Sunday, May 29, 2016

homework and exercises - Rod sliding on a frictionless surface



A uniform rod$(m,l)$ is standing vertically on a horizontal frictionless surface. Gravity is downwards and uniform. I give its upper end a little push and off it goes. I want to find the Normal Force(and all other variables) as a function of time.



Here's what I do :



$mg-N=ma$


Where a is downward acceleration(and the only) of centre of mass wrt ground.


Next, I think of conservation of energy. I assume it rotates $\theta$ from vertical


$mg\frac{l}{2}(1-\cos\theta)=K.E.$


I have problem writing expression of its Kinetic Energy. I have studied about instantaneous centre of rotation.


So, I can write its speed and rotational kinetic energy about it as $K.E.$? Also How do I find relation between $\omega$ (angular velocity about that instantaneous centre) and velocity of rod?



Answer



This is what I did and I think is simple and right :


Assume $v$ linear speed of centre of mass downwards and $\omega$ angular speed around it. Use the fact the bottom point has no vertical speed to find relation between $v$ and $\omega$. And I am done.


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