Equilibrium of Force

Laboratory 1 Equilibrium of Force

KAS 1331

Introduction The design of all type of structures is base on the effect of their own weight, the forces acting on them, and whether the structure is moving. Normally in structural design there is no motion and the bodies are said to be in static equilibrium. A system of forces in equilibrium satisfies two principles: the resultant of all the forces must be zero: and the moment of all the forces about any point must be zero. The usual mathematical way of writing this for a three dimensional system is with six conditions:ΣPx = ΣPy = ΣPz = 0 ΣMx = ΣMy = ΣMz = 0 If the forces act solely in one plane then the three conditions of equilibrium are:ΣPx = ΣPy = 0 ΣMx = 0 List of parts See Packing List at back of Instruction Manual.

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Laboratory 1 Equilibrium of Force

KAS 1331

Theory Definition: The equilibrant of any number of forces is the single force required to produce equilibrium

In the figure the resultant of

and

is shown in red. The equilibrant of

and

then the vector opposite in direction to this resultant with the same magnitude (i.e.

•

,

and

•

is the equilibrant of

•

and

is ).

are in equilibrium

and

are kept in equilibrium by

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Laboratory 1 Equilibrium of Force

KAS 1331

The triangle below illustrates relationships of polygon.

There is an important principle which emanates from some of the trigonometric calculations performed above. The principle is that as the angle with the horizontal increases, the amount of tensional force required to hold the sign at equilibrium decreases. For find the total of internal angles = (n-2) x 180 Degree accuracy = different between theory and experiment / theory internal angles

Objective •

To study the equilibrium of a set of forces acting in a vertical plane

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In the first part the case of concurrent forces is to be investigated and checked by the graphical solutions of a triangle of forces (three forces) or a closed polygon for more than three forces.

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The second part deals with non-concurrent forces and the use of a link polygon.

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Laboratory 1 Equilibrium of Force

KAS 1331

Apparatus •

The set of accessories is mounted in a reaction frame HST1.

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The forces board

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A1 size drawing paper

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Loading hanger

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Loading

Procedure Two sheets of drawing paper size (841 x 594mm) and three or four colours of pen or pencil are used. To commence each of the experiment a clean sheet of the drawing paper is to be fitted on the work surface of the force board using the four clips. Either cut a hole in the middle of the paper to fit over the centre peg, or remove the peg to fit the paper and then make a hole in order to replace the peg. Part 1: Concurrent Forces •

The cord ring is taken and three load cord assemblies are attached.

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The cord ring temporarily is placed on the centre peg and three pulley brackets are fixed.

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The ring is allowed to find its position of equilibrium due to the weight of each load hanger.

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The loads are added to the hangers, how the cord moves are noted to a new equilibrium position each time an extra load is added.

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The line marker gauge to transfer any equilibrium state onto the drawing paper two points on each of the three load cords radiating from the ring. The set of six points is identified and the total loads in the cord are noted.

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Now start again with four or five load cords attached to the ring and with an equal number of pulley brackets round the frame.

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Laboratory 1 Equilibrium of Force •

KAS 1331

When a suitable equilibrium position is attained mark the lines of action and forces in the cords with a new colour.

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Finally, the procedure is repeated with all six load cords attached to the ring cord.

Part 2: Non-concurrent Forces •

The double cord ring is used and attaches five or six cords, three on one ring and the rest on the other.

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One of the rings temporarily on the centre peg while attaching the pulley brackets round the frame.

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The load cords are draped over the pulleys and attach load hangers.

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Suitable weights are added to the hangers and carefully release the cord ring from the centre peg.

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When a convenient state of equilibrium is attained transfer the lines of action and known forces to the drawing paper and mark the system with a colour.

