The figure shows a graph of the angular velocity of a rotatingwheel as a functio

Question

The figure shows a graph of the angular velocity of a rotatingwheel as a function of time. Although not shown in the graph, theangular velocity continues to increase at the same rate untilt = 9.1 s. What is the angular displacement of the wheelfrom 0 to 9.1 s?

Number Units No unitssmm/sm/s^2Nradrad/srad/s^2 The figure shows a graph of the angular velocity of a rotatingwheel as a function of time. Although not shown in the graph, theangular velocity continues to increase at the same rate untilt = 9.1 s. What is the angular displacement of the wheelfrom 0 to 9.1 s? Number Units

Explanation / Answer

We need this graph to determine angularacceleration. Since graph is straight line it means that angularacceleration is constant. As we can see, during period of 5 secondsangular velocity changes from -9.0 to +6.0 rad/s. So here is data: t = 5 s = 6.0 rad/s - (-9.0) rad/s = 15.0 rad/s (note that you can use any period of time and corresponding angularvelocity change) (also note that sign isn't multiplied with t nor , butmeans change in something) = / t = (15.0 rad/s) / 5 s = 3.0rad/s So during period of first 3.0 seconds angulardisplacement will be: 1 = 0t1-t12/2 = {-9.0 * 3.0 - [3.0 *(3.0)2]/2} rad = -4.5 rad After those 3.0 seconds till 9.1 seconds wheel rotates in differentdirection. Wee will use t2 to refer to this period oftime. 2 = (t2)2/2= {[3.0 * (9.1 -3.0)2]/2} rad= [(3.0 *6.12)/2] rad= 55.82 rad Resulting angular displacement is: = 1 + 2 = -4.5 rad + 55.82 rad = 51.32 rad but since 2 radians is equal to zero displacement, we need tocalculate the real displacement. / 2 rad = number of rotations = 8.17 rotations These 8 full rotations are equal to zero displacement, but that0.17 is displacement that matters. Number 8 stands for 8 full rotations, so number 1 would stand forsingle full rotation. According to this logic 0.17 would stand for0.17 of single rotations, or 17% of single rotation. To calculateit we need to find 17% of full rotation. One full rotations is2 rad. So: = 17% 2 rad = 17/100 * 2 rad = 0.34 rad If you want to convert to degrees: rad = 180 deg, so: 1 rad = 180/ deg = 180/3.14 deg = 57.32 deg 1 rad = 57.32 deg, so: 0.34 rad = 0.34 * 57.32 deg = 19.49 deg Finally: = 19.49 degrees This would be it. I hope you got it. Note that you should do the calculations again yourself because itis always possible that I have accidentally made calculationmistake. Cheers.

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