A pile driver drives a post into the ground.The mass of the pile driver is 2 600 kg and it is dropped through a height of 9.0 m on each stroke.If the resisting force of the ground is 4.0 × 10⁶ N,how far is the post driven in on each stroke? A)9.2E+1 cm B)5.9E-3 cm C)7.1E-4 cm D)5.7E-2 cm

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Quizplus · Accepted Answer

To find how far the post is driven into the ground on each stroke, we can use the work-energy principle. The work done by the pile driver on the post is equal to the change in potential energy of the pile driver as it falls. This work is then used to overcome the resisting force of the ground.The potential energy lost by the pile driver is given by $PE = mgh$, where $m$ is the mass of the pile driver, $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$), and $h$ is the height from which the pile driver is dropped.Substituting the given values, we get:$$PE = 2600 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 9.0 \, \text{m} = 229320 \, \text{J}$$This energy is used to overcome the resisting force of the ground and drive the post in. The work done against the resisting force is given by $W = Fd$, where $F$ is the resisting force and $d$ is the distance the post is driven in. Rearranging for $d$, we get:$$d = \frac{W}{F} = \frac{PE}{F}$$Substituting the values, we get:$$d = \frac{229320 \, \text{J}}{4.0 \times 10^6 \, \text{N}} = 0.0573 \, \text{m}$$Converting to centimeters (since 1 m = 100 cm), we get:$$d = 0.0573 \, \text{m} \times 100 = 5.73 \, \text{cm}$$Therefore, the post is driven in 5.73 cm on each stroke, which corresponds to choice D (5.7E-2 cm).

A Pile Driver Drives a Post into the Ground