Analyzing Friction in a Hanging Mass System

In summary: The Attempt at a SolutionTo find the friction force between the hook and the string, you would need to solve for T1 and T2 separately. T1 would be the tension in the shorter side of the string, the side closer to the pole and inclined at a higher angle. T2 would be the tension in the other side. If the string were at the hook, the two tensions would be the same and there would be no friction. However, since there is friction which prevents the hook from sliding to the low point, the friction force must be greater than the lower tension, T2.
  • #1
darkSun
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0

Homework Statement


A mass of 5 kg has a hook attached to it, and it is placed on an initially slack string that is suspended from two poles.

Due to friction between the rope and the hook, the mass does not slide to the middle of the two poles; it hangs closer to one side so that the string makes an angle of 50 degrees with the horizontal on that side, and 30 degrees with the other side.

Question is: What is the force of friction between the hook and the string?

Is this description clear enough?


Homework Equations


Is this correct? Friction between the string and the hook = (mu*Tension)


The Attempt at a Solution


The first thing I'm wondering is whether the tensions in the string on either sides of the mass will be the same. They are both the same string, but... I'm not sure.

Also, when separating tension into components, ie

T1sin 50 + T2 sin 30 = 5g
and
T1cos 50 = T2cos 30

T1 is the tension in the shorter side of the string, the side closer to the pole and inclined at a higher angle. T2 is the tension in the other side.
Would friction be included in these equations? I think so, and I think it would be along the same direction as T1. (so replace T1 by T1 + Friction)

But then I get two equations in three unknows, namely T1, T2, and Friction. I think I am going about this the wrong way...
 
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  • #2
I didn't mention that the hook is cylindrical, but apparently that has a great deal to do with the problem.

Basically I have to find an expression for friction on a curved surface... I think that expression looks like e^(mu) or something similar (friends told me). Let's see if I can do that now!
 
  • #3
darkSun said:

Homework Statement


A mass of 5 kg has a hook attached to it, and it is placed on an initially slack string that is suspended from two poles.

Due to friction between the rope and the hook, the mass does not slide to the middle of the two poles; it hangs closer to one side so that the string makes an angle of 50 degrees with the horizontal on that side, and 30 degrees with the other side.

Question is: What is the force of friction between the hook and the string?

Is this description clear enough?


Homework Equations


Is this correct? Friction between the string and the hook = (mu*Tension)
, no friction is a function of the normal bearing force between the hook and string. The lower tension, T2, must lie within the range of the higher tension,T1 (no friction) and [tex]T_1e^{u_sk}[/tex], but that equation is not needed in this problem.

The Attempt at a Solution


The first thing I'm wondering is whether the tensions in the string on either sides of the mass will be the same. They are both the same string, but... I'm not sure.
they are not the same; friction accounts for the difference. If they were the same, the hook would slide to the low point of the wire, and there would be no friction.
Also, when separating tension into components, ie

T1sin 50 + T2 sin 30 = 5g
and
T1cos 50 = T2cos 30

T1 is the tension in the shorter side of the string, the side closer to the pole and inclined at a higher angle. T2 is the tension in the other side.
yes
Would friction be included in these equations? I think so, and I think it would be along the same direction as T1. (so replace T1 by T1 + Friction)
the equations do not include friction which is internal to the system. T1 is T1 everywhere on the shorter side, and T2 is T2 everywhere on the longer side, regardless of whether the string is at the hook or away from it. Rather, the resulting tension differential is due to friction.
But then I get two equations in three unknows, namely T1, T2, and Friction. I think I am going about this the wrong way...
Calculate T1 and T2. They would be equal if there were no friction, and the hook would slide to the low point, but since there is friction which prevents that, the friction force must be________?
 
  • #4
I see, so you would just solve for the tensions like in a problem without friction, and find the difference. That would be the friction.

I'll try to think about why the friction is not included in

T1sin 50 + T2 sin 30 = 5g
and
T1cos 50 = T2cos 30

..Oh, I think I just got it. These equations are used when the mass is at rest regardless of what friction or anything else is acting, to find the tensions.

And also, could you give me a hint as to where that T1e^mu came from? That looks very interesting... and useful for the next part of the problem, which asks for the minimum coefficient of static friction that will allow the hook to sit there.

Thank you very much PhanthomJay!
 
  • #5
I'm not much into the calculus, but for the derivation of [tex]T_2 = T_1e^{\mu\beta}[/tex], see
http://ocw.mit.edu/NR/rdonlyres/Physics/8-01TFall-2004/84DD1138-93A4-47E0-8F54-CB45EBE8351D/0/exp05b.pdf [Broken]
 
Last edited by a moderator:
  • #6
Interesting link, I appreciate it PhanthomJay.
 

What is a mass hanging from a string?

A mass hanging from a string is a common physics experiment that involves suspending a weight (or mass) from a string or rope. The weight is then allowed to swing freely, creating a simple pendulum motion.

What factors affect the motion of a mass hanging from a string?

The motion of a mass hanging from a string is affected by several factors, including the mass of the weight, the length of the string, and the force of gravity. Other factors such as air resistance and friction may also play a role in the motion.

How does the length of the string affect the motion of a mass hanging from it?

The length of the string has a direct impact on the period (or time) of the pendulum's swing. The longer the string, the longer the period, meaning it takes longer for the pendulum to complete one swing. The length of the string also affects the amplitude (or distance) of the pendulum swing.

What is the formula for calculating the period of a mass hanging from a string?

The formula for calculating the period of a mass hanging from a string is T=2π√(L/g), where T is the period, L is the length of the string, and g is the force of gravity. This formula assumes that the mass is swinging in a vacuum with no air resistance or friction.

What are some real-world applications of a mass hanging from a string?

The motion of a mass hanging from a string has several real-world applications, including timekeeping devices such as pendulum clocks and metronomes. It is also used in seismology to measure the motion of the earth's surface during earthquakes. In addition, the concept of a pendulum is used in engineering to create suspension systems for buildings and bridges.

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