What Is the Minimum Initial Velocity to Roll Up a Carpet?

[bobsmith601]

Homework Statement

Imagine a very long carpet, with a very small thickness of t, width W, length L and density of ρ,
which is partially rolled up to form a cylinder of radius r. Since the carpet is very thin, it can be
assumed that the rolled-up part of the carpet is a perfect cylinder. We decide to roll up the whole
carpet by just kicking the cylindrical part of the carpet with initial velocity v.
What is the minimum initial speed v, required for the cylindrical part to roll the whole carpet?

 

[tiny-tim]

welcome to pf!

hi bobsmith601! welcome to pf!

What is the minimum initial speed v, required for the cylindrical part to roll the whole carpet?

looks to me like a conservation of energy problem …

what energy is needed to end with the carpet fully rolled at zero speed?

 

[bobsmith601]

I'm not sure...

 

[tiny-tim]

(just got up :zzz:)

for initial speed v, what are the initial kinetic energy and potential energy?

for final speed 0, what are the final kinetic energy and potential energy?

 

[Nickie]

As a physicist, I would approach this problem by first considering the conservation of momentum. The initial momentum of the cylindrical part of the carpet is equal to its mass multiplied by its initial velocity, or p = mv. As the cylindrical part rolls and gains speed, it will transfer its momentum to the rest of the carpet. Therefore, the final momentum of the whole carpet must be equal to the initial momentum of the cylindrical part, or p = mv.

Next, I would consider the dynamics of rolling motion. The rotational inertia of the cylindrical part can be calculated as I = 1/2mr^2, where m is the mass and r is the radius. The rolling motion of the cylindrical part will also have a kinetic energy of 1/2Iω^2, where ω is the angular velocity. In order for the cylindrical part to roll the whole carpet, its kinetic energy must be equal to the potential energy of the carpet, which is given by mgh, where m is the mass, g is the acceleration due to gravity, and h is the height of the rolled-up part.

Combining these equations and solving for the minimum initial velocity, we get v = √(2gh). This means that the minimum initial velocity required for the cylindrical part to roll the whole carpet is dependent on the height of the rolled-up part, the acceleration due to gravity, and the radius of the cylinder.

In conclusion, the minimum initial velocity required for the cylindrical part to roll the whole carpet can be determined by considering the conservation of momentum and the dynamics of rolling motion. This problem highlights the importance of understanding the fundamental principles of physics in solving real-world problems.