Chapter 8. Angular velocity, angular acceleration, and time for constant angular acceleration only (8-15)

Question

YUTcR8i8aEsQI7cEy+gI6dhoQ8eT4eT5yWiB1M49PlaZPD7yPcACmNmSEjpsXgf/DFB/XGx4vkFCXA6z09YnRaR43SeN+qdL/XZgDGo+S/L/sMlb6eRTuB1c745PoYdjEi53Qg==
{"title":"Angular velocity at time t of a rotating object with constant angular acceleration","description":"Correct!","type":"correct","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"4,19,40,51\"}]"} {"title":"Angular velocity at time t = 0 of the object","description":"Wrong","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"113,12,151,48\"}]"} {"title":"Constant angular acceleration of the object","description":"Incorrect","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"234,15,270,49\"}]"} {"title":"Time at which the object has angular velocity omega sub z","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"281,13,299,51\"}]"}

Question

H1GzuBM5OJY9dlHmf/dmlA9woSRRoguH81sDsZbH2nE/LKd1V5DhZbY7aMigNjY85uNlQUZmCV6AMetHDXlEquc7t/s=
{"title":"Angular velocity at time t of a rotating object with constant angular acceleration","description":"Incorrect","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"4,19,40,51\"}]"} {"title":"Angular velocity at time t = 0 of the object","description":"Correct!","type":"correct","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"113,12,151,48\"}]"} {"title":"Constant angular acceleration of the object","description":"Incorrect","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"234,15,270,49\"}]"} {"title":"Time at which the object has angular velocity omega sub z","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"281,13,299,51\"}]"}

Question

R6NVI2/fQ0KohlQ0t/WeafImF1x35FX6BngvkGu5VEbQQVnPESDroW2U/i/P0cozQ9YoqPuB7HdmxjbY
{"title":"Angular velocity at time t of a rotating object with constant angular acceleration","description":"Incorrect","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"4,19,40,51\"}]"} {"title":"Angular velocity at time t = 0 of the object","description":"Incorrect","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"113,12,151,48\"}]"} {"title":"Constant angular acceleration of the object","description":"Correct!","type":"correct","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"234,15,270,49\"}]"} {"title":"Time at which the object has angular velocity omega sub z","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"281,13,299,51\"}]"}

Question

I+auTfQEELiQTtLLhh5lhkRtA0NZgcEe2t5cdu5D2hqiBLGG69eWP7xuuuZE8O6nOgh1Xz8hxfWK2jUpwE7LWPKW4shIHRFqqs0/CA==
{"title":"Angular velocity at time t of a rotating object with constant angular acceleration","description":"Incorrect","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"4,19,40,51\"}]"} {"title":"Angular velocity at time t = 0 of the object","description":"Incorrect","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"113,12,151,48\"}]"} {"title":"Constant angular acceleration of the object","description":"Wrong","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"234,15,270,49\"}]"} {"title":"Time at which the object has angular velocity omega sub z","description":"Correct!","type":"correct","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"281,13,299,51\"}]"}

Review

It’s straightforward to write down the equations for rotational motion with constant angular acceleration. To see how, take a look at the equations for linear velocity \(v_x\) and angular velocity \(\omega_z\), and the equations for linear acceleration \(a_x\) and angular acceleration \(\alpha_z\):

Linear Motion Rotational Motion
(linear velocity) (angular velocity)
(linear acceleration) (angular acceleration)
Table 8.1: Linear and Angular Motion

Comparing these equations shows that the rotational quantities \(\theta\), \(\omega_z\), and \(\alpha_z\) are related to each \perpher in exactly the same way that \(x\), \(v_x\), and \(a_x\) are related to each \perpher. So we can take the equations for constant linear acceleration and convert them to the equations for constant angular acceleration by replacing \(x\) with \(\theta\), \(v_x\) with \(\omega_z\), and \(a_x\) with \(\alpha_z\). The equations for linear motion are:

(2-5) \(v_x = v_{0x} + a_xt\) (constant acceleration only)

(2-9) \(x = x_0 + v_{0x}t + \frac{1}{2}a_xt^2\) (constant \(x\) acceleration only)

(2-11) \(v_x^2 = v_{0x}^2 + 2a_x(x - x_0)\) (constant \(x\) acceleration only)

Hence, along with equation 8-16 and 8-17, the equation for constant angular acceleration is: