Q A rigid, uniform, horizontal bar of mass m 1 and length L is supported by two

Question

Q

A rigid, uniform, horizontal bar of mass m1 and lengthL is supported by two identical massless strings.(Figure 1) Both strings are vertical. String A is attached at a distance d<L/2 from the left end of the bar and is connected to the ceiling; string B is attached to the left end of the bar and is connected to the floor. A small block of mass m2 is supported against gravity by the bar at a distance x from the left end of the bar, as shown in the figure.

Throughout this problem positive torque is that which spins an object counterclockwise. Use g for the magnitude of the acceleration due to gravity.

http://session.masteringphysics.com/problemAsset/1010948/29/MFS_nr_4.jpg

What is the smallest possible value of x such that the bar remains stable (call it xcritical)?

Explanation / Answer

Take summation of moments about String A (and since counterclockwise torque is positive),

m2(x - d) + m1(L/2) - Tb(L) = 0

and solving for "Tb"

Tb = (1/L) + m2(x - d) + m1(L/2)


Taking summation of vertical forces (assuming that downward forces are positive).

m2(g) + m1(g) + Tb - Ta = 0

Solving for Ta,

Ta = g(m1 + m2) + Tb

and substituting for the derived value of Tb, then

Ta = g(m1 + m2) + (1/L) + m2(x - d) + m1(L/2)

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