# The height traveled by the mass center of a thrown rod

I'm learning Newtonian mechanics and want to understand the following example and the methods I can use for it.

Suppose I throw a rod in the air with length $L$. The time I'm letting go of the rod the closest edge of the rod to my hand has zero velocity. If the rod does $N$ rotation until I catch it again at the same point I threw it, show that the height its mass centre traveled in the air is $$h=\frac{\pi NL}{4}.$$

Maybe I can try Energy conservation. But not sure how to. Or just use Newton's laws but I don't know how because how it is possible for the starting velocity to be zero since I throw something in the air like a projectile?

• Interesting question. You have some ideas how you might solve it - so why don't you try and show us your calculation? I think you will have to use the eqns for projectile motion to relate time of flight and max. height to initial velocity u of the CM. You can also relate u to the constant rate of rotation - ie the number of rotations during the flight time. (I haven't worked through the problem myself yet, but I think this will work.) – sammy gerbil Sep 13 '16 at 23:34
• The second sentence in your question is not much clear. Can you elaborate more? – Kosala Sep 15 '16 at 16:10
• @Kosala i mean my had is moving i dont drop the object i throw it how can it have zero velocity? – Jam Sep 15 '16 at 17:04
• The end of the rod which is in your hand has zero velocity, but the centre of mass of the rod does not have zero velocity. – sammy gerbil Oct 1 '16 at 14:46