Question

What is the minimum speed needed for a $2\,kg$ mass to stay in motion in a vertical circle of radius $1.45\,m$?

Answer 1

In order to solve this question, we will look upon the concept of centrifugal and centripetal force. Also we will look upon the forces due to tension.

Formula used:
Centripetal acceleration: ${{\vec{F}}_{c}}=\dfrac{m{{v}^{2}}}{r}$
Where $m$ is the mass, $v$ is the velocity and $r$ is the radius.
Force due to gravity: ${{\vec{F}}_{g}}=mg$
Where $m$ is the mass and $g$ is the gravitational acceleration.

Complete step by step answer:
First of all we will learn about what we mean by centrifugal and centripetal forces.
A centripetal force is a net amount of force that acts on an object which is required to keep the object moving along a circular path. Whereas, a centrifugal force is the force which arises due to the inertia of the body which is moving in a circular path. The centripetal force is directed towards the center of the circular motion whereas the centrifugal force is directed away from the center.

DATA GIVEN:
The mass of the object: $m=2\,kg$
The radius of the circle: $r=1.45\,m$
We write the net force acting on the mass as
${{\vec{F}}_{net}}={{\vec{F}}_{g}}+T$
Where $T$ is the tension of the spring. But in this problem, we will consider the tension on the string to be zero and also the resultant or the net force is the centripetal force.

Therefore, in order for the mass to continue its motion along the circular path, both the centrifugal force and the centripetal force must be equal. Therefore,
${{\vec{F}}_{c}}={{\vec{F}}_{g}}$
Where $c$ represents centripetal and $g$ represents centrifugal force which is due to gravity. Using
${{\vec{F}}_{c}}=\dfrac{m{{v}^{2}}}{r}$and${{\vec{F}}_{g}}=mg$
$\Rightarrow \dfrac{m{{v}^{2}}}{r}=mg \\\ \Rightarrow \dfrac{{{v}^{2}}}{r}=g \\\ \Rightarrow {{v}^{2}}=rg \\\ \Rightarrow v=\sqrt{rg} \\\ \Rightarrow v=\sqrt{1.45(9.8)} \\\ \Rightarrow v=3.77\,m/s$
Cancelling the mass on both sides of the equation will not affect the answer as it is constant. We took the gravitational acceleration to me $9.8\,m/s$.

Therefore, the minimum velocity with which the mass can continue its circular motion is $3.77\,m/s$.

Note: When an object moves in a vertical circle, it moves against gravity for the first half of the circle and along with gravity for the second half, indicating that it has a maximum speed at the bottom and a minimum speed at the top of its course. However, it must have a crucial speed at the bottom in order to repeat the loop.