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Introductory Mechanics Problems Laboratory
Sears Tower
Goals: Identify steps in a problem. Use reasonable guesses to make approximations. Use unit conversion to get an answer.
PROBLEM The Sears Tower in Chicago is the tallest building in North America. As a scientist riding with a tour group you are asked if you know how many people work and visit there each day. Another tourist asks about how many liters of water are used each minute by the Tower during the day.
You don’t have a web site handy, but you know a couple of facts. The tour guide tells you that the building is 110 stories high and sits on one city block (1/16 mile square). From the bus you see this view of the Tower: (picture from Trizec Properties).
2 Sears Tower
PROBLEM SKILLS There are two questions directly posed by the problem. The first is estimating the num- ber of workers in the building. The second is estimating the amount of water used by those workers.
In general a step represents a simple mathematical expression. This can be a simple addition, subtraction, multiplication or division. Sometimes a step will correspond to a single formula. In estimation problems a step probably includes rounding or order of magnitude approximations.
How do we break down steps? Usually there is a formula that we think applies to the problem. That formula will have some number of variables. If you look at the formula you should identify which variables are already known and which are unknown. If there is more than one unknown variable in the formula, you will need to add a preceding step to find a value for one unknown variable.
When all but one of the variables in a step are known, the step can be completed. You may have to rearrange the formula to solve for the remaining unknown variable. After finding the unknown value, you should check the precision of the result. You should use the same number of significant figures as for the least accurate value that goes into the formula.
BACKGROUND INFORMATION
In analyzing a question about a building you probably need to know the dimensions of the building - length, width, and height. One mile is 5280 feet or 1760 yards. A yard can be useful because when approximating it is close to one meter, 0.914 m to be more exact. The typical height between floors in a commercial builing is about 12 feet or about 4 meters.
Inside the building one can ask about the volume or the floor area. If a building is per- fectly rectangular then the volume is length times width times height. The floor area is equal to the area of one floor (length times width) times the number of floors. For non- rectangular buildings that are smaller at the top, one has to estimate how large the aver- age floor is.
Here is a typical floor plan from the Sears Tower. Notice the amount of space used for
elevators and rest rooms. There must also be space for halls, storage, and conference rooms. It’s not all office space.
