Experiment 4. Vector Addition: The Force Table
As we have learned in lecture, to the extent that pulleys are massless and frictionless, they can change the direction but not the magnitude of the tension force associated with (negligible mass) strings which are hung over them. This fact forms the basis for a force table. The force table allows one to redirect the force of gravity to produce two-dimensional force vectors. We will see that we can use the force table to experimentally determine the sum of vectors, or resolve vectors into their components, or to solve physical problems involving vector quantities that do not involve forces at all.
As you can see from the illustration which follows, weights are hung from the strings (on convenient weight hangers). Their weight provides the tension in the string, which in turn provides the forces which we will manipulate. All of the pulleys are movable so that the forces may be directed in any direction desired in the two dimensional plane of the table.
1. Resolving the Components of a Vector
Begin by removing two of the four hangers. Move two pulleys to the 30◦ and 210◦ positions
(remember that we are measuring angles counter-clockwise from the 0◦position. Add weights
to the hangers so that each hanger has a total of 400. grams. As you know Fg = mg, so that
this translates to a force of (0.400 kg)g = 3.92 N. (We will assume g = 9.80 m/s2 is known to
3 significant digits in this experiment.) The forces represented by these two sets of weights should be equal in magnitude but opposite in direction. As a result, the center pin in the force table should be centered within the central ring to which the strings are attached. We assume that the forces are in equilibrium, and the net force is 0.
Now, looking down on the force table, imagine that you have superposed a rectangular coordinate system, with the positive x axis extending from the center pin to the 0◦ position,
the positive y axis extending from the center pin to the 90◦ position, the −x axis extending
from the center pin to the 180◦ position, and the −y axis extending from the center pin to
the 270◦ position. With that in mind the two forces may be represented as below:
For any two dimensional vector, we may resolve it into x and y components. We will now do this experimentally for vector ~F1 which points toward the 30◦ location. As we have seen ~F
1
balances ~F2, the force vector pointed toward the 210◦location. Remove all weights (including
the hanger!) from the string at 30◦. We are going to replace this vector force ~F
1 with its
x and y components. Position pulleys at 0◦ and 90◦. Weights hung on the strings at these
Figure 4–1: 2 forces in equilibrium
weights at these positions until the center circle of the force table is once again centered on the center pin. We have now replaced ~F1 with two vectors, one acting along the +x direction (F1,x) and one acting along the +y direction (F1,y).
Record the mass needed at each position, and convert this to the corresponding force. Is your result consistent with theoretical expectations?
2. Experimental Solution of Force Problems
Consider the following problem. A 0.450 kg sign hangs from two strings, as shown below. What is the tension in each of the strings?
At this point, this is a problem that you can solve easily from a theoretical viewpoint. Here, you must solve this problem experimentally. One can put a pulley and weights representing the sign, at say, 0◦. Where should one then put pulleys and weights representing the tensions
in string 1 and string 2? After that is determined, add the appropriate amount of weights at these positions to bring the system into balance (the center ring centered on the pin). Describe your procedure, record the needed mass at each position, and convert this to the corresponding force. Is your result consistent with theoretical expectations?
3. Adding Vectors in General
While a force table facilitates the manipulation of two dimensional force vectors, it may also be used to represent the combination of any type of vectors. One may associate a scaling with the weight (or mass) associated with a direction. For example, 300 m north may be represented by 300 grams at 90◦, or 34.5 m/s at 20◦ west of north may be represented by
345 grams at 110◦. It is straightforward to experimentally determine the vector sum of two
or more vectors.
Some degree of care is needed however. When using the force table to add vectors, it is important to remember that the force which balances the vectors to be summed is known as the equilibriant ~E. The equilibriant has the same magnitude but the opposite direction to the vector sum. Effectively, 180◦ must be subtracted from the direction of the equilibriant
to give the direction of the vector sum. A schematic example shows this.
Figure 4–3: Adding Vectors
A plane can fly at 350. miles/hour with respect to the air, and the pilot aims the plane due east. It is in a region where the air is blowing at 300. miles/hour due north with respect to the ground. What is the plane’s velocity with respect to the ground?
As you are aware, one can define vectors ~vpa ≡ velocity of plane with respect to air, ~vag ≡
velocity of air with respect to ground, and ~vpg ≡ velocity of plane with respect to ground.
The first two quantities are known, and the third may be found by vector addition: ~vpg = ~vpa+ ~vag.
Describe your procedure to solve this problem experimentally, and give your result. Please provide both magnitude and direction. A naive way to add quantities might be simply to add magnitudes so that the magnitude of vpg = 650 miles/hour (Note: This may NOT be
a good idea!!). Is this appropriate here? Is your experimental result consistent with this? Is there another way to mathematically combine the magnitudes that gives a result more consistent with experiment?
If your results are not consistent with theoretical expectations, can you suggest any reasons why? Can you suggest any improvements to the procedure?
Procedure Notes
All of the provided weights will be hung on hangers which have mass 50 grams. Remember to include the mass of the hanger when calculating the total mass at a position!
When you are not using a string and pulley, remember to remove the hanger. The weight associated with the hanger can affect the balance.
Most of the force tables have two sets of angles. For these tables, you are to use the inner set of angles which increases from 0◦ in the counter-clockwise direction. Two force tables
have only the one set of angles which increases in the counter-clockwise direction.
Keep in mind that these are experimental procedures. It is not expected that you duplicate theoretical expectations. This is especially true since the theoretical calculations assume no friction and massless pulleys and strings. The assumptions are not strictly true in the lab. Still, I expect that the experimental results will not be far from the theoretical expectations. You will be expected to provide these theoretical expectations.
Include your
theoretical solutions to these problems
with your lab report.
The students at each station should participate in and understand each of the 3 procedures. All of the students at a given station may share their data. You will not need to repeat your force table measurements.
When trying to find the unknown position of an equilibriant vector, it is often convenient to loosen the pulley and move it along, occasionally pulling on the string. When it appears that pulling on the string will center the center pin in the circle, you have found your location. You can then tighten the pulley and begin adding weights.
Measuring Apparatus
The Force Table A picture of a Pacific Science Supplies, Inc. Student Force Table from the Instructional Manual is reproduced below. The majority of the force tables are identical to this, and the others are similar.
