8. A heavy pump casing with a mass m is to be lifted off the ground using a crane. For simplicity, the motion is assumed to be two-dimensional, and the pump casing is represented by a rectangle having side dimensions ab (see figure). A cable of length L1 is attached to the crane (at point P) and the pump casing (at point O). The crane pulls up vertically on the cable with a constant velocity Vp. The center of mass G of the pump casing is assumed to lie in the center of the rectangle. It is located at a distance L2 from point O. The right side of the pump casing is located at a horizontal distance c from the vertical line passing though point P. Find the maximum cable tension during the lift, for the part of the lift before the pump casing loses contact with the ground (lift off). Evaluate for a specific case where: a = 0.4 m b = 0.6 m c = 0.2 m L1 = 3 m m = 200 kg Assume: • The friction between the pump casing and ground is high enough so that the pump casing does not slide along the ground (towards the right), before lift off occurs. • Before lift off occurs, dynamic effects are negligible and this can be treated as a statics problem. • The mass of the cable can be neglected

To determine the maximum cable tension during the lift of the pump casing before it loses contact with the ground, we can analyze the forces acting on the system using principles of statics. Given the parameters provided, we can break down the problem step by step.

Understanding the Forces Involved

When the crane lifts the pump casing, two main forces act on it:

• Weight of the pump casing (W): This is the force due to gravity acting downward, calculated as W = m * g, where g is the acceleration due to gravity (approximately 9.81 m/s²).
• Tension in the cable (T): This is the force exerted by the crane through the cable, acting upward.

Calculating the Weight of the Pump Casing

First, we need to calculate the weight of the pump casing:

Given:

• Mass (m) = 200 kg
• Acceleration due to gravity (g) = 9.81 m/s²

Now, we can calculate the weight:

W = m * g = 200 kg * 9.81 m/s² = 1962 N

Analyzing the Static Conditions

Before the pump casing lifts off, it remains in static equilibrium. This means that the sum of the vertical forces must equal zero. Therefore, we can set up the following equation:

T - W = 0

From this, we can express the tension in the cable:

T = W

Calculating the Maximum Tension

Substituting the weight we calculated earlier:

T = 1962 N

Considering the Horizontal Forces

Since the pump casing does not slide, we also need to consider the horizontal forces acting on it. The horizontal distance (c) from the vertical line through point P to the right side of the pump casing creates a moment about point O. However, since the problem states that the friction is high enough to prevent sliding, we can assume that the horizontal forces do not affect the vertical tension directly before lift-off.

Final Consideration of Moments

To ensure that the pump casing does not tip over before lift-off, we can analyze the moments about point O. The moment due to the weight of the pump casing about point O must be balanced by the moment due to the tension in the cable:

Moment due to weight (W) = W * (a/2) = 1962 N * (0.4 m / 2) = 392.4 Nm

Moment due to tension (T) = T * L2

Since L2 is not provided, we can assume it is the vertical distance from point O to the center of mass G, which is half the height of the pump casing (b/2 = 0.6 m / 2 = 0.3 m). Thus, we can express:

Moment due to tension = T * 0.3 m

Setting the moments equal gives:

392.4 Nm = T * 0.3 m

Solving for T gives:

T = 392.4 Nm / 0.3 m = 1308 N

Conclusion on Maximum Tension

In this specific case, the maximum tension in the cable before the pump casing loses contact with the ground is 1962 N, as this is the force required to lift the weight of the casing. However, considering the moments and the static equilibrium, the effective tension that maintains stability without tipping is 1308 N. Thus, the maximum tension in the cable before lift-off occurs is determined primarily by the weight of the pump casing, which is 1962 N, but the system must also ensure stability, which could lead to a lower effective tension depending on the configuration.