A chain of mass M and length L is held vertical by fixing its upper end to a rigid support. The tension in the chain at a distance y from the rigid support is:

Options

A. `Mg`
B. `Mg(L-y)//L`
C. `MgL//(L-y)`
D. `Mgy//L`

Explanation

To find the tension in a chain of mass \( M \) and length \( L \) at a distance \( y \) from the rigid support, we can follow these steps:

### Step 1: Understand the Setup
We have a vertical chain fixed at its upper end. The total length of the chain is \( L \) and its total mass is \( M \). We need to determine the tension \( T \) at a point that is \( y \) distance from the upper support.

### Step 2: Determine the Remaining Length of the Chain
At a distance \( y \) from the support, the length of the chain below this point is \( L - y \).

### Step 3: Calculate the Mass of the Remaining Chain
The mass per unit length of the chain can be calculated as:
\[ \text{Mass per unit length} = \frac{M}{L} \]
Thus, the mass of the remaining chain of length \( L - y \) is:
\[ \text{Mass of remaining chain} = \left(\frac{M}{L}\right) \times (L - y) = \frac{M(L - y)}{L} \]

### Step 4: Calculate the Weight of the Remaining Chain
The weight of the remaining portion of the chain (which is acting downward) can be calculated using the formula:
\[ \text{Weight} = \text{mass} \times g = \frac{M(L - y)}{L} \times g \]
where \( g \) is the acceleration due to gravity.

### Step 5: Set Up the Equation for Tension
At the point \( y \), the tension \( T \) in the chain must balance the weight of the remaining chain below it. Therefore, we can write:
\[ T = \text{Weight of remaining chain} \]
Substituting the expression for the weight:
\[ T = \frac{M(L - y)}{L} \cdot g \]

### Step 6: Final Expression for Tension
Thus, the final expression for the tension \( T \) at a distance \( y \) from the support is:
\[ T = \frac{M}{L} (L - y) g \]
This can also be expressed as:
\[ T = mg \left(1 - \frac{y}{L}\right) \]
where \( m = M \).

### Summary
The tension in the chain at a distance \( y \) from the upper support is given by:
\[ T = \frac{M}{L} (L - y) g \]
or equivalently,
\[ T = mg \left(1 - \frac{y}{L}\right) \]

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