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During time interval; flywheel with constant angula acceleratian rurns_ through 419 radians and acquires an angular velocity 146 rad/s. (Assume the flywheel rotatin...

Question

During time interval; flywheel with constant angula acceleratian rurns_ through 419 radians and acquires an angular velocity 146 rad/s. (Assume the flywheel rotating in the positive direction_ Indicate the direction with the signs Your wnswers;What the angular veloclty (In rad/s) at the beginnlng tne 4.1 s? rdysWhat the angular acceleration of the flywheel (in rad/s2)? rad/s2

During time interval; flywheel with constant angula acceleratian rurns_ through 419 radians and acquires an angular velocity 146 rad/s. (Assume the flywheel rotating in the positive direction_ Indicate the direction with the signs Your wnswers; What the angular veloclty (In rad/s) at the beginnlng tne 4.1 s? rdys What the angular acceleration of the flywheel (in rad/s2)? rad/s2



Answers

A flywheel has angular acceleration $\alpha_{z}(t)=$$8.60 \mathrm{rad} / \mathrm{s}^{2}-\left(2.30 \mathrm{rad} / \mathrm{s}^{3}\right) t,$ where counterclockwise rotation is positive. (a) If the flywheel is at rest at $t=0,$ what is its angular velocity at 5.00 s? (b) Through what angle (in radians) does the
flywheel turn in the time interval from $t=0$ to $t=5.00$ s?

Solving party. So I will use the formula. Dale Omega is equal to omega, minus omega one. And we also know that they'll omega is equal to Alford. Lt. So I can write omega minus omega one is equal to Alpha dirty 60 to minus 31. So just putting the value in the question I can write Omega to edge minus 1.33 Radiant person can be square multiplication one It is again plus 4 20 radiant per second. So on simplification, I get the final value of omega to watch 1. 81 80.6. They didn't tell us again. So this is the angular velocity. At T is equal to one. It is again now solving part b. Okay, so we will use the formula. Dirty ties equal to omega one dirty plus, uh, alpha dirty is quite This is the standard formula, so I can ride. The value of delta theta is equal to 4. 20 Didion purse again. Multiplication. One eighties again minus one by two. Multiplication minus 1.33 Radiant per second square multiplication. One 82nd square. One simplification I get Delta theta is equal to 54054 Radiant now calculating the number of revolutions. So, as we all know, a formula that is number of revolutions and is equal to anger displacement. Delta to buy two pies. So just putting the value I can write. 54054 Radian by two Pi Radian. So finally I get N is equal to 8602 Revolution

This problem, we're given the angular position um or the rotation of an oscillating flywheel um as a function of time. So that's this expression here and plastics um calculate the the angular coordinate the angular velocity and the angular acceleration of the flywheel at T equals zero and T equals 0.125 seconds. So this really comes down to just being a calculus problem um where we know the angular velocity is the time derivative of the angle her um displacement. So we can go through the calculus and get this. I'm not going to go through all the details of the calculus, I'm assuming that you can do that or use a computer to do it. Um And then the angular acceleration is the time derivative, the angular velocity. And we can go through math and calculus to get this expression here, We're told that data not is 0.5 radiance. So at this point it's just a matter of um plugging in our values. So at T equals zero. Co sign is zero is one. E. Is one. Each of the zero is one. So we get data is um 0.5 radiance. So state a state or not and that's why we call it data not because that's data when The T. equals zero. Um plugging in Daytona and T. Into this expression here And just punching through the the Algebra was crunching through the math. We get that Omega is -4.7 radiance per second. And I'm plugging everything in the acceleration we get to the angular acceleration is -34.5 radiance per second squared. Now if we use um t equals 0.125 Which is 1/8, plug that into here. We see that that we wind up getting um uh no sign of pi over two which is zero. So that means that the angular displacement at this time is zero. We can plug everything into here. Um and again, this term is going to be zero, but this term is gonna, this sign term is going to be one. And then, you know, this is gonna be something and everything gets kind of messy. But just go through the math, Plug it on the calculator computer. And we get that omega equals one is -1.5 1.9 radiance per second, plugging everything into here. Again, we can just crank through the calculations. Um and we get that the angular acceleration is 36.7 radiance per second squared mm hmm.

This problem covers the concept of the rotational paramedics and for party from the equation traditional canna medics. Then we'll acceleration equals the final and the velocity minus the initial velocity. Upon the time t. I understood the value the final angular velocity is zero When the wheel stops minus the initial is 25-12 radiance for a second. Upon the time T. That is 19 .7 seconds. So the angular acceleration, the constant angular acceleration is -1.28 Radiance or 2nd square. No party from the charismatic question. The final english speech squared equals the shelling this brief square Plus two and 2. The angular acceleration and two. The greatest placement from this equation we can right the angular displacement phi equals the final english B squared Minister initially. Unless b squared upon to into the angular acceleration, The final language speed is zero so we can neglect the storm no substitute. So there are five equals zero minus 25.2 radiance four seconds square Upon to a two. The angular acceleration is -1.28. The radiance or 2nd square. So the first topping the angular displacement is 248 radius. Albert C the angular displacement in revolution equals 248 radiance into one revolutions. Upon Dubai radiant are the angle is angular displacement is um 39 0.5 revolutions

Let's use the rotational cinematic equations. Help us solve this problem. The time of the process is four seconds. The angle traveled. They did. My stay or not is equal to 162 Radiance. The final angular velocity is 108 radiance a second and our desired unknown is the initial your velocity. To do this, we're going to use this cinematic equation. They don't mind stated on Siegel to omega Non puzzle Megha over two times the time solving this for Omega not gives omega non equals to fade him. Want to stay the night over tea minus omega, which eagles negative 27. Praise for second. Whenever you plug in all of these values over here and that's the answer to party, Barbie is asking for the angular acceleration. And so, for this we can use Omega is equal to Omega night, plus Alfa Time, Steve solving for Alfa Omega by something and I over tea. And then we can just plug in these which we found on the previous page or were given to us originally. Yeah, so you get 100 eh? Minus native 27. This is what we found in party over four seconds and that equals 33 0.75 radiance per second squared, and that's the final answer to problem.


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