a pile drive has a 413 kg block that is lifted 1.4m and then dropped onto the pile being driven into the ground. suppose that it takes 24ms to stop when it hits the pile. what is the average force exerted on the pile by the pile driver?
Force x distance driven =
Potential energy of the lifted weight
= (413)(9.8)(1.4) J
The distance X that the pile gets driven each time is (1/2)*(pile impact velocity)*time
The impact velocity of the pike is
V = sqrt(2gH)
Compute V, X, and then solve for the force.
I just realized that this problem can be solved more directly using momentum considerations. The answer should be the same.
(Force)(stopping time)
= momentum at impact = M V
F = M V/(0.024 s)
V = sqrt(2gH), as before
A spelunker (cave explorer) drops a stone from rest into a hole. The speed of sound is 343 m/s in air, and the sound of the stone striking the bottom is heard 3.87 s after the stone is dropped. How deep is the hole?
To find the average force exerted on the pile by the pile driver, we can use the principle of work and energy.
1. First, let's calculate the potential energy of the block when it is lifted to a height of 1.4m. The potential energy formula is given by: PE = mgh, where m is the mass of the block (413 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (1.4m).
PE = (413 kg) * (9.8 m/s²) * (1.4m)
PE ≈ 5791 J (Joules)
2. When the block is dropped onto the pile, the potential energy is converted into kinetic energy. The formula for kinetic energy is: KE = (1/2)mv², where m is the mass of the block and v is the velocity of the block just before it hits the pile.
We can find the velocity using the formula: v = d / t, where d is the distance (height) the block falls and t is the time it takes to fall.
v = (1.4m) / (24 ms) = (1.4m) / (0.024s)
v ≈ 58.33 m/s
Now, let's calculate the kinetic energy:
KE = (1/2) * (413 kg) * (58.33 m/s)²
KE ≈ 282,409 J
3. The average force can be found using the work-energy principle, which states that the work done on an object is equal to the change in its energy.
The work done (W) on the block is equal to the change in energy, which is the difference between the potential energy (PE) and the kinetic energy (KE).
W = ΔE = KE - PE
W = 282,409 J - 5791 J
W ≈ 276,618 J
4. Finally, the average force (F) exerted on the pile by the pile driver can be calculated using the formula: F = W / d, where W is the work done, and d is the distance over which the force is exerted (in this case, the distance the pile is driven into the ground).
F = 276,618 J / (unknown distance)
Unfortunately, without knowing the distance the pile is driven, we cannot determine the exact average force exerted on the pile.