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Old Nov 21st 2017, 03:41 PM   #1
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Rotational Dynamics Problems

Problem 1
A small cube of mass m is placed on the inside surface of a funnel rotating about a vertical axis with constant
angular velocity ω. The wall of the funnel makes angle θ with the horizontal. The coefficient of static friction
between the cube and the funnel is s , and the cube is at distance r from the axis of rotation. Find both the largest
and smallest values of ω for which the cube will not move with respect to the funnel.
Given Diagram
Rotational Dynamics Problems-maxminw.png
Attempt at a solution
$\displaystyle r:nsinθ-F_scosθ=mw^2r$
$\displaystyle z:ncosθ+F_ssinθ-mg=0$
$\displaystyle F_s≤_sn$
$\displaystyle n=mg\(cosθ+_ssinθ)$
$\displaystyle (mgsinθ-_scosθ)/(cosθ+_ssinθ)=mw^2r$
Solving for w, I got the max value of w to be:
$\displaystyle w_max=\sqrt{(gsinθ-_sgcosθ)/r(cosθ+_ssinθ)}$
I'm just not sure that that is the correct answer, because after searching online I saw a few answers in which the answer I got was multiplied by $\displaystyle 1/2π$ for some reason. I am also not entirely sure how to find the minimum value.

Problem 2
In the figure below, a ball of mass m is attached to a rotating shaft by means of two strings of length L. The system rotates with period τ. Assume the motion occurs in intergalactic space so there is no gravitational force acting on the ball. Show that tension T in each string is given by $\displaystyle T=(2π ^2mL)/(τ^2)$, an answer independent of the angle.

Given Diagram
Rotational Dynamics Problems-angle-problem.png

Attempt at a solution
$\displaystyle r:2Tsinθ=m(vt^2/r)$
$\displaystyle z:Tcosθ-Tcosθ=0$
$\displaystyle vt^2/r=(4π^2L/τ^2)$
$\displaystyle 2Tsinθ=(4π^2mL/τ^2)$
$\displaystyle Tsinθ=2π^2mL/τ^2$

The problem I'm having with this is that I don't know how to eliminate the angle in the problem.
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