A shelf of uniform density is supported by two brackets at a distance of #1/8# and #1/4# of the total length, #L#, from each end respectively. Find the ratio of the reaction forces from the brackets on the shelf?

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1 Answer
Cosmic Defect
Nov 12, 2017

Translation Equilibrium Condition: #F_1+F_2=Mg# ........... (1)
Rotational Equilibrium Condition : #F_1+6F_2=4Mg# ...... (2)

#F_1=2/5Mg; \quad F_2=3/5Mg; \qquad F_1/F_2 = 2/3#

Explanation:

Translational Equilibrium Condition: #\sum_k vec F_k = vec 0#
#vec F_1 + vec F_2 + Mvec g = 0 \qquad \rightarrow F_1 + F_2 -Mg = 0#

#F_1 + F_2 = Mg# ........ (1)

Rotational Equilibrium Condition: #\sum_k vec \tau_k = vec 0#
#vec \tau_1 + vec \tau_2 + vec \tau_w = 0#

#vec \tau_1# : Torque due to the force #vec F_1#
#vec \tau_2# : Torque due to the force #vec F_2#
#vec \tau_w# : Torque due to the weight of the bar #vec w#

Calculating the torques about the left end,

#vec \tau_1 = +F_1.L/8; \qquad vec \tau_2 = +F_2.(L-L/4)=3/4F_2.L#
#vec \tau_w = -Mg.L/2#

#1/8F_1.L + 6/8F_2.L-Mg.L/2 = 0#

#F_1 + 6 F_2 = 4Mg# ....... (2)

We have two equations [(1) and (2)] and two unknowns [#F_1# and #F_2#]. Solving for #F_1# and #F_2# we get,

#F_1 = 2/5Mg; \quad F_2 = 3/5Mg; \qquad F_1/F_2 = 2/3#

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