A block of mass M in slides on a frictionless surface. Find an expression for the tension in the string. Express your answer in terms of the variables  m ,  M , and appropriate constants.

In this problem, we are likely dealing with two blocks connected by a string where one block is on a frictionless surface and the other is hanging over the edge, influenced by gravity. Here’s a common setup to derive the expression for the tension in the string:

### Problem Setup:
- A block of mass \( M \) is on a frictionless horizontal surface.
- Another block of mass \( m \) is hanging vertically, connected to \( M \) by a string passing over a frictionless pulley.
- Both blocks are connected by the string, and as the block \( m \) falls under gravity, it accelerates both blocks.

### Step-by-Step Solution:

1. **Identify the forces acting on each block**:
   - For the block of mass \( M \) (on the horizontal surface):
     - It experiences no friction, so the only horizontal force is the tension \( T \) in the string.
     - Newton's second law for \( M \): 
     \[
     T = M a \tag{1}
     \]
     where \( a \) is the acceleration of the system (the same for both blocks since they are connected by the string).

   - For the block of mass \( m \) (hanging vertically):
     - It experiences two forces: the downward gravitational force \( mg \) and the upward tension \( T \) in the string.
     - Newton's second law for \( m \):
     \[
     mg - T = m a \tag{2}
     \]

2. **Combine the equations**:
   From equation (1), we have \( T = M a \), and from equation (2), \( mg - T = m a \). Substituting \( T = M a \) into equation (2), we get:
   \[
   mg - M a = m a
   \]

3. **Solve for acceleration \( a \)**:
   \[
   mg = (M + m) a
   \]
   \[
   a = \frac{mg}{M + m} \tag{3}
   \]

4. **Find the tension \( T \)**:
   Now, substitute the value of \( a \) from equation (3) into \( T = M a \):
   \[
   T = M \left( \frac{mg}{M + m} \right)
   \]
   \[
   T = \frac{M m g}{M + m}
   \]

### Final Answer:
The tension in the string is given by:
\[
T = \frac{M m g}{M + m}
\]

This expression is in terms of the masses \( M \), \( m \), and the gravitational acceleration \( g \).

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Would you like more details on this solution or have any questions? 

Here are 5 related questions you might find useful to explore:
1. How would the tension change if the mass \( m \) is much larger than \( M \)?
2. What happens to the acceleration if \( M \) becomes very large compared to \( m \)?
3. How would the solution change if there was friction on the surface?
4. What would the tension be if both masses were equal?
5. How does the tension change if the string is not massless?

**Tip**: When solving problems with connected objects, always treat the acceleration of the system as the same for all connected parts, unless stated otherwise.

A block of mass M slides on a frictionless surface. Find an expression for the tension in the string. Express your answer in terms of the variables m, M, and appropriate constants.

Discover how to find the tension in a string connecting two blocks, with one on a frictionless surface and the other hanging vertically. Using Newton's Second Law, this problem explores tension, forces, and acceleration. Suitable for Grades 9-12 students.

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