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Assume we have some stationary point mass that would slide, if not for some coefficient of friction, along a light, inextensible string, attached to two points at different heights. Considering forces on the mass whilst it is stationary, which direction does the frictional force act?

My first thought is that it will act along the steeper of the two strings, but the concept of a reaction force (which surely friction must still be acting perpendicular to) seems less clear for a string?

enter image description here

This diagram gives an idea of the kind of set up i'm thinking about (from a random webpage with a sufficiently useful diagram, not specific to this problem), although the string can be of any length and thus don't assume that either string is horizontal.

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    $\begingroup$ A diagram can help here. $\endgroup$
    – John Alexiou
    Commented Dec 16, 2013 at 14:38
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    $\begingroup$ Friction opposes motion. Find out which way it is going to slide and friction will be opposite of that. $\endgroup$
    – John Alexiou
    Commented Dec 16, 2013 at 15:43
  • $\begingroup$ As a general rule, I agree of course, but there are a few caveats here, for one thing motion isn't linear, for the mass to move along the string it must trace out some function (a parabola?), does this mean friction should act to oppose it on this path? Also how do you reconcile this with the direction of the normal reaction force? $\endgroup$
    – zephyr
    Commented Dec 16, 2013 at 15:56
  • $\begingroup$ If force is a force, then it opposes a tangentially linear motion, even if it is an angular speed at a radius. $\endgroup$
    – John Alexiou
    Commented Dec 16, 2013 at 17:03
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    $\begingroup$ I understand that with the idealization of a point mass on a string it creates a "kink" where tangency is not defined. Is that your real question here? $\endgroup$
    – John Alexiou
    Commented Dec 16, 2013 at 17:07

2 Answers 2

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The string contacts the point on two infinitesimally close points with different slopes. Imagine a small pulley end the two points are the entry and exit point of the string.

If the string is between points A on the left and point B on the right (with B lower) then we call the angles of the string from horizontal $\theta_A$ and $\theta_B$. If the mass is moving to the right, then the balance of forces are:

FBD

I forgot to add the weight of the mass, but you can imagine what that would look like above.

The zoomed in sketch of point C shows that there are two friction forces acting tangentially each. Here $N_A$ and $N_B$ are the contact forces, and $\vec{v}_C$ is the velocity vector of the mass. To solve this problem you need to define the kinematics of C which always lies on an ellipse. You have to choose one independent coordinate (like $\ell_A$) and calculate all other variables (and derivates) from this value (and its derivatives).

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  • $\begingroup$ This looks great, but can we draw any surer conclusions in the case that the mass is stationary, as in the original question? $\endgroup$
    – zephyr
    Commented Dec 16, 2013 at 22:07
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Not sure i understand fully your question but in general friction can be seen as a force (vector) pointing in the opposite direction of motion with (if there is a motion). Moreover, the force is tangent to the surface of contact. Thus for an object with spherical symmetries (like a pulley, cylinder, sphere...), the force is perpendicular to the radius.

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  • $\begingroup$ Sorry, should have specified that the mass is stationary, i.e. it's a statics question, if what you say about force being perpendicular to the pulley is true though then does that mean the reaction force should be acting at an angle such as to bisect the the angle between the two lengths of string? $\endgroup$
    – zephyr
    Commented Dec 16, 2013 at 14:56
  • $\begingroup$ If its a static problem then the mass will stay where the energy is minimized. So I guess that you are looking for this position? $\endgroup$
    – PinkFloyd
    Commented Dec 16, 2013 at 16:33
  • $\begingroup$ Not necessarily, there will be a family of points at which the force of gravity is not able to overcome static friction. Ignoring work done against friction the lowest energy point is simply that which minimises the gravitational potential, i.e. the lowest point attainable for a given length of string. But tha's really not what my questions asking, i want to know for a given position in which direction friction acts and how the reaction force relates to this. $\endgroup$
    – zephyr
    Commented Dec 16, 2013 at 16:45
  • $\begingroup$ now i understand ! so the friction will indeed be in the same direction of one side of the string, pointing in the opposite direction than the motion. $\endgroup$
    – PinkFloyd
    Commented Dec 16, 2013 at 17:47

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