Mathematical Methods And Statistical Techniques Assignment
Mathematical Methods and Statistical Techniques Assignment
Analyze and apply mathematical and statistical methods to solve engineering problems, interpret data, and validate formulas through dimensional analysis, probability calculations, and data interpretation, using appropriate software tools where necessary.
Paper For Above instruction
The assignment requires a comprehensive analysis of various mathematical and statistical applications in engineering contexts. It involves applying dimensional analysis, interpreting data statistically, calculating probabilities, and visualizing results for non-technical audiences. The core of the task is to demonstrate understanding of the relevance and application of mathematical methods to engineering problems, and to communicate findings clearly and accurately.
Introduction
Mathematics and statistics play a pivotal role in engineering, providing tools to interpret, analyze, and validate physical phenomena and data. This paper addresses multiple scenarios where these methods are essential, focusing on dimensional analysis, probability calculations, and data presentation. The examples span electrical circuits, mechanical systems, signal processing, and data analysis, illustrating the broad utility of mathematical techniques in engineering practice.
Part 1: Application of Mathematical Methods
a) Dimensional Analysis of Power Dissipation in Resistors
The power dissipated in a resistor when subjected to a voltage V is given by the formula: P = V^2 / R. To find the dimensions of R, we analyze the dimensions of each component. Power (P) has dimensions of energy over time, which is [ML^2T^-3], voltage (V) is electric potential with dimensions [ML^2T^-3I^-1], where I stands for electric current, and R is resistance with an unknown dimension. Equating the dimensions on both sides, we get:
[ML^2T^-3] = ([ML^2T^-3I^-1])^2 / [Resistance]
which simplifies to:
[ML^2T^-3] = [M^2L^4T^-6I^-2] / [Resistance]
Thus, the dimensional formula for resistance R becomes:
[R] = [M^1L^2T^-3I^-2]
This matches the known dimensional formula for electrical resistance in SI units, confirming the consistency of the formula from a dimensional perspective.
b) Dimensional Analysis of Vibration Period of a Guitar String
The proposed formula is t = 2π√(μ / F), where μ is mass per unit length and F is tension. According to dimensional analysis, μ has dimensions [M L^-1], and F has dimensions [MLT^-2]. To test if the formula is dimensionally correct:
√(μ / F) has dimensions √([M L^-1] / [M L T^-2]) = √([1 / L^0 T^-2]) = √(T^2) = T
Therefore, the entire formula has dimensions of time, matching the period's dimensionality, indicating the formula is dimensionally consistent. However, if the formula contained incorrect variables or incorrect exponents, the dimensional analysis would reveal inconsistency. The colleague’s proposed formula's inadequacy can be shown by substituting wrong variables and checking for dimensional inconsistency.
c) Summation of Voltage Samples in an ADC
The voltage samples form an arithmetic sequence: 2, 4, 6, ..., 40. To find the sum over these 20 terms:
The sum of an arithmetic series is:
S = n/2 (first term + last term) = 20/2 (2 + 40) = 10 * 42 = 420 mV
This sum provides the total voltage measurement over the samples, useful in assessing signal strength or for further digital processing.
d) Calculation of the 9th Count of a Digital Counter
The sequence doubles each time: 1024, 2048, 4096, 8192.... Recognizing this as a geometric sequence with first term a = 1024 and common ratio r = 2, the nth term is given by:
t_n = a * r^{(n-1)}
For n=9:
t_9 = 1024 2^{8} = 1024 256 = 262,144
This calculation is critical for understanding counting processes in digital electronics, such as timing and addressing in microprocessors.
e) Determining Capacitance in an RC Circuit
The voltage across the capacitor in an RC circuit is described by Vc = Vs(1 - e^{-t/RC}). Given Vc = 2V after t=4 seconds, and Vs=12V, solving for C:
2 = 12(1 - e^{-4/RC})
=> 1/6 = 1 - e^{-4/RC}
=> e^{-4/RC} = 1 - 1/6 = 5/6
Taking natural logarithm:
-4/RC = ln(5/6)
=> C = -4 / (R * ln(5/6))
Assuming R = 1 MΩ (as given), the capacitance C is approximately:
C ≈ -4 / (1,000,000 (-0.182)) ≈ 21.98 10^{-6} F or approximately 22 μF.
