Required tension of rope around a cylinder to hold a object.

[kaar]

please see attached drawing. I am trying to understand how this would work.

Imagine a vertical cylinder of diameter "D". I would like to tie a mass M to it, using a rope. what is the tension "T" that is required on the rope?

I assume the force required will depend on the coefficient of friction between the cylinder and the mass, let us assume that to be "mu".

but in this case, friction force = mu * force perpendicular to the cylinder surface

how to calculate this perpendicular force? is it just equal to the tension T? I am confused because when a rope goes though a pully we always assume tension to be tangential to the pully at any given point...

any clarifications of my understanding is welcome!

 

[Spinnor]

I have redrawn the problem a bit, I think the physics remains the same. Hope this helps.

 

[Spinnor]

You wrote,

is it just equal to the tension T?

No in general. I think you would have to accurate dimensions of the mass M to calculate what fraction of T presses mass M against the cylinder. My sketch above takes liberty with the dimensions as none were given.

 

[kaar]

That clarified my doubt..thanks.

 

[asteriatic]

The required tension of the rope around a cylinder to hold an object will depend on several factors, including the mass of the object, the diameter of the cylinder, and the coefficient of friction between the cylinder and the object. The tension required can be calculated using the formula T = μmg, where μ is the coefficient of friction, m is the mass of the object, and g is the acceleration due to gravity. This formula assumes that the tension in the rope is equal to the perpendicular force exerted on the cylinder by the object.

In this case, the perpendicular force would be equal to the tension in the rope, as long as the rope is parallel to the surface of the cylinder. If the rope is at an angle, the perpendicular force would be equal to the tension multiplied by the cosine of the angle between the rope and the cylinder surface.

It is important to note that this formula assumes a static situation, where the object is not moving. If the object is in motion, the tension in the rope may need to be adjusted to account for the acceleration and friction forces acting on the object.

Additionally, the assumption of the tension being tangential to the pulley is only applicable in a simplistic scenario where the rope is moving smoothly over the pulley. In reality, there may be other factors at play, such as the weight of the pulley itself, or the angle at which the rope is wrapped around the pulley. These factors would need to be taken into account when calculating the required tension in the rope.

Overall, the required tension in the rope around a cylinder to hold an object is a complex calculation that will vary depending on the specific circumstances. It is important to consider all relevant factors, such as mass, coefficient of friction, and angle of the rope, in order to accurately determine the tension needed.