Question

In vertical motion, the ratio of kinetic energy to potential energy at the horizontal position is
A. 5:2
B. 2:1
C. 3:2
D. 2:3

Answer 1

Hint: We have to assume that the object just manages to complete one vertical circle. We should basically have knowledge of kinetic and potential energy of a particle in a circular motion. Firstly we will find the kinetic energy and potential energy then we will take the ratio.

Formula used:
\[{{V}_{TOP}}=\sqrt{gR}\]
Total energy at top most point = \[\dfrac{1}{2}m{{\left( \sqrt{gR} \right)}^{2}}+2mgR\]
\[PE=mgR\]

Complete answer:
We have to assume that the particle of mass (m) just manages to complete the one vertical circle. In that case the velocity of the particle when it was at top \[{{V}_{TOP}}\].
\[{{V}_{TOP}}=\sqrt{gR}\]
We will find the total energy at the topmost point.

Total energy at top most point = \[\dfrac{1}{2}m{{\left( \sqrt{gR} \right)}^{2}}+2mgR\]

=\[\dfrac{5}{2}mgR\]
Now we will find the energy of particle mass m at a horizontal position.
At, the horizontal position particle possesses only potential energy.
\[PE=mgR\];
Total energy=\[\dfrac{5}{2}mgR\]
Kinetic energy =total energy-potential energy
\[\therefore K.E=\dfrac{5}{2}mgR-mgR\]
\[=\left( \dfrac{5}{2}-1 \right)mgr\]
=\[\dfrac{3}{2}mgR\]
Therefore the ratio of kinetic energy to the potential energy is
\[\dfrac{K.E}{P.E}=\dfrac{3}{2}\]
The ratio between kinetic energy to the potential energy at the horizontal position is 3:2.

So the correct option is (c).

Additional Information:
In a uniform circular motion, the tangential force is zero. This implies that the angular velocity is constant. ∴v=ωr ⇒v is constant (∵ω is constant).
In uniform circular motion is a motion in a circular path at a constant speed. An object can reach its escape velocity, if its kinetic energy, and its gravitational potential energy becomes such that they are equal and opposite in direction so that their sum is equal to zero.

Note:
The energy in any motion is conserved. So kinetic energy is the difference between the total energy and potential energy for circular motion. In a circular motion, the horizontal position particle possesses only potential energy.