A 2kg block lying on smooth table which is connected by a body of mass 1 kg by a string which passes through a pulley. Take 1 kg mass hanging vertically. Find the acceleration of block & tension in string.

Answer

To solve the problem, we need to analyze the forces acting on both the 2 kg block on the table and the 1 kg mass hanging vertically. We'll use Newton's second law of motion to find the acceleration and the tension in the string.

### Step 1: Identify the masses and forces
- Mass of the block on the table (m1) = 2 kg
- Mass of the hanging block (m2) = 1 kg
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Write the equations of motion
For the hanging mass (m2):
- The forces acting on it are the gravitational force (weight) downward and the tension (T) in the string upward.
- Using Newton's second law, we can write the equation:
\[ m_2 g - T = m_2 a \]
Substituting the values:
\[ 1 \cdot 9.8 - T = 1 \cdot a \quad \text{(1)} \]

For the block on the table (m1):
- The only horizontal force acting on it is the tension (T) in the string.
- Again using Newton's second law, we write:
\[ T = m_1 a \]
Substituting the values:
\[ T = 2a \quad \text{(2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 9.8 - T = a \)
2. \( T = 2a \)

Substituting equation (2) into equation (1):
\[ 9.8 - 2a = a \]
Combine like terms:
\[ 9.8 = 3a \]
Now, solve for \( a \):
\[ a = \frac{9.8}{3} \approx 3.27 \, \text{m/s}^2 \]

### Step 4: Find the tension in the string
Now that we have the acceleration, we can find the tension using equation (2):
\[ T = 2a = 2 \cdot 3.27 \approx 6.54 \, \text{N} \]

### Final Results
- The acceleration of the block is approximately \( 3.27 \, \text{m/s}^2 \).
- The tension in the string is approximately \( 6.54 \, \text{N} \).