A 120-MT concrete beam is lifted from the floor by two tandem heavy lift cranes up to a target height of 7.5 meters. (Refer to Diagram)
Diagram
Calculate the weight of the beam. Use g = 9.8 m/s^2.
Calculate the potential energy of the beam when resting at the floor
Calculate the potential energy of the beam when it is halfway the target height
Calculate the potential energy of the beam when it is at 7.5 meters.
Calculate the work done by each crane to lift the beam up to 9 meters given that they are synchronized and were able to divide the load equally.
To calculate the weight of the beam, we can use the equation:
Weight = mass x acceleration due to gravity
Given that the mass of the beam is 120 MT (or 120,000 kg) and the acceleration due to gravity is 9.8 m/s^2, we can calculate the weight:
Weight = 120,000 kg x 9.8 m/s^2 = 1,176,000 N
The potential energy of an object is given by the equation:
Potential energy = weight x height
When the beam is resting at the floor, the height is 0 meters, so the potential energy is:
Potential energy = 1,176,000 N x 0 m = 0 J
When the beam is halfway to the target height of 7.5 meters, the height is 7.5/2 = 3.75 meters. The potential energy is:
Potential energy = 1,176,000 N x 3.75 m = 4,410,000 J
When the beam is at the target height of 7.5 meters, the potential energy is:
Potential energy = 1,176,000 N x 7.5 m = 8,820,000 J
To calculate the work done by each crane, we need to calculate the total work done and then divide it equally between the two cranes. The work done is given by the equation:
Work = force x distance
The force is equal to the weight of the beam, which is 1,176,000 N. The distance lifted is 7.5 - 0 = 7.5 meters. So the total work done is:
Total work = 1,176,000 N x 7.5 m = 8,820,000 J
Since the work is divided equally between the two cranes, each crane would do:
Work per crane = Total work / 2 = 8,820,000 J / 2 = 4,410,000 J
Therefore, each crane would do 4,410,000 J of work to lift the beam up to 9 meters.