A long, uniform rod of mass M and length L is supported at the left end by a hor

Question

A long, uniform rod of mass M and length L is supported at the left end by a horizontal axis into the page and perpendicular to the rod, as shown. The right end is connected to the ceiling by a thin vertical thread so that the rod is horizontal. Express the answers to all parts of this question in terms of some or all of the quantities M, L , and g, and any necessary constants.

The thread is burned by a match. For the time immediately after the thread breaks, determine:

(a) the angular acceleration of the rod about the axis shown

(b) the translational acceleration of the center of mass of the rod

(c) the force exerted on the end of the rod by the axis

I found the answer to all of these correctly (a = (3g)/(2L), b = (3/4)g, c = (1/4)mg). What I'm struggling with is why the force exerted on the end of the rod by the axis DECREASES after the string is burned. (the force exerted on the end of the rod by the axis before the string is burned is (1/2)mg.) It seems to me that it should only INCREASE.

Although the problem doesn't specify this detail, my professor said that the axis and the rod are connected.

Please provide an intuitive and thorough answer. Thanks!

Explanation / Answer

a)

When the thread is burnt,

The net torque about the left end(about the axis):

Mg*(L/2) = I*A

where I = moment of inertia about the left end = ML^2/3

A = angular acceleration,

So, A = MgL/2/(ML^2/3) = 3g/2L <-------answer

b)

Now, A = a/(L/2)

So, 3g/2L = a/(L/2)

So, a = 3g/4 = (3/4)g <-------answer

c)

Net force on the rod = Fnet= M*a = M*(3/4)*g

From force equation, Fnet = Mg - F'

where F' = force exerted by the axis

So, Mg - F' = 3Mg/4

So, F = Mg - 3Mg/4 = Mg/4 <--------answer

--------- NOTE : -------

<> And for your doubt, the force exerted by the axis has do decrease.

We see from the free abody diagram there are three forces acting on the rod,

It is intuitive because after burning the thread, the net acceleration is non-zero, so the force from the axis has to decrease so as to increase the net force and thus to have a net acceleration to be non-zero.

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