PHY 3221 Classical Mechanics, Homework #2, due 8:30 am, 02/06/12
1 A spherical shot of lead (diameter D = 2 mm and density ρPb = 11.342 g/cm3 ) is dropped into a
container of corn syrup at 20◦ C, with density ρcs = 1.38 g/cm3 and a viscosity ηcs = 13.8 N·s/m2 .
a) Assuming the linear drag force at velocity v is given by 3πηDv, find the characteristic time τ
associated with the descent of the lead shot in the corn syrup.
b) By include the effects of buoyancy, calculate the terminal speed vter of the lead shot in the corn
syrup.
c) How long after it is dropped will the ball reach 90% of its terminal speed?
d) Using a quadratic drag force of ρcs Av 2 /4, where A is the cross-sectional area of the lead shot,
calculate the ratio of the quadratic to linear drag forces.
e) Explain whether the quadratic drag force could be ignored for the lead shot falling in air (at
25◦ C) of density ρair = 1.16 × 10−3 g/cm3 and viscosity ηair = 1.85 × 10−5 N·s/m2 .
1
2 Under the influence of a magnetic field, B, which points in the positive z-direction, a particle of
mass m and charge q moves in the (x, y)-plane.
a) Find the frequency of the circular motion executed by the charge (in terms of m, q and B).
b) Assuming the particle is an electron and that B is the magnetic field strength at the North Pole
(6.5 × 104 ) nT (nanoteslas), calculate the period of the electron’s circular motion.
c) If the speed v of the electron is 3 × 105 m/s, find the radius of the electron’s orbit?
d) If there is also an electric field E of 1000 N/C in the positive z-direction, by how much would the
vertical component of the electron’s speed change during one orbital period.
e) Explain whether the electron will be traveling clockwise or counterclockwise around the North
Pole.
2
3 In a certain European ball game, a small hard ball, with a mass of 160 gm and 7.25 cm in
diameter, is projected over a 20 m horizontal distance at an initial speed of v = 45 m/s.
a) Assuming air resistance is a quadratic function of the ball’s speed given by ρAv 2 /4 (where ρ is
the density of air = 1.29 kg/m3 and A is the cross sectional area of the ball), and ignoring, to
begin with, any vertical motion, calculate the time the ball takes to travel to its destination.
b) Using the same assumptions as in part (a), find the final horizontal velocity of the ball.
c) Given the time of flight calculated in part (a) and assuming the ball starts and finishes at the
same vertical height, calculate the maximum vertical extent of the ball’s flight path by ignoring
any vertical drag force.
d) Find the maximum vertical speed of the ball using the same assumptions as in part (c).
e) By considering that a linear viscous drag force affects the vertical motion of the ball and is given
by 3πηDv (where η is the viscosity of air = 1.70 × 10−5 N·s/m2 and D is the diameter of the
ball), calculate the change in the vertical height of the trajectory assuming the maximum speed
obtained in part (d).
3