This showcases the application of exponential decay models in circuit analysis.
f) Time Calculation for Sine Wave Signal
The signal is v(t) = 6 sin(2π 10^{6} t). To find t when v(t) = +3V:
3 = 6 sin(2π 10^{6} t)
sin(2π 10^{6} t) = 0.5
=> 2π 10^{6} t = π/6 or 5π/6 (since sine is positive in first and second quadrants)
couting for the first positive time:
t = (π/6) / (2π 10^{6}) = 1 / (12 10^{6}) ≈ 8.33 * 10^{-8} seconds
Plotting over two cycles using software such as MATLAB or Python's Matplotlib, one can visualize the wave and mark the first crossing point, improving understanding for non-technical audiences.
g) Hyperbolic Cosine Curve Calculations
Given y = 60 cosh(60 x):
- When x = 10^4 (x=10000): y = 60 cosh(6010000), which is extremely large, indicating a large y-value.
- When y = 180: Solving for x: x = arcosh(y/60) / 60 = arcosh(3) / 60 ≈ 0.0583 / 60 ≈ 0.000972 rad.
This exemplifies how hyperbolic functions model complex physical curves, such as cable shapes or thermal profiles.
h) Formula for Pendulum Period via Dimensional Analysis
The period t depends on mass m, length l, and gravity g. Using dimensional analysis:
t = k l^a g^b * m^c, where k is dimensionless.
Since the period of a simple pendulum depends primarily on length and gravity, and is independent of mass, the dominant relationship is:
t ∝ √(l / g), i.e., t = 2π√(l / g), confirming classical pendulum formula.
i) Speed of Sound in Gas Influenced by Physical Properties
Assuming dependency on pressure p, density r, and g, using dimensional analysis:
u ∝ p^a r^b g^c.
Since the speed of sound depends primarily on the medium's properties, more accurately, u = √(γ * p / r), where γ is the adiabatic index, indicating dependence primarily on pressure and density, with gravity g being negligible in this context.
Part 2: Data Analysis and Probability
a) Statistical Analysis of Transmit Power
The sample powers are: 18.1, 19.2, 18.4, 18.1, 19.9, 18.1, 17.4, 19.1, 18.1, 17.4 dBm.
Calculating the mean:
Mean = (Sum of all readings) / 10 = (18.1 + 19.2 + 18.4 + 18.1 + 19.9 + 18.1 + 17.4 + 19.1 + 18.1 + 17.4) / 10 ≈ 18.38 dBm.
Standard deviation is computed using the formula:
SD = √[ Σ( xi - mean )^2 / (n - 1) ] ≈ 0.85 dBm.
Frequencies of power measurements can be tabulated, revealing the distribution of measured powers, assisting quality control assessment.
b) Probability of Tolerance Violations
The probability that a bolt exceeds the tolerance is 6%, given as 1 - 0.94. For six bolts, the binomial probability distribution applies.
(i) Exactly two bolts exceeding the tolerance:
P = C(6, 2) (0.06)^2 (0.94)^4 ≈ 0.027.
(ii) More than two bolts exceeding the tolerance:
P = 1 - P(0) - P(1) - P(2), which sums the probabilities for 0, 1, 2, and subtracts from 1, yielding approximately 0.094.
c) Normal Distribution of Capacitor Capacitance
Given mean = 100 μF, SD = 7 μF. Using standard normal distribution tables, the proportion of capacitors between 90 and 110 μF corresponds to Z-scores:
Z1 = (90 - 100)/7 ≈ -1.43, Z2 = (110 - 100)/7 ≈ 1.43.
From Z-table, the probability between these Z-scores is approximately 0.852. Therefore, about 852 capacitors out of 400 are expected to fall within this range, indicating most capacitors meet their specifications.
d) Evaluating the Effect of a Fuel Additive Using Z-Tests
Sample mean = 48 mpg, SD = 13 mpg, n=100; initial mean = 44 mpg.
Null hypothesis: no effect; alternative hypothesis: additive influences mpg.
The z-score:
z = (48 - 44) / (13/√100) = 4 / (1.3) ≈ 3.08.
Referring to the z-table, p-value
Conclusion
The application of dimensional analysis, probability theory, and data visualization demonstrates that mathematical methods are integral to engineering analysis. These techniques enable engineers to verify formulas, interpret experimental data, and make informed decisions. Employing software tools enhances clarity and supports communication of complex concepts to diverse audiences.
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