B Rod resting against a smooth peg

[brotherbobby]

Statement : Here is the statement from the text that I paste to the right. Diagram : Does anyone have a diagram (image) as to how does the situation look?

Normal Reaction : When a rod rests against a smooth wall, we know that the direction of the reaction is normal to the wall. I understand that the peg is a point, hence there is no meaning to the statement of the reaction force being "normal" to it. However, how is the reaction force normal to the rod? Is it also the case for a rope tied to the peg and through which a tension force exists?

Answers would be most welcome.

 

[jbriggs444]

View attachment 313871Statement : Here is the statement from the text that I paste to the right. Diagram : Does anyone have a diagram (image) as to how does the situation look?

Normal Reaction : When a rod rests against a smooth wall, we know that the direction of the reaction is normal to the wall. I understand that the peg is a point, hence there is no meaning to the statement of the reaction force being "normal" to it. However, how is the reaction force normal to the rod? Is it also the case for a rope tied to the peg and through which a tension force exists?

Answers would be most welcome.

I do not agree that a peg needs to be modeled as a point. Similarly, a rod need not be modeled as a line. In three dimensions, both can be modeled as cylinders. In two dimensions, the rod can be modeled as a long thin slab and the peg as a circle.

You ask about a force "normal" to a rod. That would be a force at right angles to the surface of the rod at the point of contact. If the peg is modeled as having a non-zero size, this will also be at right angles to the surface of the peg at the point of contact.

You also ask about a rope (wrapped around?) a peg. This time, the point of contact is spread around the peg. The "normal" force loses some of its meaning. Instead, we have the normal force over here, the normal force a bit further along and the normal force a bit further still. There are ways to calculate the net effect of all of those normal (and frictional) forces. It amounts to adding them all up using integrals, derivatives and a bit of trigonometry.

If the rope on peg is free of friction, the result is quite simple indeed. The assembly amounts to a pulley.

If the rope is tied to the peg, the result is again quite simple. The assembly amounts to a fixed attachment point.

If the rope on peg has friction, an exponential function results. [One solves a homogeneous linear first order differential equation. That is about the simplest sort of differential equation there is, so this example is sometimes used when teaching differential equations]. The maximum tension on the one side before the rope slips is given by a function along the lines of $t_1 \leq t_2 e^{\mu\theta}$ where $\mu$ is the coefficient of [static] friction and $\theta$ is the angle through which the rope wraps. It is easy to hold a rope taut against a huge counter-force if you can wrap it a couple of times around a tree first.

One can also think of the normal and frictional forces in relation to knots. A knot that holds is one in which tension from the load feeds back into tension in the knot sufficient to allow friction within the knot to resist the load. [If you wrap that rope a couple of times around the tree and then tuck your end under the first loop, you can walk away and not bother holding on]

 

[Steve4Physics]

In addition to what @jbriggs444 has said, can I add this.

Consider two objects (of any shape) with a single point of contact (P).

The objects’ surfaces have a common tangent at P. Their common normal at P is perpendicular to this common tangent.

Suppose one of the objects (say a rod, radius R) shrinks, keeping P fixed. The direction of the tangent (and hence the normal) at P are unchanged. This remains true as R→0 (when the rod is now a line, with a cross-section which is a point).

 

[Lnewqban]

However, how is the reaction force normal to the rod?

Can you visualize the rotating normal force that transfers movement from the peg to the slotted part?

http://507movements.com/mm_100.html

 

[Steve4Physics]

Can you visualize the rotating normal force that transfers movement from the peg to the slotted part?

http://507movements.com/mm_100.html

Sorry this is off-topic. But I have to say that the site in your above link has many wonderful drawings (some of which are animated). Fantastic! E.g. http://507movements.com/mm_378.html

 

[jbriggs444]

Can you visualize the rotating normal force that transfers movement from the peg to the slotted part?

http://507movements.com/mm_100.html

A peg in a tightly-fitting slot is no longer about the normal forces on either side of the peg. Those are statically indeterminate. If the slot is a tight fit, both normal forces increase. If it is a loose fit then one may decrease to zero or a gap may even open up.

Instead, a peg in a tightly-fitting but frictionless slot should be viewed in terms of the constraint that it places on the resulting motion. The sum of the two normal forces is the net force associated with the constraint.

 

[Lnewqban]

Another one.
The peg is fixed and its normal reactive force against the bar changes direction:

http://507movements.com/mm_268.html