The mass of a metallic beam of uniform thickness and of length `6m` is `60kg`. The beam is horizontally and symmetrically lies on two vertical pillars which are separated by a distance `3m`. A person of mass `75kg` is walking on this beam. The closest distance to which the person can approach one end of the beam so that the beam does not tilt down is (neglect thickness of pillars)

Options

A. `30 cm`
B. `20 cm`
C. `15 cm`
D. `10 cm`

Explanation

To solve the problem step by step, we will analyze the situation using the principles of equilibrium and moments.

### Step 1: Understand the System
We have a metallic beam of length 6 m and mass 60 kg, supported symmetrically on two pillars that are 3 m apart. A person with a mass of 75 kg is walking on this beam. We need to find the closest distance that the person can approach one end of the beam without causing it to tilt.

### Step 2: Identify Key Points
- Let the beam be positioned horizontally with points A and D at the ends, and E and F as the pillars.
- The midpoint of the beam (C) is at 3 m from either end (A or D).
- The distance between the two pillars (E and F) is 3 m.

### Step 3: Set Up the Free Body Diagram
- The weight of the beam acts downwards at its center (C), which is 60 kg × g.
- The weight of the person acts downwards at a distance \( x \) from point A.
- Let \( R_E \) be the reaction force at pillar E and \( R_F \) be the reaction force at pillar F.

### Step 4: Conditions for Equilibrium
For the beam to be in equilibrium:
1. The sum of vertical forces must be zero:
\[ R_E + R_F = 60g + 75g \]
2. The sum of moments about any point must be zero. We will take moments about point E.

### Step 5: Calculate Moments about Pillar E
Taking moments about E (clockwise moments are positive):
- Moment due to the weight of the beam (60 kg) at a distance of 1.5 m from E (half of 3 m):
\[ \text{Moment due to beam} = 60g \times 1.5 \]
- Moment due to the person (75 kg) at a distance of \( 3 - x \) from E:
\[ \text{Moment due to person} = 75g \times (3 - x) \]

Setting the sum of moments about E to zero:
\[ 60g \times 1.5 = 75g \times (3 - x) \]

### Step 6: Simplify the Equation
We can cancel \( g \) from both sides:
\[ 60 \times 1.5 = 75 \times (3 - x) \]
Calculating the left side:
\[ 90 = 75 \times (3 - x) \]

### Step 7: Solve for \( x \)
Expanding the right side:
\[ 90 = 225 - 75x \]
Rearranging gives:
\[ 75x = 225 - 90 \]
\[ 75x = 135 \]
\[ x = \frac{135}{75} = 1.8 \text{ m} \]

### Step 8: Find the Closest Distance to the End
The closest distance to end A is \( x = 1.8 \text{ m} \).

### Step 9: Convert to Centimeters
To convert meters to centimeters:
\[ 1.8 \text{ m} = 180 \text{ cm} \]

### Final Answer
The closest distance to which the person can approach one end of the beam without causing it to tilt is **180 cm**.