Final answer:
To find the work done lifting a leaky 10-kg bucket with a weight-dependent rope, we use a Riemann sum to account for the variable weight as water leaks out. At each height, the total weight is calculated, and the work over a small interval is the product of this weight and the height increment. Summing these products from ground level to the final height gives an approximation of the total work.
Step-by-step explanation:
The work done lifting an object can be calculated using the formula Work = Force x Distance. In this case, the force is the weight of the leaky bucket and its remaining water as well as the rope. Since the bucket and water weight decrease as the bucket is lifted, we can use a Riemann sum to approximate the total work.
To set up the Riemann sum, consider dividing the height into small intervals Δh. At each interval, the force is roughly constant, and we can calculate the work done over that small interval as the weight at that height times Δh. Summing these over all intervals from the ground to the height of 11m will approximate the total work done.
At a point h meters off the ground, the bucket still has (33 - 3h) kg of water (since it leaks at a rate such that it is empty at 11m). The bucket's weight is constant at 10 kg. The rope above height h weighs 0.8(11 - h) kg. Hence, the total weight at height h is 98 N * (10 + 33 - 3h + 0.8(11 - h)) kg. The work done lifting from h to h + Δh is this weight times Δh (since speed is constant and acceleration due to gravity is 9.8 m/s²).
The total work done can be approximated by the sum of all these small works: Σ (from h=0 to h=11) of 98 * (10 + 33 - 3h + 0.8(11 - h)) * Δh.