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

`3.77m/s`

Explanation:

The net force acting on a mass that is travelling in a vertical circle is composed of the force of gravity and the tension in the string.

`vecF_"net" = vecF_g+vecT`

Since this net force results in centripetal motion (the mass travels in a circle at constant speed), the net force is acting as a centripetal force:

`vecF_c=vecF_g+vecT`

To "stay in motion" as you say, we mean the mass must continue to follow the circle, and this means the string must not go slack.

Here is the important understanding: it is at the point of going slack when `T=0`. This condition is put into the equation to make it:

`vecF_c=vecF_g`

Using `F_c= (mv^2)/r` and `F_g=mg`

`(mv^2)/r=mg`

Cancel the `m`

`(v^2)/r=g`

`v^2=gr`

`v=sqrt(gr) = sqrt((9.8)(1.45)) = 3.77m/s`

Note that the mass, by cancelling out of the problem, does not effect the answer!

Answer 2

`v_(critical)=3.8m/s`

Explanation:

The minimum or critical speed is given by `v_(critical)=sqrt(rg)`. This is the point where the normal (or tension, frictional, etc.) force is `0` and the only thing keeping the object in (circular) motion is the force of gravity.

Therefore, `v_(critical)=sqrt(1.45m*9.8m/s^2)=3.8m/s`

As to where the equation for the critical speed comes from (if you are interested), consider the net force in circular motion. We know this is always the centripetal force, which points radially inward. The centripetal force causes centripetal acceleration and is necessary for the object to keep moving in a circle. Note that this is not a new force, but is supplied by forces such as tension, gravity, friction, etc. By Newton's second law, we also know that `F=ma`.

`F=F_c=m*a_c=m*(v^2)/r`

Note that centripetal acceleration (`a_c`) is also called radial acceleration.

At the top of the motion (vertical circle), if all we have is the force of gravity as the centripetal force which keeps the object moving in a circle,

`F_c=F_G=m*v^2/r`

As `F_G=mg`,

`=>mg=m*v^2/r`

`=>g=v^2/r`

Solving for `v`,

`=>v^2=rg`

`=>v=sqrt(rg)`

Hope that helps!