass, start from rest at the same height $H$ above the bottom of a hill and move down it. The marble rolls without slipping, but the ice slides without friction. (a) Find the speed ofeach of these objects when it reaches the bottom of the hill. (b) Which object is moving faster at the bottom, the ice or the marble? (c) Which object has more kinetic energy at the bottom, the ice or the marble?

a) $v=1.19 \sqrt{g H}$$v_{\mathrm{ice}}=1.41 \sqrt{\mathrm{g} H}$b) The velocity of the marble is 1.19$\sqrt{g H}$The speed of the block of ice is 1.41$\sqrt{g H}$Thus, the block of ice is moving faster at the bottom.c) both the ice and marble have same kinetic energy.

Physics 101 Mechanics

Chapter 9

Rotational Motion

Physics Basics

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Hope College

University of Sheffield

University of Winnipeg

McMaster University

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in this problem, we're going to apply the conservation of energy to both the marble and block of ice. The mo inertia of the marvel since it's a solid sphere is equal to two fists times the mass of marble times are squared. We have the relationship that the linear velocity of the marble is equal to the radius of the marble times the angular velocity, the marble. Since that rolls without slipping during this problem, gonna let the plus Why direction be upward? I'm going to let why is equal to zero be the value at the bottom. This fact implies that the initial height is H and the final height zero. And so now I can apply the conservation of energy for both the objects. It's equal to the initial connect energy. Let's see. Initial potential energy equals final kinetic energy, plus final potential for both objects. Initial kinetic energy is equal to zero, and the final potential is equal zero. So, really, this is the conservation of energy for a situation in hand. So what's first applying for the marble? The initial potential is just equal to M g times initial high, which is a tch so have MGH is equal to 1/2 film times the velocity of the marble square. This is the linear, kinetic energy we need to add it to the rotational connect energy which is 1/2 I omega squared But Omega squared by this law here is the linear velocity over r squared. And so whenever I saw Sue that in I get it 1/2 own VM square. This term doesn't change. And then I z equal to two fist. So I get that to scan sold by the way and then, Oh, make us where it is, Veum, Over our square. Now we have in our square in the denominator, so that cancels with this r squared. And then we have 1/5 m b m squared that combines with the 1/2 of'em squared and it makes seven tense. When you get that common denominator 7/10 film beams were. And so if ldh is equal to this, we can solve this for the and we did 1.2 spirit of the age of that. This is for the marble. We need to do the same thing for the block of ice. In this case, the initial potential energy is the same. It's MGH, since they both have the same mass in the real story at the same height. In this case, there's no rotational kinetic energy. There's only linear, kinetic energy, so this is just equal to 1/2 in velocity of the block of ice squared. And so this is easy in a cell for the block of ice was bossy. It's equal to one point for one times square of G H. And so those two things combine are the answer to part a Barbie and switch. One is moving faster at the bottom, the ice or the marble. And the answer is that the block of Isis, since one point for one, is bigger than 1.2. This may seem surprising, but it's because the block of ice has no friction, whereas the marbles rolling. And so a lot of the connect energy goes into the spinning action of the marble. But the block of ice has no friction, and it can't spend so all of the energy gets translated into the when your kinetic energy it's over, parent part be the ice moves faster at the bottom and that everywhere throughout ice moves faster for Park Si. We want to compare their final kinetic energies and Sophie recall for both of these objects. The Conservation of energy said the initial potential energy is equal to the final kinetic now, since the initial potential energy is the same, the final kinetic energy must also be the same. And so kf is same for both objects and that's final.

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