The mass of a uniform ladder of length 5 m is 20 kg. A person of mass 60 kg stands on the ladder at a height of 2 m from the bottom. The position of centre of mass of the ladder and man from the bottom nearly is

Options

A. 1 m
B. 2.5 m
C. 3.5 m
D. 2.125 m

Answer

To find the position of the center of mass of the ladder and the person standing on it, we can follow these steps:

### Step 1: Identify the masses and their positions
- Mass of the ladder (m1) = 20 kg
- Length of the ladder = 5 m
- Center of mass of the ladder is at its midpoint = 5 m / 2 = 2.5 m from the bottom.
- Mass of the person (m2) = 60 kg
- Height of the person from the bottom = 2 m

### Step 2: Use the formula for the center of mass
The center of mass (CM) for a system of particles can be calculated using the formula:

\[ h_{CM} = \frac{m_1 \cdot h_1 + m_2 \cdot h_2}{m_1 + m_2} \]

Where:
- \(h_{CM}\) = height of the center of mass
- \(m_1\) = mass of the ladder
- \(h_1\) = height of the center of mass of the ladder
- \(m_2\) = mass of the person
- \(h_2\) = height of the person

### Step 3: Substitute the values into the formula
Substituting the known values:

\[ h_{CM} = \frac{(20 \, \text{kg} \cdot 2.5 \, \text{m}) + (60 \, \text{kg} \cdot 2 \, \text{m})}{20 \, \text{kg} + 60 \, \text{kg}} \]

### Step 4: Calculate the numerator and denominator
Calculate the numerator:

\[ 20 \cdot 2.5 = 50 \, \text{kg m} \]
\[ 60 \cdot 2 = 120 \, \text{kg m} \]
\[ \text{Total} = 50 + 120 = 170 \, \text{kg m} \]

Calculate the denominator:

\[ 20 + 60 = 80 \, \text{kg} \]

### Step 5: Calculate the center of mass
Now, substitute back into the formula:

\[ h_{CM} = \frac{170 \, \text{kg m}}{80 \, \text{kg}} = 2.125 \, \text{m} \]

### Step 6: Conclusion
The position of the center of mass of the ladder and the man from the bottom is approximately **2.125 m**.

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