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Physics / Chapter 12: Equilibrium and Elasticity / Section 12-2

The Requirements of Equilibrium — Question 7

Section 12-2 · Physics pages 1804–1805

Question 7 A negative unilateral support reaction rejects a candidate equilibrium A weightless rigid beam rests on upward-only supports at x = 0 and x = 2 m. A 60 N downward load acts at x = 3 m on its overhang. There are no anchors. 60 N 0 2 m 3 m Schematic; labels, not drawing lengths, specify the data. (a) Solve the formal equilibrium equations for the reactions. (b) Is the proposed two-contact state physically admissible?
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Question 7 - Solution A negative unilateral support reaction rejects a candidate equilibrium 60 N 0 2 m 3 m (a) Derivation and calculation Formal force and moment balance give RA + RB = 60, 2RB −3(60) = 0. Therefore RB = 90 N , RA = 60 −90 = −30 N . (b) Result and interpretation The unanchored supports require RA ≥0, RB ≥0. Since RA = −30 N violates this condition, the two-contact equilibrium is inadmissible. If left contact is lost, the load moment about B is τB = −60(3 −2) = −60 N m, which promotes clockwise tipping. Physical check A negative required reaction signals failure of the contact assumption, not a permissible downward force from the support.

Question 7

A negative unilateral support reaction rejects a candidate equilibrium

A weightless rigid beam rests on upward-only supports at x=0x=0 and x=2mx=2\,\mathrm m. A 60N60\,\mathrm N downward load acts at x=3mx=3\,\mathrm m on its overhang. There are no anchors.

Diagram shown in the original SVG worksheet.

  1. Solve the formal equilibrium equations for the reactions.

  2. Is the proposed two-contact state physically admissible?

Question 7 - Solution

A negative unilateral support reaction rejects a candidate equilibrium

Diagram shown in the original SVG worksheet.

(a) Derivation and calculation

Formal force and moment balance give RA+RB=60,2RB−3(60)=0.R_A+R_B=60,\qquad2R_B-3(60)=0. Therefore RB=90N,RA=60−90=−30N.\boxed{R_B=90\ \mathrm N},\qquad\boxed{R_A=60-90=-30\ \mathrm N}.

(b) Result and interpretation

The unanchored supports require RA≥0,RB≥0.R_A\ge0,\qquad R_B\ge0. Since RA=−30R_A=-30 N violates this condition, the two-contact equilibrium is inadmissible. If left contact is lost, the load moment about BB is τB=−60(3−2)=−60Nm,\tau_B=-60(3-2)=-60\ \mathrm{N\,m}, which promotes clockwise tipping.

Physical check

A negative required reaction signals failure of the contact assumption, not a permissible downward force from the support.

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