Problem 3: A uniform beam of length L and mass mB is supported by two pillars lo

Question

Problem 3: A uniform beam of length L and mass mB is supported by two pillars located L/3 from either end, as shown in the figure. A duck of mass mDstands on one end. A scale is placed under each pillar. The entire system is in equilibrium.

Part (a) Enter an expression, in terms of defined quantities and g, for the force that the scale under right pillar shows.

Part (b) Enter an expression, in terms of defined quantities and g, for the force that the scale under left pillar shows

Part (c) Enter an expression, in terms of defined quantities and g, for the sum or the scale readings.

Explanation / Answer

Let the Force from Right Pillar = Fr
Let the Force from Left Pillar = Fl

Calculating Moment at Left Pillar -
As the system is in equilibrium =
mD *g * L/3 - mB *g* (L/2 - L/3) + Fr* (L - L/3 - L/3) = 0
mD *g * L/3 + Fr* (L/3) = mB *g* L/6
(mD*g)/3 + Fr*/3 = (mB *g)/6
Fr = ((mB *g)/6 - (mD*g)/3) * 3
Fr = (mB/2 - mD)*g

Calculating Moment at Right Pillar -
As the system is in equilibrium =
mD *g * (L-L/3) + mB *g* (L/2 - L/3) - Fl* (L - L/3 - L/3) = 0
mD *g * (2L/3) + mB *g* (L/6) - Fl* (L/3) = 0
Fl = 3*(mD *g * (2/3) + mB *g* (1/6))
Fl = (2mD + mB/2) *g

Part(a)
Force that the scale under Right pillar shows , Fr =  (mB/2 - mD)*g

Part(b)
Force that the scale under left pillar shows , Fl = (2mD + mB/2) *g

Part (c)

Sum of Scale = Fr + Fl
Sum of Scale = (mB/2 - mD)*g + (2mD + mB/2) *g
Sum of Scale = (mB + mD)*g

Sum of the scale Readings, = (mB + mD) * g

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