Posted: February 13th, 2023

Experiment: Forces as Vectors

[Equipment list: Force Table, Mass Hangers, Various Gram Masses, Dry-Erase Pen, Eraser]

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The object of this experiment is to demonstrate the vector property of forces and to gain experience in the addition of vector quantities.

**Theory:** If many concurrent forces (forces that have the same point of application) are acting simultaneously on a body, they can be replaced by a single force called the resultant. In this experiment, you will determine a force that brings these concurrent forces into equilibrium. This force is equal in magnitude but opposite in direction to the resultant and is called the equilibrant.

The original forces and the equilibrant add to give a net force of zero – the body they act on is in equilibrium. In the case of the force table, if several arbitrary forces are applied to a small ring at the center of the table, in general the ring will not be in equilibrium.

The ring will be in equilibrium only when the equilibrant is applied to it. Since the resultant is equal in magnitude and opposite in direction to the equilibrant, this is an experimental method for finding the resultant.

To find the resultant of a number of vectors analytically, we first resolve each vector into components along two mutually perpendicular axes. The components along each axis are added algebraically to give the components of the resultant. The procedure is easier if the axes are chosen such that one of the vectors lies along an axis.

**As an example**, consider two component vectors **A** and **B**, which have an angle θ between them. For convenience, the positive x-axis is chosen to lie along **B**. The decomposition of the vectors into their components along with the resultant **R** and equilibrant **E** are shown below:

y

A

x

B

θ

The magnitude of the resultant is obtained analytically

by finding its rectangular components and applying

the Pythagorean Theorem:

In this particular case cos θ, and therefore Ax, is negative. The direction of the resultant is specified by the angle φ that it makes with the positive x-axis.

The equilibrant is equal in magnitude to the resultant but makes an angle of with the positive x-axis. This method can be extended to any number of concurrent vectors.

**Procedure: For each section of the procedure, you will find the results by three different methods (experimentally, graphically, and mathematically).**

**Part A:** Hang a mass of 200 grams over a pulley positioned at 30° on the force table. Use dry erase markers to label your coordinate system. To balance the x-component of this force (200 grams multiplied by the acceleration due to gravity) we must use a force with a magnitude equal to the x-component, but directed in the negative x-direction. Hang a weight from a pulley located at 180° in order to do this. To balance the y-component of the force, we must hang weight from a pulley at 270°. Add weight to the hangers for the x-component and the y-component until the forces are in equilibrium. The values of the components are determined from the weight over each pulley. Do not forget to include the mass of the hanger when determining these components. Record the magnitudes of these components onto the Excel Worksheet for this experiment. After you have recorded the magnitudes of the components, determine the uncertainties in both the magnitude and direction of each component. For the uncertainty in the magnitude, add mass in increments of 5 gram to one of the pulleys. When the ring moves the system is out of equilibrium and the amount of weight added is the uncertainty in magnitude. Remove the additional weight and repeat the process for the other pulley. To find the uncertainty in the direction move one pulley, with its correct weight, in increments of one degree until the ring moves. Do this to both the left and right of the initial angle to get a range of uncertainty. Return the pulley to its original position and find the directional uncertainty in the other pulley. Record these uncertainties onto the Excel Worksheet.

Find the components of the vector graphically. Plot the force to scale on the graph found on the Excel Worksheet for this experiment and determine its components. Draw vector arrows found in the “Shapes” option of the “Insert” drop-down menu. Go to “Insert”, click on “Illustrations”, then on “Shapes”. Choose the arrow symbol. Draw each vector, horizontally, to a length corresponding to the number of grams (), using the scale that you have designated. Right-click on the arrow and choose “Size and Position”. From there you can then adjust the angle of the arrow to the desired value. Once this has been done, you can then move the arrow to the desired position on the graph. Determine the components and record them on the Excel Worksheet.

Find the components of the vector mathematically. Calculate these in the designated cells on the Excel Worksheet. Make sure that the angle is in Radians in your equation for the proper calculation using Excel.

**Part B:** Leave the 200 grams at 30° and repeat the procedure outlined in part A for a new coordinate system in which the +x-axis is at 80° on the force table. Erase and remark to define the new coordinate system. Determine the x-component and the y-component of the weight of the 200-gram mass in the new coordinate system.

Find the components of the vector graphically on the Excel Worksheet.

Find the components of the vector mathematically on the Excel Worksheet.

**Part C:** Chose two forces between 100 grams (g) and 300 grams (g). Erase and remark to the original axes. Place the pulleys on the table so that the angle between them is not close to 0°, 90°, 180°, or 270°. A third force, called the equilibrant, can balance the two forces. The resultant is equal in magnitude, but opposite in direction, to the experimentally found equilibrant. Find the equilibrant and resultant of your two-force system.

Add the two forces graphically and mathematically. Compare the results of all three methods.

**Part D:** Repeat the procedure outline in C, but use three forces between 100 grams (g) and 300 grams (g). Place the pulleys so that the angles between the forces are not close to 0°, 90°, 180°, or 270°.

Add the three forces graphically and mathematically on the Excel Worksheet.

Compare the results of all three methods (experimental, graphical, and mathematical) in the Results section of the lab report.

Questions to Answer:

1.

a. In which procedure do you have the best agreement among the three different methods? State how close the results are and give some reasons why the results are better in this procedure than in the others.

b. In which procedure do you have the least agreement among the three methods? State how close the results are and give some reasons why the results are worse in this procedure than in the others.

c. Discuss the agreement and accuracy of the two procedures not discussed above.

2.

In procedure A, the pulleys were placed on the negative x-axis and the negative y-axis to find the positive x component and the positive y component of the vector. Where did you place the pulleys in procedure B? In what directions were the actual components? Explain the difference between the pulley placement and the actual component direction.

3.

How did you determine that equilibrium was established on the force table?

4.

Discuss the different types of uncertainty that affect the experiment and calculations.

SOLUTION

The experiment can be carried out as follows:

- Set up the force table by attaching it to a sturdy table or stand.
- Suspend one mass hanger from the center ring of the force table and place various gram masses on it to create a balanced load.
- Draw a vector diagram to represent the forces acting on the load. Each force is represented by an arrow pointing in the direction of the force, and its magnitude is proportional to the length of the arrow.
- Add two or more forces to the load and repeat the vector diagram. Observe the resulting force and compare it to the vector sum of the individual forces.
- Repeat the above steps for different combinations of forces.
- To demonstrate the principle of vector addition, apply forces at different angles to the load and observe the effect on the load’s motion. The net force can be determined by drawing a vector diagram and finding the vector sum of all forces.
- Repeat the steps for different forces and angles to observe the effect of vector addition on the load’s motion.
- Clean up the equipment by removing all masses from the hangers and storing them properly. Wipe the force table and dry-erase pen with an eraser to remove any marks.

By conducting this experiment, you will gain a better understanding of how forces can be added together to create a single net force. You will also see how the direction and magnitude of the net force are determined by the direction and magnitude of the individual forces.

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