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. Calculate the weight of the beam. 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 need to convert the mass into weight. The weight of an object can be calculated by multiplying its mass by the acceleration due to gravity.
Weight = mass * acceleration due to gravity
Weight = 120 MT * 9.8 m/s^2
Weight = 1176 kN
(Note: 1 metric ton = 1000 kg = 9.8 kN)
The potential energy of an object is given by the equation:
Potential energy = weight * height
Potential energy when resting at the floor:
Potential energy = 1176 kN * 0 m
Potential energy = 0 kN-m
Potential energy when halfway the target height (3.75 meters):
Potential energy = 1176 kN * 3.75 m
Potential energy = 4410 kN-m
Potential energy when it is at 7.5 meters:
Potential energy = 1176 kN * 7.5 m
Potential energy = 8820 kN-m
To calculate the work done by each crane to lift the beam up to 9 meters, we need to calculate the change in potential energy. The work done is equal to the change in potential energy.
Work done = potential energy at final position - potential energy at initial position
Work done = (1176 kN * 9 m) - (1176 kN * 7.5 m)
Work done = 10,584 kN-m
Since the cranes are synchronized and the load is divided equally, each crane would have done half of the work.
Work done by each crane = (10,584 kN-m) / 2
Work done by each crane = 5292 kN-m
To calculate the weight of the concrete beam, we simply need to convert the mass from metric tons (MT) to kilograms (kg) and then multiply it by the acceleration due to gravity.
Weight = Mass * Acceleration due to gravity
1 metric ton = 1000 kilograms
Acceleration due to gravity ≈ 9.8 m/s²
Weight = 120 MT * 1000 kg/MT * 9.8 m/s²
Weight = 1,176,000 kg * 9.8 m/s²
Weight = 11,500,800 N
Therefore, the weight of the concrete beam is 11,500,800 Newtons (N).
Now let's calculate the potential energy (PE) of the beam at various points.
Potential Energy = Mass * Gravitational Acceleration * Height
1. Potential energy at the floor:
Since the beam is resting at the floor, the height is 0 meters.
PE(floor) = 120 MT * 1000 kg/MT * 9.8 m/s² * 0 m
PE(floor) = 0 Joules (J)
2. Potential energy when the beam is halfway to the target height (3.75 meters):
PE(halfway) = 120 MT * 1000 kg/MT * 9.8 m/s² * 3.75 m
PE(halfway) = 4,117,500 J
3. Potential energy when the beam is at the target height (7.5 meters):
PE(target) = 120 MT * 1000 kg/MT * 9.8 m/s² * 7.5 m
PE(target) = 8,235,000 J
Now let's calculate the work done by each crane to lift the beam up to 9 meters, assuming they divide the load equally.
Work = Force * Distance
The force each crane exerts is half the weight of the beam, which is 11,500,800 N / 2 = 5,750,400 N.
Work = 5,750,400 N * 9 m
Work = 51,753,600 J
Therefore, each crane would have done 51,753,600 Joules of work to lift the beam up to 9 meters.
To calculate the weight of the beam, you can use the formula:
Weight = mass * acceleration due to gravity.
Given that the mass of the beam is 120 MT (which is equivalent to 120,000 kg) and the acceleration due to gravity is 9.8 m/s^2, you can calculate the weight as follows:
Weight = 120,000 kg * 9.8 m/s^2 = 1,176,000 Newtons.
Next, to calculate the potential energy of the beam when resting at the floor, you can use the formula:
Potential energy = weight * height.
Since the beam is at a height of 0 meters when resting on the floor, the potential energy is given by:
Potential energy = 1,176,000 N * 0 m = 0 Joules.
To calculate the potential energy of the beam when it is halfway to the target height (3.75 meters), you can use the same formula:
Potential energy = weight * height.
The height in this case is 3.75 meters, so the potential energy is:
Potential energy = 1,176,000 N * 3.75 m = 4,410,000 Joules.
Similarly, when the beam is at the target height of 7.5 meters, the potential energy is:
Potential energy = weight * height.
The height in this case is 7.5 meters, so the potential energy is:
Potential energy = 1,176,000 N * 7.5 m = 8,820,000 Joules.
Lastly, to calculate the work done by each crane to lift the beam up to 9 meters, you first need to determine the change in potential energy.
Change in potential energy = final potential energy - initial potential energy.
The final potential energy when the beam is at 9 meters is:
Potential energy = weight * height.
The height in this case is 9 meters, so the potential energy is:
Potential energy = 1,176,000 N * 9 m = 10,584,000 Joules.
Since the initial potential energy was 0 Joules (when the beam was at the floor), the change in potential energy is:
Change in potential energy = 10,584,000 J - 0 J = 10,584,000 Joules.
Since the load is divided equally between the two cranes, each crane has to do half of the work. Therefore, the work done by each crane is:
Work done = Change in potential energy / 2.
Work done = 10,584,000 J / 2 = 5,292,000 Joules.