The centre of mass of a system of two uniform spherical masses of 5 kg and 35 kg with centres of them 0.7 m apart is

Options

A. 0.6125 m from 35 kg
B. 0.6125 m from 5 kg
C. 0.35 m from 35 kg
D. 0.35 m from 5 kg

Answer

To find the center of mass of a system of two uniform spherical masses, we can follow these steps:

### Step 1: Identify the masses and their positions
Let:
- \( M_1 = 5 \, \text{kg} \) (mass 1)
- \( M_2 = 35 \, \text{kg} \) (mass 2)
- The distance between the centers of the two masses, \( d = 0.7 \, \text{m} \)

### Step 2: Define a coordinate system
Assume the position of mass \( M_1 \) (5 kg) is at \( x_1 = 0 \, \text{m} \) and the position of mass \( M_2 \) (35 kg) is at \( x_2 = 0.7 \, \text{m} \).

### Step 3: Use the formula for the center of mass
The formula for the center of mass \( x_{cm} \) of a two-mass system is given by:
\[ x_{cm} = \frac{M_1 x_1 + M_2 x_2}{M_1 + M_2} \]

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ x_{cm} = \frac{(5 \, \text{kg} \cdot 0 \, \text{m}) + (35 \, \text{kg} \cdot 0.7 \, \text{m})}{5 \, \text{kg} + 35 \, \text{kg}} \]

### Step 5: Calculate the numerator and denominator
Calculate the numerator:
\[ = 0 + (35 \cdot 0.7) = 24.5 \, \text{kg} \cdot \text{m} \]
Calculate the denominator:
\[ = 5 + 35 = 40 \, \text{kg} \]

### Step 6: Calculate the center of mass
Now, substituting back into the equation:
\[ x_{cm} = \frac{24.5}{40} = 0.6125 \, \text{m} \]

### Step 7: Interpret the result
Since we took the position of \( M_1 \) (5 kg) as our reference point (0 m), the center of mass is located 0.6125 m from the position of \( M_1 \).

### Final Answer
The center of mass of the system is \( 0.6125 \, \text{m} \) from the 5 kg mass.

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