A uniform rope of mass M and length L is fixed at its upper end vertically from a rigid support. Then the tension in the rope at the distance l from the rigid support is

(1)MgL/L+l

(2)Mg/L(L-l)

(3)Mg

(4)l/L Mg

To determine the tension in a uniform rope of mass M and length L at a distance l from the rigid support, we need to consider the forces acting on the rope and how they change along its length. The rope is hanging vertically, and the tension varies depending on how far you are from the top. Let's break this down step by step.

Understanding the Forces in Play

The rope has a uniform mass distribution, meaning that its weight is evenly spread along its length. The total weight of the rope is given by:

• Weight (W) = Mg, where g is the acceleration due to gravity.

Analyzing the Segment of the Rope

When we look at a segment of the rope, specifically the portion that is below the point where we are measuring the tension (at distance l), we can see that the tension at that point must support not only the weight of the rope below it but also the weight of the segment above it. The length of the rope below the point of interest is (L - l).

Calculating the Weight of the Rope Below

The weight of the segment of the rope that is below the point at distance l can be calculated as follows:

• Weight of the segment below = (mass per unit length) × (length below)
• The mass per unit length of the rope is M/L.
• Thus, the weight of the segment below is W_b = (M/L) × (L - l) × g.

Setting Up the Equation for Tension

The tension T at the distance l from the top must balance the weight of the segment below it. Therefore, we can express this relationship as:

• T = W_b = (M/L) × (L - l) × g

Final Expression for Tension

Now, substituting the expression for the weight of the segment below into our equation gives us:

• T = (Mg/L) × (L - l)

Identifying the Correct Option

Now, let's compare our derived expression with the options provided:

• (1) MgL/(L + l)
• (2) Mg/L(L - l)
• (3) Mg
• (4) l/L Mg

The correct expression we derived is equivalent to option (2): Mg/L(L - l).

Conclusion

Thus, the tension in the rope at a distance l from the rigid support is given by option (2): Mg/L(L - l). This analysis shows how the tension in a hanging rope varies with distance from the support, reflecting the weight of the rope below that point.