Question

There is a hemispherical smooth vessel particle of mass m is projected downward with the speed 5m/s from the edge of the vessel. What will be its angular momentum with respect to point O, when it is at lowest point?

Answer 1

In this question, we will use the equation of total energy, which is the sum of potential energy and the kinetic energy. Further, by substituting the values in the equation of angular momentum, will give us the required result. Also, we will discuss the basics of energy and law of conservation of energy.

Formula used:
$E = mgh + \dfrac{1}{2}m{v^2}$
$L = mvr$

Complete step-by-step answer:
Let us suppose that the velocity is v at the bottom point, and then by applying law of energy conservation we get:
$E = mgh + \dfrac{1}{2}m{v^2}$
Now, by substituting the given values, we get:
$\dfrac{1}{2}m{(5)^2} = - mgR + \dfrac{1}{2}m{v^2}$
$\Rightarrow \dfrac{{25}}{2} = - 10 \times 10 + \dfrac{1}{2} \times {v^2}$
\eqalign{& \Rightarrow {v^2} = 225 \cr & \Rightarrow v = 15 \cr}
Now, about point O, we calculate the angular momentum:
As we know, Angular momentum is the rotational equivalent to linear momentum, given by:
$L = mvr$
Here, L is linear momentum, m is mass, v is velocity and r is the radius of an object.
\eqalign{& L = m \times 15 \times 20 \cr & \therefore L = 300m \cr}
Therefore, we get the required answer of the angular momentum.

Additional Information: As we know that the energy is defined as the ability to do work. Energy can be found in many things and can take different forms, like-kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure. Total energy of a system is given by the sum of kinetic energy and potential energy.
We also know that, according to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another. Here, in this question electrical energy is transferred to heat energy.
We should also know about the power. It is the amount of energy transferred or converted per unit time. In the International System or S.I Unit of power is the watt. The S.I unit of power can also be written as one joule per second. Power is a scalar quantity, which means it has only magnitude not direction.

Note: We should remember that the laws of conservation of energy can never be violated in any case. According to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another.