An overhead crane is being directed to construct a multistoried building. The horizontal arm of the crane has a linear mass density of 100 kg/m and is 50 m in length. Short arm on the opposite side of the support is 50 m long. A Pulley block on the long arm ,which can be moved along the arm ,weights 500 kilogram. Ignore the mass of the vertical frame. The vertical frame will twist and break if there is an excess imbalance of more than 10%. What is the minimum counter balance required on the short arm which is to be installed on a permanent basis.?

To determine the minimum counterbalance required on the short arm of the crane, we need to analyze the forces and moments acting on the crane system. The goal is to ensure that the moments about the pivot point remain balanced, especially considering the maximum allowable imbalance of 10%. Let's break this down step by step.

Understanding the System

The crane consists of a long arm and a short arm, both measuring 50 meters. The long arm has a linear mass density of 100 kg/m, and there is a movable pulley block weighing 500 kg attached to it. The short arm will need a counterbalance to maintain stability.

Calculating the Weight of the Long Arm

First, we need to calculate the total weight of the long arm:

• Length of the long arm = 50 m
• Linear mass density = 100 kg/m
• Total mass of the long arm = 100 kg/m * 50 m = 5000 kg

Finding the Total Moment on the Long Arm

The total weight acting on the long arm includes both the weight of the arm itself and the weight of the pulley block:

• Weight of the long arm = 5000 kg * 9.81 m/s² = 49050 N
• Weight of the pulley block = 500 kg * 9.81 m/s² = 4905 N
• Total downward force on the long arm = 49050 N + 4905 N = 53955 N

The moment about the pivot point due to this downward force can be calculated as:

• Moment = Force * Distance
• Distance from pivot to the center of mass of the long arm = 50 m / 2 = 25 m
• Moment due to the long arm = 49050 N * 25 m = 1226250 N·m
• Moment due to the pulley block = 4905 N * 50 m = 245250 N·m
• Total moment on the long arm = 1226250 N·m + 245250 N·m = 1471500 N·m

Calculating the Required Counterbalance Moment

To maintain balance, the moment created by the counterbalance on the short arm must equal the total moment from the long arm. The short arm is also 50 m long, so we can express the required counterbalance weight:

• Let the counterbalance weight be W (in Newtons).
• Moment due to the counterbalance = W * 50 m

Setting Up the Equation

For balance, we set the moments equal:

• W * 50 m = 1471500 N·m
• W = 1471500 N·m / 50 m = 29430 N

Converting Weight to Mass

To find the mass of the counterbalance, we convert the weight back to mass:

• Weight (W) = mass (m) * g (acceleration due to gravity)
• m = W / g = 29430 N / 9.81 m/s² ≈ 2995 kg

Considering the 10% Imbalance Limit

Since the vertical frame can only tolerate a 10% imbalance, we need to calculate the maximum allowable moment:

• Maximum allowable moment = 1471500 N·m * 1.1 = 1618650 N·m

Now, we can recalculate the required counterbalance weight:

• W' * 50 m = 1618650 N·m
• W' = 1618650 N·m / 50 m = 32373 N
• Mass for maximum allowable moment = 32373 N / 9.81 m/s² ≈ 3295 kg

Final Recommendation

To ensure the crane remains stable and within the allowable limits, a counterbalance of approximately 3295 kg should be installed on the short arm. This will provide the necessary support to prevent any twisting or breaking of the vertical frame during operation.