A force of 9 N pulls a block of 4 kg through a rope of mass `0.5` kg. The block is resting on a smooth surface. What is the force of reaction exerted by the block on the rope?

Answer

To solve the problem step by step, we will analyze the forces acting on both the block and the rope, and then use Newton's second law to find the force of reaction exerted by the block on the rope.

### Step 1: Identify the Forces Acting on the Rope
- The rope has a mass of 0.5 kg and is being pulled by a force of 9 N.
- The normal reaction force (N) exerted by the block on the rope acts upwards on the rope.
- The net force acting on the rope can be expressed as:
\[ F_{\text{net, rope}} = 9 \, \text{N} - N \]

### Step 2: Apply Newton's Second Law to the Rope
- According to Newton's second law, the net force is equal to the mass of the rope multiplied by its acceleration (A):
\[ F_{\text{net, rope}} = m_{\text{rope}} \cdot A \]
- Substituting the values, we have:
\[ 9 - N = 0.5 \cdot A \]
- Rearranging gives us:
\[ A = \frac{9 - N}{0.5} \]

### Step 3: Identify the Forces Acting on the Block
- The block has a mass of 4 kg and experiences the normal force (N) from the rope.
- The acceleration of the block is also A since both the block and rope are moving together.

### Step 4: Apply Newton's Second Law to the Block
- For the block, the equation can be expressed as:
\[ N = m_{\text{block}} \cdot A \]
- Substituting the mass of the block:
\[ N = 4 \cdot A \]

### Step 5: Substitute the Expression for A into the Block's Equation
- From the equation for A derived from the rope, substitute into the block's equation:
\[ N = 4 \cdot \left(\frac{9 - N}{0.5}\right) \]
- Simplifying gives:
\[ N = 4 \cdot (18 - 2N) \]
\[ N = 72 - 8N \]
\[ N + 8N = 72 \]
\[ 9N = 72 \]
\[ N = \frac{72}{9} = 8 \, \text{N} \]

### Step 6: Conclusion
- The force of reaction exerted by the block on the rope is **8 N**.