A light rigid rod of length l is constrained to move in a vertical plane, so that its ends are along the x and y axes respectively. Find the instantaneous axis of rotation of the rod when it makes an angle `theta` with horizontal.

Explanation

When the rod inclines at an angle `theta` with vertical, let the contact forces at A & B are `N_(A) & N_(B)` respectively as shown in the figure.
`rArr` Equation of motion of c.m. of the rod can be given by,
`F_(x) = ma_(x) " "rArr" "N_(A) = ma_(x)" "...(i)`
& `F_(y) = ma_(y)" "N_(B) - mg = ma_(y)" "...(ii)`
where `a_(x) & a_(y)` are the horizontal & vertical acceleration of the c.m. of the rod respectively. The moment of `N_(A) & N_(B)` anticlockwise about G is given by [G is COM of rod]
`tau(G) = -N_(A) (l//2) cos theta + N_(B) (l//2) sin theta`
`rArr" "tau_(G) = I_(G) alpha,"where I"_(G)` = M.I. of the rod about G.
`I_(G) = ml^(2)//12 and alpha` = angular accleration of the rod `=(d^(2)theta)/(dt^(2))`
`rArr" "(-N_(A) cos theta + N_(B) sin theta) l//2 = ml^(2) alpha //12" "...(iii)`

Now we have three equations and four unknowns `N_(A), N_(B), a_(x) & a_(y)`. Therefore we need another two equations by using kinematics, x = (l/2) sin `theta` & y = (l/2) cos `theta`, by differentiating x and y, we obtain
`a_(x) = (d^(2)x)/(dt^(2))=(l)/(2) (alpha sin theta - omega^(2) cos theta)" "...(iv)`
`a_(y) = (d^(2)y)/(dt^(2)) =-(l)/(2)(alpha sin theta + omega^(2) cos theta)" "...(v)`
Using (i) & (iv), find `N_(A)` & using (ii) & (v) find `N_(B)`, then put `N_(A) & N_(B)` in equation (iii) to obtain `alpha = (3g//2l) sin theta`
Putting `alpha = omega (d omega)/(d theta)` we obtain,
`omega d omega = (3g)/(2l) sin theta d theta rArr int_(0)^(omega) omega d omega = (3g)/(2l) int_(0)^(theta) sin theta " "rArr omega = sqrt((3g(1-cos theta))/(l))`
Alter
`I = (ml^(2))/(12) + (ml^(2))/(4) " "(because y = l//2)`
`I = (ml^(2))/(3)`
By law of energy conservation,
`(1)/(2)I omega^(2) = "mg" (l)/(2)(1-cos theta) rArr omega = sqrt((3g)/(l)(1-cos theta))`
`alpha = (domega)/(dt) = (3g)/(2l) sin theta`