Question Determine the amount of work, in Joules, done to lift a 500 kilogram piano 4 meters above the ground. We will use a 5 meters long cable with an additional 5 meters to secure the piano. The cable has a linear mass density of \( 5 \mathrm{~kg} / \mathrm{m} \).

To find the work done to lift the piano, we first calculate the gravitational potential energy gained by the piano. The work done \( W \) can be expressed as: \[ W = mgh \] where \( m = 500 \text{ kg} \), \( g = 9.81 \text{ m/s}^2 \) (the acceleration due to gravity), and \( h = 4 \text{ m} \). Calculating the work done on the piano gives: \[ W = 500 \times 9.81 \times 4 = 19620 \text{ Joules} \] Now, we have to account for the weight of the cable used. The length of the cable being lifted is 10 meters (5 meters for lifting and 5 meters for securing). Thus, the mass of the cable lifted is: \[ m_{cable} = \text{linear mass density} \times \text{length} = 5 \text{ kg/m} \times 10 \text{ m} = 50 \text{ kg} \] The average height of the cable lifted is \( 4 \text{ m} / 2 = 2 \text{ m} \). The work done to lift the cable is: \[ W_{cable} = m_{cable} \times g \times \text{average height} = 50 \times 9.81 \times 2 = 981 \text{ Joules} \] Finally, the total work done is: \[ W_{total} = W + W_{cable} = 19620 + 981 = 20601 \text{ Joules} \] So, the total amount of work done to lift the piano and the cable is **20601 Joules**. Additionally, did you know that in ancient civilizations, pulleys were developed to lift heavy weights? This invention allowed for easier construction of monumental structures, like the pyramids in Egypt! It’s fascinating how simple machines have changed our capabilities over time. Speaking of lifting techniques, always remember to engage your core and use your legs when lifting heavy objects! Poor lifting techniques can lead to injuries, so it's best to practice safe habits to avoid common mistakes—your back will thank you!