/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 A body of mass \(m\) is dropped ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 43

A body of mass \(m\) is dropped from a height \(h\) on a sand floor. If the body penetrates \(x \mathrm{~m}\) into the sand, the average resistance offered by the sand to the body is (A) \(m g\left(\frac{h}{x}\right)\) (B) \(m g\left(1+\frac{h}{x}\right)\) (C) \(m g h+m g x\) (D) \(m g\left(1-\frac{h}{x}\right)\)

Short Answer

Expert verified
(A) \(mg\left(\frac{h}{x}\right)\)

Step by step solution

01

List down given values and choose correct formula

We're given the mass m of the body, the height h from which it is dropped, and the penetration depth x. We will use the work-energy principle to relate the work done on the object to the change in its kinetic energy: \[ W = \Delta KE \]
02

Calculate initial and final kinetic energies

The initial kinetic energy (KE_initial) is 0 because the body starts from rest. The final kinetic energy (KE_final) is also 0 because the body comes to a stop after penetrating the sand. Therefore, \(\Delta KE = KE_{final} - KE_{initial} = 0\).
03

Calculate work done by gravity and sand

The work done on the body consists of two components: work done by gravity (W_gravity) as it falls from the height h and work done by sand (W_sand) when it penetrates x meters into the sand floor. To get the total work done on the object (W), we add these two components: \[ W = W_{gravity} + W_{sand} \]
04

Calculate work done by gravity

The work done by gravity (W_gravity) is the product of the force of gravity (mg) and the height h from which the body was dropped: \[ W_{gravity} = mgh \]
05

Calculate work done by sand and average resistance

We know that the total work done (W) is 0 and the work done by gravity is mgh. We can now solve for the work done by the sand (W_sand): \[ W_{sand} = - mgh \] Finally, to calculate the average resistance (R_average) offered by the sand, we divide the work done by the sand by the penetration depth x: \[ R_{average} = \frac{-W_{sand}}{x} = \frac{mgh}{x} \] So, the average resistance offered by the sand is: (A) \(mg\left(\frac{h}{x}\right)\)

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