Mathematical Investigations IV

Name:

Vectors

Getting To the Point

Hanging in Suspense

In order for an object to be in a state of equilibrium, that is, in a state where the object is not being pushed in any direction by the forces acting on it. The sum of the forces acting on the object must be zero. Force is a vector.

For example, if you hang a picture or put a shelf on a wall, good craftsmanship demands that building materials and methods are chosen in such a way that the picture or shelf do not move much if bumped. If the picture to be hung has a wire on the back to be put over a nail (or other appropriate hardware), forces acting on the picture are gravity and the tension of the wire.

Gravity is a vector

magnitude = g = 9.8 m/s2
direction = down (–j)

1. Let’s consider a 20 kg mass suspended in mid-air by two cords. Note that we do not want to confuse the length of the cords (the physical object) with the magnitude of the force (the length of the vector).

Using vectors, we can find the tension in each cord. The vector diagram (called a “free-body diagram” in physics), indicating the forces acting on the weight, looks like this:


We know the direction but not the magnitude of both T1 and T2.

Express T1 and T2 in terms of their x- and y-components. Fill in the blanks:

T1 = < | T1| cos 150o, | T1| sin 150o >

[Measuring the angle from of the positive x-axis]

T2 = < , >

-mgj = < 0 , >

Find | T1| and | T2| such that T1 + T2 + (-mgj) = 0. This will assure that the forces sum to 0 so that the weight is hanging in equilibrium.

2. Find the tension in the two cords holding a 10 kg mass suspended as shown.


Schematic /
Free-Body Diagram

T1 =

T2 =

3. Let’s consider what might happen to the tension in the two cords when the shape of the triangle made by the ceiling and the cords is changed?

a. Under what conditions would | T1| = | T2|?

b. Under what conditions could | T1| = 0 or | T2| = 0?

Vectors. 9.1 Rev. F08