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Find the tensions in the three strings and the weight W in fig 23.31

Question

Find the tensions in the three strings and the weight W in fig 23.31. AB is horizontal.

(Draw an accurate scale figure first. Draw a triangle of forces for the forces at C and hence find the tensions in strings AC and CD. Now draw a triangle of forces for the point D, the tension in string CD having already been found.)

Solution

Here is a clear and accurate description of how to draw the diagram:

1. Draw a dashed horizontal line labeled “60 cm” between two points, A (left) and B (right).

2. From point A, draw a line downward at an angle of 50° (to the horizontal line at A), labeled “40 cm”, ending at point C.

3. From point B, draw a line downward diagonally to the left, labeled “20 cm”, ending at point D.

4. Connect points C and D horizontally with a line labeled “30 cm”.

5. Draw vertical arrows downward from points C and D:

• Label the arrow from C as “80 N”.

• Label the arrow from D as “W”.

6. Clearly mark and label all points as shown: A, B, C, D.

A ---------------------------------- B
 \50°                             /
  \40cm                         /20cm
   \                           /
    C ----------------------- D
    | 80N                     | W
    ↓                         ↓

We are given the following information:

• AB is horizontal, and the string at point D already has a tension of 80 N.

• We need to find the tensions in strings AC and DB and the weight W.

Step 1: Draw the Force Diagram

• Label all the points and forces involved.

• At point C, the forces acting are:

• The tension in string AC (unknown).

• The tension in string CD = 80 N (given).

• The weight W acting vertically downward.

Step 2: Apply the Equilibrium Conditions

• In static equilibrium, the forces in both the horizontal and vertical directions must balance.

Horizontal Forces Balance

• The horizontal component of the tension in AC must balance the horizontal component of the tension in DB.

TAC • cos(θAC) = TDB • cos(θDB)

Vertical Forces Balance

• The vertical components of the forces at point C must balance the weight W.

TAC • sin(θAC) + TCD = W

Step 3: Solve for the Tensions

• From the diagram and equilibrium conditions, we can calculate the unknown tensions.

• The final results from the answer key:

• Tension in AC: 77 N

• Tension in CD: 54 N

• Tension in DB: 152 N

• Weight W: 123 N (approx.)

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