Result On the Part sheet of drawing paper use a long straight edge to draw the line of action of each set pf forces in equilibrium. The lines, lengthened as required, ought to interest where the centres of the cord ring were. For the first experiment using only three cords, construct a triangle of forces, as described in the text book, by drawing lines parallel to the experimental lines of action. Measure the lengths of all three sides a, b and c. compare these lengths with the forces represented by these sides, that is an alternative construction should be made by first deciding a scale for the vector Fa (for example 1 cm = 1N)so that it can be set off along its line of action. Fa

:Fb

:Fc

A

:b

:c

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Laboratory 1 Equilibrium of Force

KAS 1331

The triangle of forces is continued by setting off vector Fb to scale in its line of action. The third side can then be compared. 1. in direction with the experimental line of action, 2. in magnitude with the known forces Alternatively the third vector Fc is drawn to scale in its line of action, and if it does not complete the triangle then that is the effect of experiment error. Where there are more than three forces in equilibrium this method of drawing the vectors sequentially is the best way of using the results to obtain a force polygon. One would expect the polygon to nearly close as a simple experiment of this kind does not usually produce much error. The Part 2 sheet of drawing paper is used to determine the direction and magnitude of sets of non-concurrent forces in equilibrium. For each set of forces polygon should close, and this can be tested by drawing the vectors sequentially clear of their lines of action on the drawing. The next step is to construct a link polygon on the actual lines of action of sets of forces. Place a polar point 0 anywhere on the forces polygon and work backwards from the forces polygon onto the lines of action. This will provide a link polygon which should close on the last line of force of the set, since this would be the so-called equilibrium (the reverse of the resultant). Repeat the graphical construction for each set of non-concurrent forces in equilibrium. Also check that the use of a different polar point “0” (using another colour) still leads to a closed (different) link polygon.

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Laboratory 1 Equilibrium of Force

KAS 1331

Discussion The objective of the experiment was achieved. This experiment is separated to two parts: First part-concurrent forces Second part-nonconcurrent forces From the experiment, the values of internal angle which have been calculated are different from the theory for each part where Part 1 Total of internal angle Load

Theory

Experiment

2N,2N,2N 2N,2N,2N,1N Part 2 Total of internal angle Load

Theory

Experiment

2N,1N,4N,1N,1N,3N

From the table assume an error has occurred in the experiment 1(b), 1(c) and 2. This is caused by some factors. The errors that exist are: •

Error while taking reading-the reading was booked only by one person and the observer read the results once.

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Laboratory 1 Equilibrium of Force •

KAS 1331

The factor that we cannot connect the line to form a regular shape is maybe we didn’t draw the line well during the experiment.

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It may also cause by the accuracy that we do in the lab.

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In the experiment, the point must be plotted on the paper to get the real figure of the forces. However, there is an error while plotting the line of forces. This is because the sketch is different from the figure that we should get.

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Environment error must be considered as one of the factors. This can cause by accidentally hit the instrument.

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The cord is stuck at the pulley. This can affect the accuracy of reading and the equilibrium of forces.

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The other errors are the instrument does not function properly To avoid these errors, there are few steps that must be taken while doing the

experiment. First we must ensure that the instrument is in good condition and has been setup properly. This is to get the accuracy of reading in this experiment. In the process of booking and reading the result must been taken more than one reading in order to avoid parallax error. Make sure that the environment is safe from any disturbance and vibration from external attraction. Before plotting, make sure that all the pulleys are not stuck. Ensure that the cords are being pressed while sketching the line of forces. Conclusion In this experiment it was mentioned that resultant forces cause objects to accelerate. If an object is stationary or moving at constant velocity then either: •

No forces are acting on the object, or

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The forces acting on that object are exactly balanced.

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Laboratory 1 Equilibrium of Force

KAS 1331

A resultant force would cause a stationary object to start moving or an object moving with a given velocity to speed up or slow down or change direction such that the velocity of the object changes. In other words, for stationary objects or objects moving with constant velocity, the resultant force acting on the object is zero. The object is said to be in equilibrium.

Appendix

Example of the load sketch

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Laboratory 1 Equilibrium of Force

KAS 1331

References

Strength of Materials (Mechanics of Solids), R.K Rajput, S. Chand. Mechanics of Material Ferdinand P. Beer, E. Russell Johnston, Jr. and John T. DeWolf. McGrawHill Mechanics of Engineering Materials Crawford, C.G and Armstrong Pearson Prentice Hall Mechanics of Materials James M.Gere Thomson Brooks Cole Structural Engineering Chyuan-Shen Lee and J.J Lee McGrawHill

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Laboratory 1 Equilibrium of Force

KAS 1331

Engineering Mechanics Statics (SI EDITION) Anthony Bedford and Wallace Fowler Pearson Prentice Hall

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