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Explain angular velocity and angular acceleration about fixed axis and derive the equation of rotational motion and write the analogy between the equations of linear motion and rotational motion. |
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Answer» Solution :As shown in the FIGURE a RIGID body rotating about fixed Z-axis in XYZ cartesion co-ordinate system. Any particle P of the body circulating in XY-plane. The angular position of this particle P at `t=0` time is `theta_(0)` and `t=t` time it is `theta_(0)+theta` `therefore` In time t its angular displacement is `theta`. Now select X. and Y. parallel to X and Y. Z-axis is already fixed. Intrateneous angular velocity `omega` = time RATE of change of angular displacement `therefore omega=(d theta)/(dt)`, is in the direction of fixed Z-axis so it can be taken as scaler. Angular acceleration `alpha` = time rate of change of angular velocity. `alpha=(domega)/(dt)`, is in the direction of fixed axis so it can be also taken as scalar. EQUATIONS of pure linear motion `v=v_(0)+at` `x=x_(0)+v_(0)t+(1)/(2)at^(2)` `v^(2)=v_(0)^(2)+2a(x-x_(0))` where `x_(0)` = initial position x = final position `v_(0)` = initial velocity v = final velocity a = acceleration, t = time Equation of pure rotational motion `omega=omega_(0)+at` `theta=theta_(0)+omega_(0)t+(1)/(2)alphat^(2)` `omega^(2)=omega_(0)^(2)+2alpha(theta-theta_(0))` where `theta_(0)` = Initial angular position `theta` = Final angular position `omega_(0)` =Initial angular velocity `omega` = Final angular velocity `alpha` = Angular acceleration t = time |
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