   Chapter 5, Problem 23P

Chapter
Section
Textbook Problem

A 2.10 × 103-kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5.00 m before coming into contact with the top of the beam, and it drives the beam 12.0 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.

To determine
The average force the beam exerts on the pile driver while the pile driver is brought to rest.

Explanation

Section 1:

To determine: the work done that the beam did on the pile driver

Answer: The work done that the beam did on the pile driver is F(0.120m) .

Explanation:

Given Info:

The mass of the pile driver is 2.10×103kg .

The height that the pile driver falls before coming into contact with the top of the beam is 5.0m

The displacement undergone by the driver while in contact with the beam is 12.0cm .

Formula to calculate the work done that the beam did on the pile driver is,

W=(Fcosθ)d

• F is the average force the beam exerts on the pile
• θ is the angle between the force vector and the displacement vector
• d is the magnitude of displacement

Since, the force exerted on the driver and the displacement undergone by the driver is opposite in direction; the angle θ is 180° .

Substitute 180° for θ and 12.0cm for d to find the work done,

W=(Fcos180°)d=F(0.120m)

The work done that the beam did on the pile driver is W=F(0.120m) .

Section 2:

To determine: the work done that the beam did on the pile driver

Answer: Thus, the work done that the beam did on the pile driver is

W=12m(00)+(2100kg)(9.8m/s)(0.120m5.0m)

Explanation:

Given Info:

The mass of the pile driver is 2.10×103kg .

The height that the pile driver falls before coming into contact with the top of the beam is 5.0m

The displacement undergone by the driver while in contact with the beam is 12.0cm .

From the Work-Energy theorem, the work done that the beam did on the pile driver is,

W=(KEfKEi)+(PEfPEi)=12m(vf2vi2)+mg(hfhi)

• m is the mass of the pile driver
• vf is the final velocity of the driver
• vi is the initial velocity of the driver
• hf is the final height
• hi is the initial height

Since the pile driver is starting from rest and ending in rest, both vf and vi are zero

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