# Calculating resultant force of friction from rope and bar.

Hello,

I hope this is a suitable forum, it's a million miles from homework but it may be on the same sort of level.

A picture is worth a thousand words so please have a look at the attached picture. When the rope is pulled there isn't just the issue of gravity, but also of the friction of the rope over the metal bar. What I am wanting to calculate is the actual force that must be applied to stop the weight from falling, taking into account the friction.

I really don't know where to start with this, but from the searching that I've done, this page seems to be using diagrams and words that seems applicable. http://www.engineeringtoolbox.com/bollard-force-d_1296.html

The equation given is:
S = F e-μα
Where:
S = effort force in the rope (N)
e = 2.718
μ = friction coefficient (approximately 0.3 - 0.5 is common for a rope around a steel or cast iron bollard)
α = angle where the rope is in contact with the bollard (radians)

So, plugging in the following data (assuming the rope is only in contact with half the bollard):

F = 300N
e = 2.718
μ = 0.4
α = ∏

I get:

300 * ( 2.718 -0.4 * ∏ ) = 85.39 N

Which seems to imply that the force that is needed to stop the weight from falling is 214.6 1N (300-85.39). However, I've got a feeling that I might be messing up the calculation, would somebody be kind enough to take a look for me?

Thank you.

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gneill
Mentor
Hello,

I hope this is a suitable forum, it's a million miles from homework but it may be on the same sort of level.

A picture is worth a thousand words so please have a look at the attached picture. When the rope is pulled there isn't just the issue of gravity, but also of the friction of the rope over the metal bar. What I am wanting to calculate is the actual force that must be applied to stop the weight from falling, taking into account the friction.

I really don't know where to start with this, but from the searching that I've done, this page seems to be using diagrams and words that seems applicable. http://www.engineeringtoolbox.com/bollard-force-d_1296.html

The equation given is:
S = F e-μα
Where:
S = effort force in the rope (N)
e = 2.718
μ = friction coefficient (approximately 0.3 - 0.5 is common for a rope around a steel or cast iron bollard)
α = angle where the rope is in contact with the bollard (radians)

So, plugging in the following data (assuming the rope is only in contact with half the bollard):

F = 300N
e = 2.718
μ = 0.4
α = ∏

I get:

300 * ( 2.718 -0.4 * ∏ ) = 85.39 N

Which seems to imply that the force that is needed to stop the weight from falling is 214.6 1N (300-85.39). However, I've got a feeling that I might be messing up the calculation, would somebody be kind enough to take a look for me?

Thank you.
Your work seems okay to me. Note that the equation in question is also known as the Capstan Equation; The Wikipedia article for it looks pretty clear if you're looking for additional info.

Thank you very much! :)