A railway engine of mass 50 tons is pulling a wagon of mass 40 tons with a force of 4500N. The resistance force acting is 1N per ton. The tension in the coupling between the engine and the wagon is 

Options

A. 1600 N
B. 2000 N
C. 2500 N
D. 1500 N

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the masses
- The mass of the railway engine \( m_1 = 50 \) tons.
- The mass of the wagon \( m_2 = 40 \) tons.
- Convert these masses into kilograms:
\[ m_1 = 50 \times 10^3 \, \text{kg} = 50000 \, \text{kg} \]
\[ m_2 = 40 \times 10^3 \, \text{kg} = 40000 \, \text{kg} \]

### Step 2: Calculate the total mass
- The total mass \( m \) of the system (engine + wagon) is:
\[ m = m_1 + m_2 = 50000 \, \text{kg} + 40000 \, \text{kg} = 90000 \, \text{kg} \]

### Step 3: Calculate the resistance force
- The resistance force is given as \( 1 \, \text{N per ton} \).
- Total resistance force \( R \) acting on the system:
\[ R = (m_1 + m_2) \times 1 \, \text{N/ton} = (50 + 40) \, \text{N} = 90 \, \text{N} \]

### Step 4: Calculate the net force
- The net force \( F_{\text{net}} \) acting on the system is given as \( 4500 \, \text{N} \).

### Step 5: Calculate the acceleration of the system
- Using Newton's second law \( F = ma \):
\[ F_{\text{net}} = m \cdot a \]
Rearranging for acceleration \( a \):
\[ a = \frac{F_{\text{net}} - R}{m} \]
Substituting the values:
\[ a = \frac{4500 \, \text{N} - 90 \, \text{N}}{90000 \, \text{kg}} = \frac{4410 \, \text{N}}{90000 \, \text{kg}} \approx 0.049 \, \text{m/s}^2 \]

### Step 6: Set up the equation for the engine
- For the engine, the forces acting on it can be expressed as:
\[ F_{\text{net}} - T - R_{\text{engine}} = m_1 \cdot a \]
where \( T \) is the tension in the coupling and \( R_{\text{engine}} \) is the resistance force acting on the engine:
\[ R_{\text{engine}} = 1 \, \text{N/ton} \times 50 \, \text{tons} = 50 \, \text{N} \]

### Step 7: Substitute values into the equation
- Substitute the known values into the equation:
\[ 4500 \, \text{N} - T - 50 \, \text{N} = 50000 \, \text{kg} \cdot 0.049 \, \text{m/s}^2 \]
\[ 4500 - T - 50 = 2450 \]
\[ 4450 - T = 2450 \]

### Step 8: Solve for Tension \( T \)
- Rearranging gives:
\[ T = 4450 - 2450 = 2000 \, \text{N} \]

### Final Answer
The tension in the coupling between the engine and the wagon is \( T = 2000 \, \text{N} \).

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