A 2,300-kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 3.00 m before coming into contact with the top of the beam, and it drives the beam 9.4 cm farther into the ground as it comes to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.
mgh=mv²/2
v²=2gh,
a= v²/2s=2gh/2s
F=ma
To calculate the average force exerted by the beam on the pile driver, we can use the principle of conservation of energy.
The potential energy of the pile driver before it falls is given by the equation:
PE_initial = m * g * h
Where:
m = mass of the pile driver = 2300 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height fallen by the pile driver = 3.00 m
PE_initial = 2300 kg * 9.8 m/s^2 * 3.00 m
PE_initial = 67,860 J
The potential energy of the pile driver after it has driven the beam further into the ground is given by:
PE_final = m * g * h'
Where:
h' = additional height the pile driver drives the beam into the ground = 9.4 cm = 0.094 m
PE_final = 2300 kg * 9.8 m/s^2 * 0.094 m
PE_final = 20,972 J
The difference in potential energy is equal to the work done by the beam on the pile driver:
Work = PE_initial - PE_final
Work = 67,860 J - 20,972 J
Work = 46,888 J
The average force exerted by the beam on the pile driver is equal to the work divided by the distance traveled:
Average Force = Work / Distance
Distance = 3.00 m + 0.094 m = 3.094 m
Average Force = 46,888 J / 3.094 m
Average Force ≈ 15,150.49 N
Therefore, the average force exerted by the beam on the pile driver is approximately 15,150.49 Newtons.
To calculate the average force exerted by the beam on the pile driver, we can use the principle of conservation of energy.
The potential energy of the pile driver is converted into kinetic energy as it falls, and then into potential energy again as it compresses the beam and comes to rest. The work done by the beam to bring the pile driver to rest is equal to the change in potential energy of the pile driver.
First, let's calculate the potential energy of the pile driver before it falls. The potential energy of an object near the surface of the Earth is given by the equation:
Potential Energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
PE = 2300 kg * 9.8 m/s^2 * 3.00 m
PE = 67620 J
Next, let's calculate the potential energy of the pile driver after it comes to rest. The change in potential energy is equal to the work done by the beam.
Change in Potential Energy = Work Done by the Beam
Change in Potential Energy = PE_final - PE_initial
Change in Potential Energy = 0 - 67620 J
Change in Potential Energy = -67620 J
Now, we know that the change in potential energy is equal to the work done by the beam. The work done is given by the equation:
Work = Force * Distance
Therefore, we can rearrange the equation to solve for the force:
Force = Work / Distance
Force = -67620 J / 0.094 m
Force = -719361 J/m
The negative sign indicates that the force is acting in the opposite direction of the displacement. Therefore, the average force exerted by the beam on the pile driver while bringing it to rest is approximately 719,361 newtons (N).