Rotational Dynamics
Rotational Dynamics Rotational Dynamics – Falling Rod? A thin uniform rod is initially positioned in the vertical direction, with its lower end attached to a frictionless axis that is mounted...
Rotational Dynamics
Rotational Dynamics – Falling Rod?
A thin uniform rod is initially positioned in the vertical direction, with its lower end attached to a frictionless axis that is mounted on the floor. The rod has a length of 3.00 m and is allowed to fall, starting from rest. Find the tangential speed of the free end of the rod, just before the rod hits the floor after rotating through 90°.
I know the formula for total energy is:
mghi + 1/2 * m * vi^2 + 1/2 * I * wi^2 = mghf + 1/2 * m * vf^2 + 1/2 * I * wf^2
However, I do not think it applies because every particle has a different initial height.
The moment of inertia for a rod with axis at the end : I = 1/3 * m * length
Thank you all, especially rijim2001, that came out to be the right answer. I never thought of using the L / 2 to deal with the mass all at the center. Thanks again, you just saved me and the rest of my class.
Firstly the moment of inertia for a rod with axis at the end : I = 1/3 * m * length squared.
Take the total mass m to be at its center of mass which is at L/2. That mass has potential energy of mgh = mg(L/2). When it falls to the ground this PE is converted to rotational
kinetic energy = (1/2) I (omega)^2.
So:
mg(L/2) = (1/2) I (omega)^2
mgL = I (omega)^2
but I = (1/3) m L^2
mgL = (1/3) m L^2 (omega)^2
g = (1/3)L(omega)^2
(omega)^2 = 3g/L
omega = sqrt(3g/L)
omega = sqrt[ 3(9.8)/3]
omega = sqrt [29.4/3] = sqrt( 9.8)= 3.13049517
but omega = v/L
so v = L (omega) = 3 (3.13049517)= 9.39148551 m/s
v= 9.39 m/s
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About the Author:
Jimmy Rayne is currently doing research on video conference and video conferencing equipment for 2nd year project for his bachelors in computer science and technology.
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