Here I challenge you with some fascinating math and physics problems
that I have encountered, or thought of over the years. Note that I have
selected the problems based on the enjoyment they offer in the solution
process, rather than their difficulty. Enjoy them!
Problems
Find the surface area and the volume of an \(n\)-dimensional sphere.
Determine the number of degrees of freedom in an \(n\)-sided polygon.
Imagine two identical objects with heat capacity \(C\) and
temperatures \(T_1\) and \(T_2\). If we were to build an engine out of
them, what is the maximum work that this engine can do? \(^1\)
Give the shape of a fixed-volume planet (in mathematical form) that
maximizes the gravitational force acting on a person standing on it.
A ballistic missile is launched from the north pole, the target is at
the latitude \(\phi\). To have the minimal launch velocity of the
missile, at which angle (with respect to the horizon) do we need to
launch it? \(^2\)
Imagine a homogenous, liquid planet of average radius \(R \) and
density \(\rho\), spinning around itself with an angular velocity of
\(w.\) Find its eccentricity.
A point particle of mass \(m\) and charge \(q\) is moving under a
homogenous magnetic field of induction \(B\) and electrical field of
intensity \(E\), directed perpendicular to each other. Ignoring
gravity and relativity, find its trajectory.\(^1\)
Imagine a water sprinkler in the garden, sprinkling water uniformly at
each angle and in all directions at the same time. What radius from
the origin would get watered the most?
Consider a particle undergoing a one-dimensional random walk, taking
steps of length \(a\) to the right or left with equal probability.
What is the probability distribution of the particle's position after
\(n\) steps?
Consider a particle undergoing an \(n\)-dimensional random walk,
taking steps in each direction with equal probability. What is the
probability that the particle will come back to the origin?
A person is collecting a set of distinct coupons, and each time they
acquire a coupon, it is equally likely to be any one of the set.
Determine the expected number of trials needed to complete the set.
Calculate the repulsion force the light rays from the Sun apply onto
the Earth.
Describe the properties of a material that makes up a liquid planet,
so that however much mass is removed, the radius of the planet stays
the same.
Find the oscillation period of a spherical ball oscillating in
water.\(^1\)
Two small particles with opposing charges \(q\) are separated by a
distance \(L\) and left free. Find the collision time.\(^3\)
A light source of intensity \(I\) and frequency \(f_0\) falls normally
onto a mirror with mass \(m\) and initial velocity of zero. Find the
frequency of the reflected beam as a function of time.
A linearly polarised light beam transits from a medium with refraction
index of \(n_1\) to another of \(n_2\), normal to the surface. Find
the portion of the power that is reflected and refracted from the
surface.\(^3\)
Imagine that the clouds are made of tiny water droplets that hang onto
the air and distributed homogenously in the space. Find the
acceleration of a rain droplet that is falling down among them. The
rain droplet, initially of negligible size, has a density of \(\rho\),
and the water droplets in the cloud have a mean density of
\(\lambda\). Assume the rain droplet absorbs all encountered water
droplets while maintaining a spherical shape.
Consider a rigid container from which all the air has been pumped out.
Now, the valve of the container is slightly opened, and the air from
outside is slowly filling in. The air flow stops when mechanical
equilibrium is reached. Find the final temperature of the air inside.
The room temperature is T, and the heat flux through the walls is
ignored.\(^2\)
How far away from the handle of a sword of length \(L\) should you hit
an item, so that you feel the impact the least?
Imagine (1) an object (2) light ray approaching a planet of mass \(M\)
from infinity, with initial lateral distance to the center of the
planet \(R\) and speed \(v\). Find the shift in the heading direction
after it escapes the gravity of the planet.