A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator, it is scraping against metal, resulting in a friction force of 0.0200 N at that point. Find its angular acceleration.

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Welcome back everybody. We are taking a look at a toy that is designed in the following way essentially when it is supplied some sort of charge, it stores rotational energy on a spherical ball. And we are told that this spherical ball is pivoted around a frictionless axle, just like This. Now we are told a couple of different things about this situation. We are told that the radius of our sphere is 60 or .06 m. We are told that the mass of the sphere is uniformly distributed with a mass of 890 g or .89 kg. And then we are also told that somewhere in the center there's like a little dent or a little scratch here. And we are told that that causes a friction force of 0.08-0 newtons. Now we are tasked with finding what the angular acceleration is of the ball. I do want to know Tate one more thing. Before we start working with formula here, we are going to need to use an angle Theta. And what is this angle Theta. Well, this angle Theta is going to be, the angle between the axis of rotation and really the the centripetal force that is experienced um via any particles on the outside of the ball or the body of the ball. So this is simply just going to be 90 degrees. Okay, so why do I bring that up? Well, we want to find angular acceleration and I'm looking at a couple of these other values and we are going to use this formula right here. That the moment of inertia of a solid sphere times the angular acceleration is going to be equal to the radius of the sphere, times the friction force times the sine of our angle theta. Since we want our angular acceleration, I'm going to divide both sides by our my moment of inertia. These terms are gonna cancel out on the left hand side and we are left with the that the angular acceleration is equal to the radius, times the friction force, times the sine of our angle data all over the moment of inertia for a sphere. Now we have all of these terms on top but we need to find what the moment of inertia is. So let's go ahead and do that real quick. The formula for the moment of inertia for a solid sphere is going to be 2/ times the mass times the radius squared. And we know all of these values. So let's go ahead and plug it in. This is going to be equal to 2/5 times 0.89 times 0.6 squared. Which when you plug this in your calculator, we get a moment of inertia of 0. kg per meter squared or sorry, kilograms meter squared. Now that we have that value, we are ready to go ahead and use this formula right here to find our angular acceleration. So we have that our radius is 0.6. We have that the mass of our. My apologies. Our friction force is 0.82 times the sine of our angle Theta all divided by our moment of inertia that we just found, which was zero point 7957. And when you plug all of this into your calculator, you get that. Our angular acceleration of the ball is 0. radiance per second, which is equivalent to 6.18 times 10 to the negative third radiance per second corresponding to our final answer. Choice of D. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

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Key Concepts

Moment of Inertia

Torque

Angular Acceleration