Particle moving on the inside of a hemispherical shell

Understanding Motion of a Particle Inside a Hemisphere

A hemisphere is half of a sphere cut through the centre, including the circular base.
The motion of a particle projected up the inside of a hemisphere is governed by the laws of kinematics.
The particle can move along a curved path, thus gravity is acting perpendicularly to the motion.
Normal contact force (N), weight of the particle (mg), and frictional force (f) affect the motion.

The two forces working on the particle are the normal reaction force (N) from the interior of the hemisphere and gravitational force (mg) directed vertically downwards.
There is a frictional force that acts tangentially to the surface of the hemisphere opposing the motion. It adjusts its direction whether the particle is moving upwards or downwards.

The reaction force (N) of a hemisphere is not always equivalent to the weight (mg). This distinction arises because the reaction force acts perpendicular to the surface, not necessarily against gravity.
The normal contact force (N) adjusts to the decreasing vertical component of the gravitational force as the particle moves up.

Frictional force will always act in the direction to oppose motion. It acts to slow down the particle’s motion as it moves up, and slow down its return slide.
In ideal cases, we can consider the hemisphere as a smooth surface, therefore the frictional force is zero (f=0).
In a real-world situation, if the particle were to reach the top of the hemisphere and continue over the top, a sufficient frictional force would be necessary.

As the particle moves up, the angular displacement changes. Thus, the equations of motion are modified for circular motion.
The equations of motion in this case will incorporate the angular displacement, denoted by θ, and will be expressed in terms of θ, dθ/dt (the rate of change of displacement) and d²θ/dt² (the rate of change of velocity or acceleration).

Principle of conservation of energy can be applied to calculate the maximum height reached by the particle.
At the bottom of the hemisphere, the particle would possess some kinetic energy and no potential energy.
As it ascends, it loses some kinetic energy which is converted into gravitational potential energy.
At the highest point, kinetic energy is minimum and potential energy is maximum.
Total Energy (TE) remains constant throughout and is equal to the initial kinetic energy: TE = KE + PE = constant.

This topic requires a good grasp of kinematics, forces, work, energy and power.
Diagrams play a vital role for understanding the forces and their directions.
Ensure to remember the modified equations of motion for circular paths.
Being careful with the direction of forces and sign convention is crucial for solving problems.
Practice and familiarity with a variety of problems is key to becoming comfortable with this topic.