order to determine the turbine’s initial angular velocity ϖ0for part a. However, in order to make the units consistent, we will convert the angular displacement θfrom revolutions to radians before substituting its value into Equation 8.8. In part b, the elapsed time tis the only unknown quantity. We can, therefore, choose from among 0tϖϖα=+(Equation 8.4), (29102tθϖϖ=+(Equation 8.6), or 2102ttθϖα=+(Equation 8.7) to find the elapsed time. Of the three, Equation 8.4 offers the least algebraic complication, so we will solve it for the elapsed time t.SOLUTIONa. One revolution is equivalent to 2πradians, so the angular displacement of the turbine is2870 revθ=(292rad1 revπ41.8010rad=×÷Solving 2202ϖϖαθ=+(Equation 8.8) for the square of the initial angular velocity, we obtain 2202ϖϖαθ=-, or (29(29(29222402137 rad/s20.140 rad/s1.8010rad117 rad/sϖϖαθ=-=-×=b. Solving 0tϖϖα=+(Equation 8.4) for the elapsed time, we find that02137 rad/s117 rad/s140 s0.140 rad/stϖϖα--===23.REASONING AND SOLUTION a.ϖ= ϖ0+αt= 0 rad/s + (3.00 rad/s2)(18.0 s) = 54.0 rad/sb.θ= 12(ϖ0+ ϖ)t= 12(0 rad/s + 54.0 rad/s)(18.0 s) = 486 rad
148ROTATIONAL KINEMATICS24.REASONINGThe angular displacement is given by Equation 8.6 as the product of the average angular velocity and the time(29102Average angularvelocityttθϖϖϖ==+1 42 43This value for the angular displacement is greater than ϖ0t. When the angular displacement θis given by the expression θ= ϖ0t, it is assumed that the angular velocity remains constant at its initial (and smallest) value of ϖ0for the entire time, which does not, however, account for the additional angular displacement that occurs because the angular velocity is increasing.The angular displacement is also less than ϖt. When the angular displacement is given by the expression θ= ϖt, it is assumed that the angular velocity remains constant at its final (and largest) value of ϖfor the entire time, which does not account for the fact that the wheel was rotating at a smaller angular velocity during the time interval.SOLUTIONa. If the angular velocity is constant and equals the initial angular velocity ϖ0, then ϖ= ϖ0 and the angular displacement is(29(290220 rad /s10.0 s2200 radtθϖ== +=+b. If the angular velocity is constant and equals the final angular velocity ϖ, then ϖ= ϖand the angular displacement is(29(29280 rad /s10.0 s2800 radtθϖ==+=+c. Using the definition of average angular velocity, we have(29(29(2911022220 rad /s + 280 rad /s10.0 s2500 radtθϖϖ=+=+=+(8.6)
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- Physics, Angular velocity, Velocity, θ, rad/s, angular displacement