# Minimum v0 for Tension in Inelastic Collision

• Jeremymu1195
In summary, for a mass m with velocity v0 to make an inelastic collision with a second mass M suspended by a string of length L and remain under tension for a complete rotation, the minimum value of v0 is given by v0 = (m+M)√(gL/2).
Jeremymu1195

## Homework Statement

A mass m with velocity v0 makes an inelastic collision with second mass M that is suspended by a string of length L. The velocity v0 is perpendicular to the vertical string. After the collision the combined masses, m+M, rotate in a vertical plane around the point of suspension of the string. Find the minimum value v0 such that the string remains always under tension for a complete rotation.

∑Fr=mv2/r
T=1/2mv2
U=mgh

## The Attempt at a Solution

The condition for the an inflexible string to remain under tension is that it never goes slack, i.e. length=L for the entire rotation.

Since the collision is inelastic, momentum is conserved while kinetic energy is not. So,

mv0=(m+M)v' where v' is the magnitude of the x-directed velocity of the combined masses (m+M) after the collision. so v'=m/(m+M)v0.

At the bottom of the vertical circle (In Polar Coordinates),
letting m'=(m+M)∑Fr= Tb-m'g=m'v'2/L (*)

At the top,

∑Fr=Tt+m'g=m'v''2/L (**)

Using Energy,

1/2m'v'2=1/2m'v''2+m'g(2L) (***)
v'2=v''2+4gL

v''2=v'2-4gL

I am going to pause here to see if there is anything I am missing.

You're on the right track. Find v'', then v', then v0.

## 1. What is the minimum velocity required for tension to occur in an inelastic collision?

The minimum velocity required for tension to occur in an inelastic collision depends on the mass and elasticity of the objects involved. In general, the minimum velocity is calculated using the conservation of momentum and kinetic energy equations.

## 2. How does the elasticity of the objects affect the minimum velocity for tension in an inelastic collision?

The elasticity of the objects involved in an inelastic collision affects the minimum velocity for tension by determining the amount of energy lost during the collision. The more elastic the objects, the less energy will be lost and the higher the minimum velocity required for tension to occur.

## 3. Is the minimum velocity for tension in an inelastic collision the same for all types of collisions?

No, the minimum velocity for tension in an inelastic collision can vary depending on the type of collision. For example, a head-on collision will have a different minimum velocity for tension compared to a glancing collision.

## 4. Can the minimum velocity for tension be negative in an inelastic collision?

Yes, the minimum velocity for tension can be negative in an inelastic collision. This can occur when the objects involved have opposite directions of motion and the tension acts to slow down one of the objects.

## 5. How is the minimum velocity for tension in an inelastic collision related to the coefficient of restitution?

The coefficient of restitution, which measures the elasticity of a collision, is directly related to the minimum velocity for tension in an inelastic collision. A higher coefficient of restitution will result in a higher minimum velocity for tension, while a lower coefficient of restitution will result in a lower minimum velocity for tension.

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