Question

In a vertical circular motion, the ratio of the kinetic energy of a particle at the highest point to that lowest point is
A. 5
B. 2
C. 0.5
D. 0.2

Answer 1

Hint:
In this problem, we use the concept of centripetal force and work-energy theorem. First, we about circular motion are a movement of an object along the circumference of a circle. It can be uniform with constant speed, but velocity is not stable. This changing of velocity indicates the presence of centripetal acceleration, and this acceleration is produced by centripetal force.

Complete step by step answer:
Consider a particle moving in a vertical circular motion, so, at the top of a circle, the tension of a particle becomes zero, and mg provides the centripetal force to a particle that acts at the centre.
$\therefore \text{mg = }{\displaystyle \frac{\text{m}{\text{v}}^2}{\text{r}}}$ Where m is mass, v is velocity, and r is the radius, g is gravity.
$\Rightarrow \text{g}={\displaystyle \frac{{\text{v}}^2}{\text{r}}}$
$\Rightarrow \text{v = }\sqrt{\text{gr}}$
i.e., the velocity at the highest point is$\text{v = }\sqrt{\text{gr}}$
Thus kinetic energy at the highest point=${\displaystyle \frac{1}{2}}\text{mgR}$
Now the velocity at lowest point find out by applying the work energy theorem
Therefore ${\text{w}}_g+{\text{w}}_{\text{T}}=\text{K}\text{.}{\text{E}}_{\text{T}}+\text{K}\text{.}{\text{E}}_{\text{L}}$
$\Rightarrow -\text{mg(2R) + 0 = }{\displaystyle \frac{1}{2}}\text{mgR + }{\text{K}}_{\text{L}}$
${\text{K}}_{\text{L}}={\displaystyle \frac{\text{mgR}}{2}}+2\text{mgR}$
${\text{K}}_{\text{L}}={\displaystyle \frac{\text{5mgR}}{2}}$
Therefore the ratio of kinetic energy at the highest to the lowest point $={\displaystyle \frac{{\text{K}}_{\text{T}}}{{\text{K}}_{\text{L}}}}$$={\displaystyle \frac{{\displaystyle \frac{1}{2}}\text{mgR}}{{\displaystyle \frac{5}{2}}\text{mgR}}}={\displaystyle \frac{1}{5}}=0.2$

Additional Information: The kinetic energy of a particle is the product of one-half its mass and therefore the square of its speed, for non-relativistic speeds. The kinetic energy of a system is the sum of the kinetic energies of all the particles within the system.

Note: Particles are objects in motion, in order they have kinetic energy. The faster a particle moves, the more kinetic energy it has. Kinetic energy is related to heat. The faster the particles during a substance move, the warmer it is.