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phy131studiof15:lectures:chapter12 [2015/10/10 03:17] mdawber [Angular acceleration] |
phy131studiof15:lectures:chapter12 [2015/10/12 09:19] (current) mdawber [Equations of motion for rotational motion] |
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===== 12.P.008 ===== | ===== 12.P.008 ===== | ||

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+ | ===== Equations of motion for rotational motion ===== | ||

+ | |||

+ | At the beginning of the course we derived, using calculus, a set of equations for motion under constant acceleration. | ||

+ | |||

+ | $v= v_{0}+at$ | ||

+ | |||

+ | $x= x_{0}+v_{0}t+\frac{1}{2}at^2$ | ||

+ | |||

+ | $v^{2}=v_{0}^2+2a(x-x_{0})$ | ||

+ | |||

+ | We can equally derive similar equations for our rotational quantities. Indeed as we can see that the relationships between the new rotational quantities we have now are exactly the same as those between the translational quantities we can simply rewrite the translational motion equations in terms of rotational variables. | ||

+ | |||

+ | $\omega= \omega_{0}+\alpha t$ | ||

+ | |||

+ | $\theta= \theta_{0}+\omega_{0}t+\frac{1}{2}\alpha t^2$ | ||

+ | |||

+ | $\omega^{2}=\omega_{0}^2+2\alpha(\theta-\theta_{0})$ | ||

+ | |||

+ | ===== 12.P.005 ===== | ||

+ | ===== 12.P.026 ===== | ||

===== From angular to tangential quantities ===== | ===== From angular to tangential quantities ===== | ||

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$a=\alpha r$ | $a=\alpha r$ | ||

+ | |||

+ | ===== 12.P.038 ===== | ||

===== Useful relationships concerning the angular velocity ===== | ===== Useful relationships concerning the angular velocity ===== | ||

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$\arctan\frac{\alpha}{\omega^{2}}$ away from the radial direction. | $\arctan\frac{\alpha}{\omega^{2}}$ away from the radial direction. | ||

+ | |||

+ | ===== 12.P.031 ===== | ||

+ | |||

===== Pseudovector representation of angular velocity and acceleration ===== | ===== Pseudovector representation of angular velocity and acceleration ===== | ||

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- | ===== Equations of motion for rotational motion ===== | ||

- | At the beginning of the course we derived, using calculus, a set of equations for motion under constant acceleration. | ||

- | $v= v_{0}+at$ | ||

- | $x= x_{0}+v_{0}t+\frac{1}{2}at^2$ | ||

- | |||

- | $v^{2}=v_{0}^2+2a(x-x_{0})$ | ||

- | |||

- | We can equally derive similar equations for our rotational quantities. Indeed as we can see that the relationships between the new rotational quantities we have now are exactly the same as those between the translational quantities we can simply rewrite the translational motion equations in terms of rotational variables. | ||

- | |||

- | $\omega= \omega_{0}+\alpha t$ | ||

- | |||

- | $\theta= \theta_{0}+\omega_{0}t+\frac{1}{2}\alpha t^2$ | ||

- | |||

- | $\omega^{2}=\omega_{0}^2+2\alpha(\theta-\theta_{0})$ | ||

- | |||

- | ===== 12.P.005 ===== | ||

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The units of torque are $\mathrm{Nm}$ | The units of torque are $\mathrm{Nm}$ | ||

+ | |||

+ | ===== 12.P.041 ===== | ||

+ | |||

===== Production of Torque in an engine ===== | ===== Production of Torque in an engine ===== | ||

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$\frac{d}{dt}(\vec{A}\times\vec{B})=\frac{d\vec{A}}{dt}\times\vec{B}+\vec{A}\times\frac{d\vec{B}}{dt}$ | $\frac{d}{dt}(\vec{A}\times\vec{B})=\frac{d\vec{A}}{dt}\times\vec{B}+\vec{A}\times\frac{d\vec{B}}{dt}$ | ||

+ | |||

+ | |||

+ | ===== 12.P.047 ===== | ||

+ | |||

+ | ===== 12.P.048 ===== | ||

===== Acceleration due to torque ===== | ===== Acceleration due to torque ===== | ||

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We should note that the distance $R$ is the distance of each mass with respect to the axis of rotation, which need not be the center of mass, and is not an intrinsic property of an object! | We should note that the distance $R$ is the distance of each mass with respect to the axis of rotation, which need not be the center of mass, and is not an intrinsic property of an object! | ||

+ | |||

+ | ===== 12.P.053 ===== | ||

+ | |||

===== Two weights on a thin bar ===== | ===== Two weights on a thin bar ===== |