Problem:
On the surface of the earth an object is given an initial speed v on a friction less surface at latitude l. Show that the object will move in a circle and find the radius of the circle for velocities small enough that the radius is much smaller than the earth radius.
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Problem:
(a) Assume that in the northern hemisphere a stone is dropped from a height h. What is its eastward deflection upon landing to first order?
(b) Now assume that the stone is thrown upward at an initial velocity v0,
so that
Show that it has a
westward deflection upon landing four times as big as the eastward deflection of the
falling stone.
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Problem:
Two equal mass particles are connected by a spring and are executing simple harmonic motion when they are placed in a horizontal, frictionless open pipe rotating at a constant angular velocity W about a vertical axis through its midpoint. The midpoint of the spring is initially at the midpoint of the pipe, but neither the particles nor the spring are attached to the pipe.
(a) Find the maximum angular velocity for which the particles stay in the pipe.
(b) Assume (a) is satisfied and the particles stay in the pipe. Find their positions as a function of time.
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Problem:
The magnetic moment vector, m(t), of a point particle of mass m, charge q, and no intrinsic spin is proportional to its angular momentum vector, m(t)=gL(t).
(a) Evaluate the gyromagnetic ratio, g, in the case of circular motion at constant angular velocity w=(r(t)´v(t))/úr(t)| 2, where r is the radius of the circle and v is the velocity of the particle.
(b) By considering the torque produced by a magnetic field on a charged particle, write down the equation of motion for the magnetic moment vector of the particle.
(c) Solve this equation for H=H0k, where H0 is constant in space and time. Describe your solution verbally.
(d) Solve this equation for the case H(t)=H0k+H1[icoswt+jsinwt] in a laboratory frame (i,j,k).
Describe the solution verbally. (Hint: The relation between the time derivative of any
vector G in a coordinate frame in the laboratory and in a frame rotating
with respect to the lab frame with fixed angular velocity
w
is
. Find the equation of
motion for the magnetic moment m(t) in a
coordinate system rotating with the magnetic field.)
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