Question

A 402 -kg pile driver is raised 12 \(\mathrm{m}\) above ground. (a) How much work must be done to raise the pile driver? (b) How much work does gravity do on the driver as it is raised? (c) The driver is now dropped. How much work does gravity do on the driver as it falls?

Short answer

Question: Calculate the work done on the pile driver in parts a, b, and c. Answer: In part a, the work done to raise the pile driver is approximately 47,324.96 Joules. In part b, the work done by gravity on the pile driver while raising is approximately -47,324.96 Joules. In part c, the work done by gravity on the pile driver as it falls is approximately 47,324.96 Joules.

Step-by-step solution

Calculate the gravitational potential energy of the raised pile driver

Before we can calculate the work done on the pile driver while being raised, we need to determine its gravitational potential energy once it reaches the highest point. To do so, we'll use the formula for gravitational potential energy: \(PE = m \cdot g \cdot h\) where: - \(PE\) is the gravitational potential energy - \(m\) is the mass of the pile driver (402 kg) - \(g\) is the acceleration due to gravity (approximately 9.81 m/s²) - \(h\) is the height above the ground (12 m) Now let's calculate the gravitational potential energy: \(PE = 402 \, kg \cdot 9.81 \, \frac{m}{s^2} \cdot 12 \, m\)

Calculate the work done in raising the pile driver (Part a)

The work done to raise the pile driver to its highest point is equal to its gained gravitational potential energy. So using the calculated potential energy in step 1: \(W = PE\) Now substitute the values: \(W = 402 \, kg \cdot 9.81 \, \frac{m}{s^2} \cdot 12 \, m\) Calculate the work: \(W \approx 47,\!324.96 \, Joules\) The work done to raise the pile driver is approximately 47,324.96 Joules.

Calculate the work done by gravity while raising the pile driver (Part b)

When lifting an object, the work done by gravity is equal to the negative of the work done to raise the object. This is because gravity is applying a force in the opposite direction of the motion. Therefore: \(W_g = -W\) Now substitute the value of \(W\) calculated in step 2: \(W_g \approx -47,\!324.96 \, Joules\) The work done by gravity on the pile driver while raising is approximately -47,324.96 Joules.

Calculate the work done by gravity as the pile driver falls (Part c)

When the pile driver is dropped and falls to the ground, gravity is doing positive work on it. In this case, the work done by gravity will be equal to the gravitational potential energy the pile driver initially had at the highest point, which we already calculated in step 1. So: \(W_g = PE\) Using the calculated potential energy: \(W_g \approx 47,\!324.96 \, Joules\) The work done by gravity on the pile driver as it falls is approximately 47,324.96 Joules.