## S-3 LADDER AND CRANE BOOM PROBLEMS1. PURPOSE:
To investigate static equilibrium situations in which a rigid bar, tilted to the horizontal, is in equilibrium under the action of non-concurrent forces. To analyze these situations using forces and torques.
(1) (2) (3)
Read the theory section of experiment S-1.
The tension is
The rod's weight also figures in when the rod is tilted. This is very quickly analyzed by considering the longitudinal force components only, for if the rod is in equilibrium, they must add to zero. Let T1 be the longitudinal component of all forces acting on the top of the rod. This is the tension at the top of the rod. Let W be the weight of the rod, acting downward at its center of mass. Let T1 be the longitudinal component of all forces acting at the bottom of the rod. For this simple example, we will assume no other forces act on the rod. Then T
Of course, if there
The The
Two situations from "everyday life" make instructive exercises in statics. They are (1) A ladder propped up against a wall (2) A simple crane boom for lifting and relocating heavy objects.
Physics textbooks have many examples and exercises relating to these two situations. As the diagrams show, the two have a great similarity. We will take advantage of this to discuss an analysis procedure which applies to both. The figures below illustrate the similarities. In both cases the rigid, but movable element makes an angle with the horizontal, which we will call θ. The bottom of this rigid element is supported by a force which has two components which we will call V (for vertical component) and H (for horizontal component.) In the ladder problem it is usually assumed that friction at the top of the ladder is small enough to neglect. So the wall exerts a force on the ladder which is strictly horizontal.
The crane boom, set up as illustrated, has a horizontal force at the top supplied by the
tension in the horizontal cable. In both problems there are only two horizontal components
on the rigid element, so these must be equal in size and opposite in direction to produce
equilibrium. We will label their size H. Therefore |H The weight of the rigid element Finally, the rigid element may support a load. In the ladder, this is the extra weight of the man standing at some point on the ladder. In the crane boom it is the load, usually hung or attached at the upper end of the boom. In either case we will label it L, for load.
Spring balances are designed to read weights correctly when they are hanging straight down. In this experiment we may choose to use them in other positions, so we must first find out how to correct the readings for angle. (1) Use a ball-bearing pulley, weight hanger, string, and spring balance arranged as shown in the figure. The angle may be measured with a goniometer or a large protractor. (2) Load the weight hanger so the spring balance reads a value about in the middle of its range. Be sure you use a pulley strong enough for this load. Don't forget that the mass of the hanger is part of the load. (3) Take data for spring balance reading vs. angle for angles from zero to 180 . Be sure to "jiggle" the system before each reading to avoid effects of frictional drag in the pulley, and in the spring balance itself. (4) Plot this data. If you happen to have a sheet of polar coordinate graph paper, use it to make another graph of the same data. (5) The data was taken for fixed load. What would you expect to happen if the load had
been significantly larger or smaller? State how your graph could be adapted to give correct
readings for
This may be constructed with a special metal beam (which represents the ladder) suspended in a square framework built from heavy laboratory rods. Spring balances attached to wires supply forces which represent the wall thrust, and the vertical and horizontal components of the forces the ground would exert on the foot of the ladder. The weight of a person on the ladder is represented by a load hung from one of the "rungs". Set up the model as shown in Fig. 9. Do not ignore the weight of the "ladder" beam.
Remove it and weigh it on a balance. Its center of mass is marked by an indentation labeled
"C.M." Keep the load, L, within a range from 1 to 5 times the mass of the ladder. Be sure that V is exactly vertical, and H and H horizontal.
Set up the crane boom as shown in Fig. 3, being certain that the supports are rigid, and
the lower end of the boom is The boom will be loaded heavily, so pay attention to good "engineering design" to
prevent collapse of the structure. Do not suspend the load far above the floor, and keep your
feet out from under it. The impact of one kilogram falling on your toe from one meter high is
a sensation you would not soon forget. [It Keep the weight of the load between 1 and 5 times the weight of the boom. Be sure the chain attached to the top of the boom is exactly horizontal. Measure the dimensions of the system, and read the thrust tension on the boom scale. Also note the amount of the load (including the chain and hanger.) Before the system can be analyzed it is necessary to know the weight and center of mass of the boom. The boom has a built-in spring balance, and it shortens under load, thereby changing the position of its center of mass. It is necessary to locate the center of mass when the boom is compressed to the same reading it had in the experiment. This may be accomplished by using a strong wire to tie the boom in its contracted position, removing it from the system, and finding its balance point. If there is time, investigate other situations with different load, boom angle, etc. Try a
case where the upper cable is
This is similar to the ladder model, and may be analyzed in the same way. The difference lies only in the directions of the forces, and in the fact that this bar is under stretch tension rather than compressional tension. The set-up is inherently stable and therefore easier to set up. In Fig. 11 the forces are labeled with letters corresponding to those used in the other two kinds of apparatus. This will allow us to describe an analysis method suitable for all three. W represents the weight of the bar (which you must measure) acting at the bar's center of mass (which you will also determine separately). Be sure L is vertical and H and H are horizontal. Keep the load, V within a range from 1 to 5 times the mass of the bar.
For brevity we will refer to the system's rigid element (boom or ladder) as the
(1) The system must be disassembled to remove the bar and measure its mass and center of mass. This can be done before or after the remaining steps. If you are using the crane boom this is best left until last, since the center of mass must be determined when the boom is exactly the same compressed length as in the main part of the experiment. The bar's mass may be measured with a beam balance of sufficient capacity. The center of mass may be determined, as in experiment S-2A, by suspending the bar from spring balances at each end. (2) The load, (3) The force, (4) In the ladder model top end of the boom.
(5) Accurately measure the tilt angle, θ, of the bar. Usually the best way is to measure the lengths of the legs of a right triangle having the bar as hypotenuse; then use the tangent relation from trig. The length of the bar must be measured also.
We will assume the principles applicable to a body in equilibrium:
Our object will be to see how well the experimental data agree with these principles. You will do this by making certain calculations, using these principles, and using only data on L, W, H, θ, a (from center of mass measurement), C and B. (For the boom, C = B). You will then see how consistent are the results from these values with your other measurements. In each calculation, write the error equation. See whether the discrepancies between calculated and measured values are consistent with the error analysis. (1) Since H are the only forces with horizontal
components, and they are themselves horizontal, then they must be equal in size and opposite
in direction. If you used the ladder model you measured both of these and can check how
closely they were equal. From here on, we will simply refer to their size as
_{2}H.(2) Consider the vertical components. You should have (3) Choose a torque center at the bottom end if the bar and (4) Choose a torque center at the top end of the bar and (5) Look at at the forces at the very top of the bar. Find the sum of all forces acting
there, Ladder: Suspended bar and boom: Use your (6) Find the angle between (7) (
(8) Find the sum of all forces acting
Use your (9) Find the angle between (10)
But, does or, does Should it? Explain. (11) From equilibrium principles, complete the following expression by filling in the missing term(s).
How well do the data confirm this relation? Text and diagrams © 1997, 2004 by Donald E. Simanek. |