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A homogeneous hemisphere of radius $r$ is placed …

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Problem 111 Easy Difficulty

A homogeneous hemisphere of radius $r$ is placed on an incline as shown. Assuming that friction is sufficient to prevent slipping between the hemisphere and the incline, determine the angle corresponding to equilibrium when $b=10^{\circ}$.

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Physics 101 Mechanics

Vector Mechanics for Engineers: Statics and Dynamics

Chapter 10

Method of Virtual Work

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Video Transcript

Okay, so here we have our uniform solid sphere, it's going to be rolling down this incline and we want to know how steep this England needs to be. So this angle fada right here so that our acceleration down the incline is a value of 1/10 of one G. Some 10.1 G. To do this. We're going to use conservation of energy. The the energy that it has, the potential energy that it has. If we release it from a height H it's going to be M. G. H. So this is some zero potential down here and that will be the total pool of energy we have. And then we'll say that once we reach the bottom that will have a total transfer of potential energy into kinetic energy, that is the translational kinetic energy would have M. B. Square Plus the rotational kinetic energy 1/2 i omega squaring. And now we can figure out a couple of things. The first one we're gonna do is we're gonna translate these into um masses and translational velocities because that will make things cancel out for us. So The moment of inertia of a solid sphere is 2/5 M. R squared, where M. Is the mass of our object and r. Is the radius of that sphere. And then also this omega squared, this is tied into how fast it's rolling. So mega is the the rotational velocity of the spirit. That's going to be equal to the over our where B. Is the translational speed. And so if we plug these two things in here, we're going to get one half times 2/5 M R squared. That's for the eye. Then the over our quantity squared. Well if you square this quantity it becomes V squared over R squared. And if we have an r squared here and in r squared in the denominator, those are squares are gonna cancel out. So now let's rewrite this. We've got MGH on the left side, we have one half and B squared Plus 1/2 times to 5th. So we're just gonna give us um 2/10 or 1/5. Let's let's call it exist. One says M. V squared makes sense. And now you'll notice that there's an M. In each term so we can cancel that out. So the mass of this sphere is irrelevant to our energy because the mass will cancel out any turn. And so we have G. H. Is equal to one half of the squared plus 1/5. The squared, well one half. So we can in fact, throughout the V. Squared we have one half plus 1/5. So this is 5/10 and 2/10. So we get 7/10, 7/10 B squared sentence. It could be now the this gets us part of the way there. So now we have um we have an ancient here, we've got this be square. This is the final velocity reaches the bottom and we need to we need to figure out this angle still so we can use uh the the changing speed since this is our final speed and we're releasing this thing from rest, we can use that to figure out what the angle needs to be. So um just over here we're going to say that We're looking at the acceleration of this thing, that's the whole point. We wanted to get the acceleration to be .1G. So we need to figure out what that angle needs to be. So this final speed here b squared has to equal be not squared which is going to be zero since we're starting this from rest Plus two times a times delta X. So this the distance parallel distance that it travels. But this delta X. This distance that it travels along here, we can figure out from the angle and the height. So between this this is a right triangle here that dealt X. That's given to us by the sign sign of that angle. His opposite over ipod news. Well the opposite his page and high pop news is delta X. So delta X. Is actually um H. Over signed data. So we can plug that in. Let's make that could win. So this is going to be v squared is equal to two Times A. Which we want to be .1G times delta X. Which is H. Overseeing paid like. So, so this is actually point to G. H. Over the same data is V squared. So let's plug that back in here and see what we get. So we're gonna have a G. H. On the left side And we have 7/10. And then for the V. Squared we're gonna plug this in. So this is going to be 02 G. H. Over santa. Well these have gotta G. H. On both sides. Let's close that pregnancies. Got a G. H. On both sides of the dhs cancel out. And I've got 7/10 times 2/10 Or 14 100s. There was 0.14 And the one on the left side, 0.14 or 750. It's uh over signed data. So we marked by saying data by both sides. This goes away let's just clean this up a little bit. We have signed paid a equals this number 14 and we can use the inverse sine to find that angle. So we have come on, hurry up. Data is going to be equal to the inverse sine of that number 0.14, Which ends up being just about 8°. So the angle that we need, our inclined to be at in order to achieve that .1 G acceleration, it's going to be 8°. Now, part B of this problem says uh we're also going to release a block from rest. And will that have a greater than less than or equal to um acceleration? And the block will actually have a greater acceleration? The acceleration of the block will be greater than the acceleration of the sphere. And the reason has to do with this right here. The if we release a block, the only energy that it will have is this amount. So the potential energy that of the block. Let's say that they are the same mass. The potential energy of the block will release it will equal as kinetic energy, which is the bottom. And that's the end of the story for the block. As long as there's no friction, the sphere has this additional rotation that we need to account for it. So it's going to have some of this potential energy gets converted into rotational energy, rotational kinetic energy. And so its speed will be reduced because of that. And if its final speed is reduced, that means that its acceleration will be reduced. And so that block will have a greater acceleration down the ramp than the sphere.

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Ferdinand Beer, Jr., E. Russell Johnston, David Mazurek

Vector Mechanics for Engineers: Statics and Dynamics

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