Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm. b. How much energy is stored in the flywheel?

Video transcript

Yeah. Hey, everyone in this problem, we have a rotating turbine with a diameter of 2 m and a mass of kg. If the turbine is spinning at a maximum angular velocity of RP, we're asked to determine the energy stored in the turbine. OK. And we're told to assume that it is disk shaped. So we have four answer choices all in jewels. Option A 1.41 times 10 to the exponent six. Option B 2.66 times 10 to the exponent six. Option C 2.66 times 10 to the exponent five. And option D 1.41 times 10 to the exponent five. All right. So let's get into this. We are looking for an energy here and we have a rotating turbine. And so the energy that we're dealing with is gonna be rotational kinetic energy. OK. So we called that the rotational kinetic energy which we denote as Kr it's gonna be equal to one half multiplied by I multiplied by omega squared. It's really similar to that regular kinetic energy when we're dealing with linear motion and one half MV squared in this case one half I omega squared. All right. So to solve this problem, we're gonna need the moment of inertia. I, and we're gonna need that angular speed omega or the angular velocity omega. Now, we're giving the angular velocity omega. OK. But we're giving it in R PM. And let's recall that when we're using this equation for rotational kinetic energy, we want that in radiance per second, you need to get to that final unit of duel that we're interested in. So let's start by converting that to our standard unit. So we have Omega which is 1800 R PM. And we're gonna take that 1800 revolutions per minute. We're gonna multiply by one minute divided by 60 seconds. OK? Because we know that there are 60 seconds in every minute. And then we're gonna multiply by two pi radians divided by one revolution because we know in every revolution, there are two P ratings can go all the way around the circle once two pi rains. So now if we look at what we have, the unit of minute will divide it, the unit of revolutions will divide out and we're gonna be left with radiance per second, which is what we want. So we have 1800 multiplied by two pi divided by 60 radiance per second. If we simplify that, we can write this as 60 pi radiance per second. OK. So we have our omega value now in the correct events, what about the moment of inertia. Well, we're told to treat this turbine as a disk shaped object and we're told that to assume that it's disk shaped. So that gives us some information about how we're going to calculate the moment of inertia. Ok. So we're called the moment of inertia for a dish shaped object rotating about the center of mass right in the middle is gonna be one half M are squared. And that's something that you can look up in a table in your textbook or that your professor provided is that those moment of Aerts depending on the shape. Now, we're given the mass and we're given the diameter which means we can find the radius. So we have everything we need to calculate the moment of inertia here. So the moment of inertia I is going to be equal to one half multiplied by 300 kg multiplied by the radius. Now the diameter is 2 m which tells us that the radius is gonna be 2 m divided by two. OK. To get the radius from the diameter, we divide by two. So this is gonna be 2 m divided by two all squared, which gives us a moment of inertia eye of we have kilogram meters squared. So now we look at our kinetic energy or rotational kinetic energy equation. We have everything we need. We have the moment of inertia. I we have this rotation or angular velocity in the correct units. So we can go back to this equation substitute in these values and solve. We get that our rotational kinetic energy kr is gonna be one half multiplied by kg meters squared, multiplied by 60 pie radiance per second. All squared. I'm working this out. We're gonna get about 2.66 times 10 to the exponent six. And we multiply our units, we get kilograms meter squared per second squared. OK. And recall that kilogram meter squared per second squared is equivalent to a jewel. And so we get that unit that we were hoping for. We have the, the rotational kinetic energy is about 2.66 times 10 to the exponent six Jews. That is the final answer. If we compare this to the answer choices we were given, we can see that this corresponds with answer choices. B thanks everyone for watching. I hope this video helped see you in the next one.