Week 7 recitation questions
A. Preamble
In lectures, we defined the work W done by a constant force F on an object that is displaced a distance d in
the direction of the force to be W = F d. The units we use are Newton-metres; that is, joules (J). Defining
work becomes more difficult if the force is not constant.
We considered the following question.
1. Consider an inverted conical tank of radius r and height h, filled with water. How much work does it
take to pump all of the water out the top of the tank?
To answer this question, we considered the work it takes to pump a horizontal slice of water a distance y.
The advantage of this is that the force applied in this case is constant.
The thickness of the slice is dy. We determined the radius of the slice to be R = r 1 − hy . Its volume is
y 2
dy,
V = πr2 1 −
h
its mass is
y 2
dy
M = πρr2 1 −
h
where ρ is the density of water, and the force acting on it is
y 2
F = πρgr2 1 −
dy
h
where g is the acceleration due to gravity. Thus the work done on it over a distance y is
y 2
πρgr2 1 −
y dy,
h
and the work done on all the slices of water is
Z h
y 2
πρgr2 h2
πρgr2 1 −
y dy =
.
h
12
0
We also considered one other question.
2. Consider three water-filled tanks of equal volume and height: one a cylinder, one a cone of radius r, and
the last one an inverted cone of radius r. In which case is the most work done pumping the water out
the top of the tank?
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B. Questions
1. (a) Suppose one end of a 100 m steel rope weighing 91 kg is attached to the lip of the roof of a 120 m-tall
building. Calculate the work done in hoisting the rope up to the roof of the building.
(b) Suppose the same rope is attached to the lip of the roof of a 40 m-tall building. Calculate the work
done in hoisting the other end of the rope up to the roof of the building.
2. A 1 kg bucket on a rope of linear density 0.5 kg/m is drawn up a height of 40 m. Calculate the work done.
3. (a) The Earth exerts a gravitational force on an object of mass m (in kilograms) a height h (in metres)
km
above the surface of the Earth of (r+h)
2 , where k and r are positive constants (r is the radius of the
Earth). Calculate the work done to raise a 1 kg object to a height of 10000 m.
(b) Calculate the work done to raise a 1 kg object to an infinite height.
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4. Consider a spherical tank of radius 1 m filled with fluid of density 1240 kg/m . Calculate the work done
in pumping all the fluid out the top of the tank.
5. Consider a water-filled tank in the shape of an inverted pyramid of height 11 m and base side length 8 m.
Calculate the work done in pumping all the fluid out the top of the tank.
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