- 2069 Asar · 3+5 marks
FBD; tensions in strings AC, CD, BD and angle θ
Short answer
At joint C, resolving forces horizontally and vertically gives T_AC = 54.96 N and T_CD = 18.87 N. At D, using T_CD and 100 N load yields T_BD = 100.14 N and the angle θ = 10.8° from the vertical.
What is free body diagram? How is it drawn? Determine the tension on the strings AC, CD and BD and the angle θ in figure below. [Figure: string AC at 20° from point A to C; string CD between C and D at 85° at C; string DB from D to B; weight of 50 N hanging at C and 100 N hanging at D.]
Free body diagram
A free body diagram is a sketch of a body (or joint) isolated from its surroundings, with all external forces acting on it shown.
How it is drawn:
- Isolate the body or joint to be analysed.
- Show the weight and all applied loads.
- Replace each support or connected member by its force: a string or cable gives a tension pulling away from the body along the string.
- Label known and unknown forces, with angles and axes.
- Apply ΣFx = 0 and ΣFy = 0 (for a particle) or ΣM = 0 as well (for a rigid body).
Problem
Reading of the figure: string AC makes 20° with the vertical at C. String CD makes 85° with the vertical at C, so it is inclined 5° below the horizontal, going from C to D. Loads: 50 N hangs at C, 100 N hangs at D. String DB goes up to B at an angle θ to the vertical.
Let T₁ = T_AC, T₂ = T_CD, T₃ = T_BD.
Joint C (up and right positive)
T_AC pulls up-left, T_CD pulls right and slightly down, 50 N down.
x: −T1 sin20° + T2 sin85° = 0
y: T1 cos20° − T2 cos85° − 50 = 0
From x: T1 = T2 (0.9962/0.3420) = 2.913 T2. In y: 2.913 T2 × 0.9397 − 0.0872 T2 = 50 -> (2.737 − 0.0872) T2 = 50 T₂ = T_CD = 18.87 N, T₁ = T_AC = 54.96 N
Joint D
T_CD pulls D back toward C (left and slightly up), T_BD pulls up-right at θ to the vertical, 100 N down.
x: T3 sinθ = T2 sin85° = 18.87 × 0.9962 = 18.80 N
y: T3 cosθ = 100 − T2 cos85° = 100 − 1.645 = 98.36 N
T₃ = √(18.80² + 98.36²) = 100.14 N θ = tan⁻¹(18.80 / 98.36) = 10.82°
Answer: T_AC = 54.96 N, T_CD = 18.87 N, T_BD = 100.14 N, θ = 10.8°.
✓ Answer checked by Claude Opus 5.5 (high effort)
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