problem set 1 - due to the box in LeC 251 by 5pm thursday september 8th.
(1)
A cone frustum has base radius b, top radius a, and height h, as shown
in the figure. Assuming that one of the following quantities is the volume of
the frustum, which one is it? (Don't solve the problem from scratch, just
check special cases.) Justify your answer.
πh 2 2 πh 2 2 πh 2
π h a4 + b4
2
(a + b ),
(a + b ),
(a + ab + b ),
·
, π hab.
3
2
3
3 a2 + b2
______________________________
In the problems below, employ dimensional analysis (even if you know a better
way).
(2)
In the limit of small oscillations, the period of a simple pendulum depends
only on length, mass, and g. You probably know the formula - confirm it is
dimensionally correct.
(3)
In very deep water (so the depth of the water does not have to be taken into
account) , show that the speed of an individual wave of specific wavelength
depends on the square root of its wavelength (take density, g, and the wavelength
as the independent variables). [the reason I say specific wavelength is that a local
disturbance in the water does not travel at this speed, as we shall observe later in
the course. This is the topic of phase and group velocity, and if you wish to
anticipate our study of such phenomena, you should right away seek out a pond,
and throw a pebble in it]
(4)
A vertical cylinder is filled with water to a height h, the area of the surface of
the water being the cross section of the cylinder, A, and the water is permitted to
drain slowly through a small opening in its base. The area of that opening is B. Let
t be the time for it to empty. Investigate the function
t = t(h,g, A, B)
(5)
water of density ρ and surface tension σ drips (slowly) from a tap (faucet)
the radius of whose opening is r.
(i) what are the dimensions of surface tension?
ρr 2 g
(ii) confirm that the quantity
is dimensionless (independent of choice of
σ
units)
(iii) If m is the mass of the resulting droplet, what is the general form of the
function m = m(ρ, σ ,r, g) ?