2. Rock Climbing |
We have seen that forces are vectors, having a magnitude and direction. Vector addition results in a resultant force. If the forces are equal in magnitude but opposite in direction, the resultant force is zero. This does not mean that there is no force, but that the overall force is zero. We say that the forces are balanced.
If forces are in the same direction, they add up to form the resultant. If they are in opposite direction, they subtract. The direction of the resultant is in the direction of the larger force. The forces are unbalanced and result in acceleration in the direction of the unbalanced force (Newton II).
In the picture we can see the resultant of two forces. We can show that they make a vector triangle by moving Force 1.
We could do this with any number of forces to make a vector polygon.
We can see that in the case above, the forces are unbalanced and there is a resultant force.
This means that forces are balanced and the overall force is zero. The picture below shows the forces in equilibrium. We have reversed the direction of Forces 1 and 2, but kept the magnitude the same. We have kept Force 3 the same magnitude and direction.
The resultant of Forces 1 and 2 will be of the same magnitude as Force 3, but in the opposite direction. This means that the forces will be balanced and the resulting force is zero.
If any force vector polygon forms a closed loop, the forces are in equilibrium. This is the polygon of forces rule. The study of forces in equilibrium is called statics, and is an important consideration when structures are designed.
You will have done an experiment like this to show the idea:
Each mass exerts a force equivalent to its weight. We can put the force vectors tail-to-tail:
We see that they form a closed triangle and the directions form a closed loop. Therefore the forces are balanced.
Choose a scale (e.g. 2 cm = 1 N)
Use graph paper.
Use a sharp pencil.
Use a compass.
Use a protractor if angles are mentioned.
Draw the arrows in the direction specified.
Here is the equilibrium situation above represented by accurate drawing:
The angles are measured with a protractor to give the values shown.
The problem with accurate drawing is that of accuracy. If you get the answer to the nearest degree, you're doing well. And accurate drawing is not easy. If you are challenged by measuring, resolution of vectors using trigonometry is the answer.
Any vector in any direction can be resolved into vertical and horizontal components at 90 degrees to each other.
For three forces in equilibrium we can draw a force vector diagram:
For this situation we know that weight always acts vertically downwards. We can resolve the two other vectors into their horizontal and vertical components:
T1 resolves into T1 cos q1 (horizontal) and T1 sin q1 (vertical);
T2 resolves into T2 cos q2 (horizontal) and T2 sin q2 (vertical).
We know that the three forces add up to zero, so we can say:
T1 cos q1 + - T2 cos q2 = 0. This means that the forces are equal and opposite. Þ T1 cos q1 = T2 cos q2
T1 sin q1 + T2 sin q2 = W
Be careful that you don't assume that W is split evenly between T1 sin q1 and T2 sin q2. This is only true when the weight is half way between the ends.
Some key words:
Strong - a large tension is needed to break a material:
Elastic - Stretches under load and returns to its original load.
Stiffness - A material is stiffer if it stretches less for a given load compared to another material.
If a wire or rope is loaded, we find that it stretches in proportion to the load. Double the load, double the stretch. We can sum this up as:
Adding the proportionality constant gives us:
Rearranging gives us the stiffness constant:
If we plot a graph of force (vertical axis) against load (horizontal axis), the stiffness constant is the gradient.
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