Solving Tension Problems

However, one thing that does confuse students, especially since it is given in the problem, is why the length of the ropes do not matter in the problem.Take the problem given above and imagine you were a small atom in the rope on the left.

However, one thing that does confuse students, especially since it is given in the problem, is why the length of the ropes do not matter in the problem.Take the problem given above and imagine you were a small atom in the rope on the left.

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The poor atom is like a poor person stuck in a tug of war between two friends, each yanking on an arm in one direction.

Since the rope is not moving and is not tearing, these forces must be the same for every atom in the rope.

Well then the problem is no harder, it's just given us more busy work.

Notice that the components are completely horizontal or vertical, respectively.

So as because it's horizontal component would pointing in the negative direction.

Notice that negatives play no role in the magnitude of the vector. After all, exerting a fixed amount of force in a direction should be the same force no matter what direction you decide to face!The atoms in the object 'pull back' resulting in tension because they form a nice stable substance, all bound together or attracted to one another, and do not want to be moved but left in equilibrium.In reality, small differences in the tension an object feels cause shredding because some atoms are pulled in one direction more than others so they move.As in the discussion above, let's draw in the components of each vector. So the forces in the horizontal and vertical directions must be equal.We label the vertical and horizontal components of each with a angle and the bottom right angle of the triangle are alternate interior angles of the transversal and must be congruent. Vertical Direction: In the vertical direction, the only forces upwards are from the vertical components of the rope (which from above discussion, we now know how to find! So the force upwards must be Horizontal Direction: In the horizontal direction, we only have the forces of the left rope and the right rope pulling the weight to the left/right.This 'reactionary' pullback force is tension in the rope. [Notice the rope feels this force but does not exert it in a direction.The rope is not pulling, so there is no true direction to this, hence why tension is not a true force.] This is all one needs to know about tension to understand how to do the problems involving tension in Calculus III.Tension is a result of Newton's Three Laws, which summed up rather loosely are 1st Law: An object with no external force acting on it does not change velocity.2nd Law: 3rd Law: Every force has an equal and 'opposite' force - a reaction force. Well if I pulled on a rope tied to a tree with 100N of force, the rope 'pulls back' with 100N of force.Meaning, when you push a book diagonally across a table, your force vector is diagonal and the book moves diagonally.However, your diagonal force vector has two components: a vertical and horizontal one.

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