Question

You have a vacation cabin that has a nearby solar (black) water container used to provide a warm outdoor shower. For a few days last summer, your pump went out and you had to personally haul the water up the 4.4 m from the pond to the tank. Suppose your bucket has a mass of 4.7 kg and holds 14.1 kg of water when it is full. However, the bucket has a hole in it, and as you moved it vertically at a constant speed v, water leaked out at a constant rate. By the time you reached the top, only 5.4 kg of water remained.

a)  Write an expression for the mass of the bucket plus water as a function of the height above the pond surface. (Use the following as necessary: y for the height above the pond surface.)

b) Find the work done by you on the bucket for each 5.4 kg of water delivered to the tank.

PLEASE SHOW ALL WORK, THIS IS A TEST QUESTION

Answer 1

Height through which bucket is lifted = h = 4.4 m

mass of bucket = m₀ = 4.7 kg

Initial mass of water = m₁ = 14.1 kg

Final mass of water = m₂ = 5.4 kg

since rate of loss of water is constant , rate of loss R = ( m₂ - m₁ ) / h

a) hence mass of bucket + water as a function of height y is m(y) = m₀ + m₁ + ( m₂ - m₁ ) y / h

b) work done to lift this mass to a height y is $\int{dW} = \int_0^h {(m_0+m)g dy} = \int_0^h {\left(m_0+m_1 + \frac{(m_2-m_1)y}{h} \right )gdy}$

$W = (m_0+m_1)gh+ \frac{(m_2-m_1)gh}{2} =623.084 \,J$