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9.5. Homework Problems

1. Pascal's snail (named after Étienne Pascal, father of Blaise Pascal) is the shape shown in the figure, defined by R = b(1+cosθ) in polar coordinates.
(a) Make a rough visual estimate of its area from the figure.
(b) Find its area exactly, and check against your result from part (a).
(c) Show that your answer has the right units. [Thompson, 1919]

Figure H. Problem 1: Pascal's snail with b=1.

2. A cone with a curved base is defined by rb and θ≤ π/4 in spherical coordinates.
(a) Find its volume.
(b) Show that your answer has the right units.

3. Find the moment of inertia of a sphere for rotation about an axis passing through its center.

4. A jump-rope swinging in circles has the shape of a sine function. Find the volume enclosed by the swinging rope, in terms of the radius b of the circle at the rope's fattest point, and the straight-line distance ℓ between the ends.

5. A curvy-sided cone is defined in cylindrical coordinates by 0≤ zh and Rkz2. (a) What units are implied for the constant k? (b) Find the volume of the shape. (c) Check that your answer to b has the right units.

6. The discovery of nuclear fission was originally explained by modeling the atomic nucleus as a drop of liquid. Like a water balloon, the drop could spin or vibrate, and if the motion became sufficiently violent, the drop could split in half --- undergo fission. It was later learned that even the nuclei in matter under ordinary conditions are often not spherical but deformed, typically with an elongated ellipsoidal shape like an American football. One simple way of describing such a shape is with the equation

\[ r \le b[1+c(\cos^2\theta-k)] , \]

where c=0 for a sphere, c>0 for an elongated shape, and c<0 for a flattened one. Usually for nuclei in ordinary matter, c ranges from about 0 to +0.2. The constant k is introduced because without it, a change in c would entail not just a change in the shape of the nucleus, but a change in its volume as well. Observations show, on the contrary, that the nuclear fluid is highly incompressible, just like ordinary water, so the volume of the nucleus is not expected to change significantly, even in violent processes like fission. Calculate the volume of the nucleus, throwing away terms of order c2 or higher, and show that k=1/3 is required in order to keep the volume constant.

7. This problem is a continuation of problem 6, and assumes the result of that problem is already known. The nucleus 168Er has the type of elongated ellipsoidal shape described in that problem, with c>0. Its mass is 2.8×10-25 kg, it is observed to have a moment of inertia of \(2.62\times10^{-54} \text{kg}\!\cdot\!\text{m}^2\) for end-over-end rotation, and its shape is believed to be described by b ≈ 6×10-15 m and c ≈ 0.2. Assuming that it rotated rigidly, the usual equation for the moment of inertia could be applicable, but it may rotate more like a water balloon, in which case its moment of inertia would be significantly less because not all the mass would actually flow. Test which type of rotation it is by calculating its moment of inertia for end-over-end rotation and comparing with the observed moment of inertia.

8. Von Kármán found empirically that when a fluid flows turbulently through a cylindrical pipe, the velocity of flow v varies according to the “1/7 power law,” v/vo = (1-r/R)1/7, where vo is the velocity at the center of the pipe, R is the radius of the pipe, and r is the distance from the axis. Find the average velocity at which water is transported through the pipe.