A uniform horizontal beam of length 3.4 m and mass 11 kg rests on vertical supports at its two ends. A 13 kg instrument is placed 2.38 m from the left end. Use g = 9.81 m s⁻². (a) Determine both upward support forces. (b) The instrument is moved toward the right support without leaving the beam: explain which reaction increases and why.

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Left 92.21 N; right 143.2 N. Moving the instrument right increases the right reaction and decreases the left reaction. Model the beam and instrument as one stationary system. Their weights act downward; the supports exert upward reactions. The beam's own weight acts at its midpoint because its mass distribution is uniform. Take torques about the left support to eliminate its unknown reaction: RᵣL − mgL/2 − Mgx = 0. Thus Rᵣ = mg/2 + Mgx/L = 143.2 N. Vertical force balance requires Rₗ + Rᵣ = (m + M)g. Therefore Rₗ = 235.4 − 143.2 = 92.21 N. As x increases, the instrument's clockwise torque increases. The right support must supply a larger counterclockwise torque; total supported weight stays fixed, so the left reaction decreases. Both calculated reactions are positive and add to the total weight. This checks that both supports remain in contact and that translational and rotational equilibrium are satisfied together.

Support forces and rotational equilibrium · set 7

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