Mock Test (Mechanics) : USAPhO

Credit: idonthaveausername
 Problems
1. Two parallel smooth orbits are cut in the plane past the center of a
homogeneous spherical planet with no rotation. The mass of the planet
is M and the radius is R. A cannon is placed at the midpoint of one
of the orbits. The cannon fires a cannonball (the mass can be omitted
with respect to the planet), which slides frictionlessly inside the orbit.
It is ejected from the surface and moves in an elliptical orbit. After
the elliptical motion, the projectile enters another orbit with the same
velocity direction as the orbit. After that, it slides in the same way in
another orbit, exits from the surface, passes through the elliptical orbit,
enters the initial orbit, and finally slides back to the initial starting
point, completing a complete cycle of motion. The period of motion of
the cannonball is required to be exactly twice the period of an object
moving in a circular motion around the planet due to the gravitational
force of the planet on the surface. Find the velocity v at which the
cannon fires the projectile, the length l of the orbit and the maximum
height h of the projectile from the surface.

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2. As shown in the picture below, a full set of 52 cards (you may count
them carefully and find that the numbers in the pictures do not match,
because our pictures are only schematic diagrams) are stacked obliquely
one by one in the process of shuffling the pattern. Card-to-card contact
is smooth, and the ground is rough enough to keep the cards from falling
by applying a horizontal force on top of the first card. The first card
is upright. The mass of each card is known to be m, the height is l,
regardless of the thickness of the cards, the mass is evenly distributed
on the face. The equal spacing between the bottom ends of the cards
is ϵ. 52 is considered a large enough number, and ϵ/l is considered a
small enough number. The acceleration of gravity is g, please use a
reasonable approximation to answer the following questions:

(a) The top of each card will form a curve (the red part in the figure),
take the bottom of the first card as the coordinate origin, the
horizontal direction is the x-axis, and the vertical direction is the
y-axis to establish a coordinate system, and find the equation of
the curve.
(b) Find the angle of inclination of the 52nd card, that is, the angle
with the vertical direction.
(c) Find the external force F required to maintain the balance of the
system.
(d) Release this stack of cards, and the cards will not bounce back
when they fall on the table. Find the mechanical energy lost in
the process.

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3. As shown in the figure, there is a stationary elliptical cylinder in S0 ,
the semi-major and semi-minor axes are respectively a and b. On its
surface, positive and negative charges with surface densities of ±σ0
are coincidently distributed. When moving around the cylinder with
a constant velocity u, the negative charge is at rest. In this way, a
magnetic field B0 can be generated in the cylinder. If a reference frame
S that moves in the negative direction of the x-axis at a certain speed
relative to this reference frame is taken, you can see exactly a right
cylinder moving to the right.

(a) Find the relative velocity v of the magnetic field inside the cylinder
in the S0 system and the transformation of the reference frame.
(b) Find the charge density σ and the current surface density i on the
surface of the cylinder in the S system, expressed as a function of
the angle θ with the x-axis.
(c) Calculate the electromagnetic field inside the cylinder in the S
system (it is not allowed to use the transformation formula of
the electromagnetic field in different reference systems, but it is
known that the interior of the cylinder is still a uniform electric
and magnetic field, and there is no electromagnetic field outside
the cylinder).

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4. As shown in the figure, the surface of a third-order Rubik’s cube is
densely covered with resistance wires. All the resistance wires have the
same resistance per unit length. The resistance value of the resistance
wire of each side length of each small cube is r. There are no electrical
connections at the four intersection points in the face, and there is
no resistance wire inside the cube. Find the resistance between the
opposite vertices AB of the cube.

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5. A high-energy photon can spontaneously decay into a pair of positive
and negative electrons next to the proton. The static mass of the
proton, mp , is initially regarded as static, and the static masses of the
positive and negative electrons are both me . No other light radiation is
emitted during the process. What is the minimum energy of a photon?
Express your result in an expression and calculate its value. Retain
three significant figures.

6. As shown in the figure, in a cylindrical optical fiber, the light completely

propagates in the z direction with a helical light of pitch h in a thin
layer of radius r, and the refractive index of the medium on this surface
is n. The figure is showing one of the rays. In order to form such a ray,
the medium should have a refractive index that varies with the radius.

(a) Computes the derivative of the refractive index with respect to

the radius, dn/dr, at that point.
(b) After that, the light exits at the end face z = 0 and enters the
vacuum. The rays formed create the outline of a hyperboloid,
then find the equation of this hyperboloid.
(c) A thin convex lens is placed in the z = 0 plane with focal length f .
Attempt to ”focus” the hyperboloid beam. Find the minimum ra-
dius to which the hyperboloid can be focused after passing through
the convex lens and the distance from the cylindrical end face.