Static Equilibrium (Crane Boom)
To verify the condition for the static equilibrium of concurrent forces
A crane boom apparatus, two force gauges, a 1-kg weight, a gravity protractor, a few pieces of string, and a calculator
The conditions for the static equilibrium of a beam are:
ΣFx = 0, ΣFy = 0, and ΣT = 0. ( The net torque about any point of the object (boom) must be equal to zero.)
For the equilibrium of forces that pass through the same point, ΣT = 0 is already satisfied. This leaves only two conditions for the equilibrium of concurrent forces, simply
ΣFx = 0 and ΣFy = 0. (1)
In this experiment a crane boom may be set up as shown in Fig. 1, below:
Point A is the center of the hole in the ruler where the 3 forces w, T, and C pass through. Angles θC and θT may be measured by a gravity protractor and inserted into the equations. To solve for the accepted values of T and C, Equations (1) may be applied according to the following figure.
ΣFx = 0 ; C cosθC - T cos θT = 0
ΣFy = 0 ; C sin θC + T sin θT = w
1) Gather the appropriate equipment as per equipment list. For this experiment use force gauges that read in Newtons.
2) set up the crane boom apparatus according to Figure 1. Do not connect the 1-kg hanging weight to the boom yet. The screw that connects the B-end ruler (boom) to the hinge axle (Fig. 1) must be below the ruler for the free rotation of the boom about that hinge. If the screw is above the ruler, it will not allow the positioning of the boom an any desired angle.
3) Place one of the force gauges (No. 1 in the figure) between the top hinge and the top end of the ruler, end A. This measures the tensile force (T) in the string. Adjust or choose the length of the string short enough to have the A-end of the boom above its B-end according to Figure 1. If the string is too long, and the A-end is either at or below the B-end, it will cause high tension in the string causing it to break. For this reason the string that hangs the 1-kg weight from the boom at A must be long enough so that the hanging weight is not more than 2-3 inches above the floor.
4) Adjust force gauge 1 to zero while it is just bearing the weight of the ruler. You may have to repeat this a few times to finally see that the gauge shows zero while the boom lays its own weight on it. This eliminates the need for taking the weight of the beam into calculations.
5) With a long piece of string hang the 1-kg weight from point A such that the bottom of the weight is not more than 2-3 inches above the floor. Stay away from this hanging weight and make sure you have covered-toe shoes on.
6) Read the tension in the top string from the force gauge and record its value.
7) To measure the compressive force, C, in the boom, connect the second force gauge to point A of the ruler by passing its hook through the hole at A while holding the force gauge and your forearm exactly along the boom.
8) Pull this force gauge along the boom while trying to hold your forearm along the boom as well until the hinge at B starts to disengage and become loose. With this trick, you are replacing the compressive force C of the boom by its equal magnitude tension in the force gauge 2. Now force gauge 1 reads the tension in the string and force gauge 2 reads the compression in the boom. Force gauge 1 should show exactly what you recorded in step 6. If not, repeat this step until it does so.
9) Obviously, force gauges 1 and 2 give you the measured values of T and C, respectively.
10) Measure angles θC and θT while the beam is loaded and plug these values along with M = 1.0kg in Equations 1 in order to solve for the accepted values of T and C.
11) Calculate a %error on each of T and C.
12) Repeat the experiment for a different case. All you need to do is to change the position of the top hinge by loosening it and slide it up or down 1-3 inches.. The angles will change and so do the magnitudes of T and C.
g = 9.8 m/s2, M = 1.0kg
θC , θT , T and C for two cases
Comparison of the results:
Provide the percent error formula used as well as the calculated percent errors on T and C in each case.
State your conclusions of the experiment.
Provide a discussion if necessary.