Answered

Find the breadth of a beam of similar material, 3 m long and 80 mm deep, to have equal strength as a beam that is 5 m long, 40 mm broad, and 100 mm deep.

Note: The strength of a beam varies directly with its breadth and the square of its depth, and inversely with its length.



Answer :

Sure, let’s solve this step-by-step to find the requested breadth.

The strength of a beam varies according to its characteristics such as breadth ([tex]\(b\)[/tex]), depth ([tex]\(d\)[/tex]), and length ([tex]\(l\)[/tex]). Given that the strength ([tex]\(S\)[/tex]) of a beam is directly proportional to the breadth, the square of its depth, and inversely proportional to its length, we can write:
[tex]\[ S \propto \frac{b \cdot d^2}{l} \][/tex]

### Given
- Beam 1:
- Length ([tex]\(l_1\)[/tex]) = 5 meters
- Breadth ([tex]\(b_1\)[/tex]) = 40 mm
- Depth ([tex]\(d_1\)[/tex]) = 100 mm

- Beam 2:
- Length ([tex]\(l_2\)[/tex]) = 3 meters
- Depth ([tex]\(d_2\)[/tex]) = 80 mm
- Breadth ([tex]\(b_2\)[/tex]) needs to be calculated

### Step-by-Step Calculation:

1. Strength Formula for Both Beams:

[tex]\[ S_1 = \frac{b_1 \cdot d_1^2}{l_1} \][/tex]
[tex]\[ S_2 = \frac{b_2 \cdot d_2^2}{l_2} \][/tex]

2. Equal Strength Condition:

Since the beams have equal strength, we set [tex]\(S_1 = S_2\)[/tex]:

[tex]\[ \frac{b_1 \cdot d_1^2}{l_1} = \frac{b_2 \cdot d_2^2}{l_2} \][/tex]

3. Substitute Given Values:

[tex]\[ \frac{40 \cdot (100)^2}{5} = \frac{b_2 \cdot (80)^2}{3} \][/tex]

4. Simplify Left-Hand Side:

[tex]\[ \frac{40 \cdot 10000}{5} = \frac{40 \cdot 2000} = 80000 \][/tex]

5. Set Up Equality:

[tex]\[ 80000 = \frac{b_2 \cdot 6400}{3} \][/tex]

6. Solve for [tex]\(b_2\)[/tex]:

[tex]\[ 80000 \times 3 = b_2 \cdot 6400 \][/tex]
[tex]\[ 240000 = b_2 \cdot 6400 \][/tex]
[tex]\[ b_2 = \frac{240000}{6400} \][/tex]
[tex]\[ b_2 = 37.5 \text{ mm} \][/tex]

Therefore, the breadth of the second beam must be [tex]\(37.5\)[/tex] mm to have an equal strength as the first beam.