Final Exam (due on 12/8/14)
1. The force exerted by a spring is given by
F = −kx
where F is the force, in newtons; x is the displacement of the spring from the equilibrium
position, in centimeters; and k is a spring constant. Prepare an Excel worksheet, including an x-y
graph, showing force as a function of distance for two springs whose spring constants are 0.1
newtons/cm and 0.5 newtons/cm, respectively. Plot both curves on the same graph. For each
spring, consider displacements ranging from 0 to 20 cm.
2. An angular support member within a bridge is subject to a coaxial compressive force. If the
horizontal force component is 120 lbf and the vertical force component is 175 lbf, what angle
does the member make with the horizontal?
3. Four-bar linkages are commonly used to transmit mechanical movement in various ways. A
typical four-bar linkage is shown in Fig.1. Member AD is anchored to its supporting base and
does not move. Member AB rotates about point A, resulting in a reciprocating movement of
members BC and CD. Therefore, the angle θ2 changes in response to the change in θ1.
𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴 cos 𝜃𝜃1 − 𝐶𝐶𝐶𝐶 cos 𝜃𝜃2 + �(𝐵𝐵𝐵𝐵)2 − (𝐶𝐶𝐷𝐷 sin 𝜃𝜃2 − 𝐴𝐴𝐴𝐴 sin 𝜃𝜃1 )2
Suppose AB = 1.5 in, BC = 6 in, CD = 3.5 in, and AD = 5 in.
(a) Determine the value of θ2 corresponding to θ1 = 45°.
(b) By plotting several additional values of θ2 against θ1, determine the maximum value of θ2 for
this configuration.
Figure 1
4. The U.S. Environmental Protection Agency has tested the average fuel efficiency of 24 latemodel cars equipped with V-6 engines and automatic transmissions. The results obtained are
shown in attached file “final_problem4”.
Analyze the data in Excel file “final_problem4” as follows:
(a) Determine the mean, median, mode, min, max, and standard deviation.
(b) Construct a histogram, based upon a reasonable interval width.
(c) Construct a cumulative distribution. Show the cumulative distribution in a separate x-y graph.
(d) Explain the difference between the mean and the median.
(e) From an examination of the histogram, what would you conclude about the data?
(f) If a car of this type is chosen at random, what is the likelihood that the fuel efficiency of this
car will not exceed 20 mpg? What is the likelihood that the fuel efficiency will not exceed 22
mpg? What is the likelihood that the fuel efficiency will exceed 25 mpg?
5. Here is a common problem in consumer economics: Suppose you deposit P dollars in a bank
account, and allow the money to remain in the bank for n years. If the bank pays interest at an
annual interest rate i (expressed as a decimal), how much money will accumulate after n years?
The amount of money accumulated will vary, depending on the frequency of compounding. The
following equations apply:
Annual compounding:
Quarterly compounding:
F = P(1 + 𝑖𝑖)𝑛𝑛
Monthly compounding:
F = P(1 + 𝑖𝑖�4)4𝑛𝑛
Daily compounding:
F = P(1 + 𝑖𝑖�12)12𝑛𝑛
F = P(1 + 𝑖𝑖�365)365𝑛𝑛
In each of these equations, n is always the number of years, and i is the annual (yearly) interest
rate, expressed as a decimal.
Construct an Excel worksheet that will calculate a value for F, based upon the following
information entered directly into the worksheet by the user: P, i, n, and the single letter A, Q, M,
or D (indicating annual, quarterly, monthly, or daily compounding). Use an Excel formula
containing nested IF functions to obtain the correct equation, based upon the letter that
indicates the frequency of compounding.
Build the worksheet two different ways:
(a) Assuming that the frequency of compounding will always be specified by a valid uppercase
letter (A, Q, M, or D).
(b) Assuming that the frequency of compounding may be either an uppercase or a lowercase
letter, which may or may not be valid (i.e., the frequency of compounding may be some letter
other than A, Q, M, or D). Display the message “Data Error” if the letter indicating the frequency
of compounding is invalid.
Label all of the information clearly.
As a test case, determine the value of F when P = $5000, i = 0.05 (corresponding to an
interest rate of 5 percent per year), and n = 20 years. Solve for F using annual, quarterly,
monthly, and daily compounding. Does the frequency of compounding have a significant
influence on the result?