A 150-g baseball, 3.85 cm in radius, leaves the pitcher's hand with 30 m/s horizontal and 20 rad/s clockwise. Calculate the ball's linear, rotational, and total kinetic energy.

K_{L} = 67.5 J ;K_{R} = 0.018 J ;K_{tot} = 67.52 J

A 150-g baseball, 3.85 cm in radius, leaves the pitcher's hand with 30 m/s horizontal and 20 rad/s clockwise. Calculate the ball's linear, rotational, and total kinetic energy.

Everyone. So I'm filling in for Marcio today. We've got a baseball that's flying through the air while also spinning. So we're told the mass and radius of this baseball, and we're told that it leaves the pitcher's hand going 30 meters per second horizontal. That's the linear speed, but we're told that this baseball is also spinning at 20 radians per second clockwise. In other words, it has a VCM, but it also has an omega. And we're gonna use this to find the linear, rotational, and total kinetic energy. Now the key thing to remember over here in this problem is that there's actually 2 motions. So there's 2 motions, which means we're gonna have 2 kinetic energies. There's one kinetic energy from the linear speed, but there's also a rotational kinetic energy because of the fact that it's spinning. But as Marcio just mentioned over here, the key thing to remember is that because this thing is floating through the air, we actually can't use the relationship that VCM is equal to our omega. We can't use this because this doesn't work when things are flying through the air. It only works when things are rolling. Alright? So what that means, though, is that these problems actually have to give you both the VCM and Omega because it can't give you one and expect to you to know the other. Alright? Now what happens is if these were actually, by the way, be the same number by accident, that would be a different story. So, for example, if you found this was 10 and this was 10, that would just be a complete coincidence, but that would be a different story. Alright? So how do we find the k l, the kinetic energy linear? This is just gonna be 1 half m v squared. So onetwo mvcm squared is going to be onetwo times the mass. The mass is 150 grams, which is 0.15 kilograms. So That's gonna be the mass. And the velocity over here that we're gonna use is the 30 meters per second horizontal. You write this out and you calculate this, you're gonna see that this is 67.5 joules. That's the linear kinetic energy. What about the rotational? Rotational kinetic energy is gonna be different formula. It's 1 half I omega squared. So we need the moment of inertia and the omega. Now unlike a basketball, which is like a hollow thing that's filled with air, baseballs actually have stuff inside of them. So what that means here is that the moment of inertia is actually gonna be the moment of inertia of a solid sphere. The moment of inertia of a solid sphere is 2 5ths m r squared. Alright? So what we're gonna use here is we're gonna use 1 half of 2 5ths, and we're gonna use the mass, which is 0.15. And then the radius, which is given to us as 3.85 centimeters, we'll have to convert this to 0.038 5, and I won't round that. So I'm going to just use 0.0385 squared. Now we're going to take the omega and square that. So this is going to be the 20 squared over here. There's a lot of stuff to calculate in this, in this example over here, in this last little equation. So just make sure you plug everything in in carefully. But what you see here is that this kinetic energy is 0.018. Alright. So it's 0.018 joules, and that means that when you actually go ahead and plug this into your final kinetic energy, we're just going to have to add those 2 things. Right? So the total kinetic energy, k total, is just going to be kl plus kr, the two numbers we just calculated. In other words, it's going to be 67.5 plus 0.018. What you're gonna get for a total kinetic energy is 67.52 joules. Alright? So that's the final final answer. We can see here that, by the way, this, this baseball actually has much more linear kinetic energy than it does rotational. It's much more in the linear than there is from the fact that it's rotating. Alright? So that's it for this one, folks. Let me know if you have any questions, and I'll see you guys later.

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A 150-g baseball, 3.85 cm in radius, leaves the pitcher's hand with 30 m/s horizontal and 20 rad/s clockwise. Calculate the ball's linear, rotational, and total kinetic energy.

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