Math Problem Statement

If the system is used to measure the temperature of a flowing gas the varies sinusoidally from 20°C to 100° with a period of 30s: a. Find the average value of the temperature waveform. b. Find the amplitude of the sinusoidal part of the temperature waveform. c. Find the angular frequency of the sinusoidal part of the temperature waveform. d. Find the average value of the system output voltage waveform. e. Find the amplitude of the sinusoidal part of the system output voltage waveform

Solution

Let's address each part of the problem step by step.

Given:

  • The temperature T(t)T(t) varies sinusoidally from 20°C to 100°C.
  • The period TT of the sinusoidal waveform is 30 seconds.

(a) Find the average value of the temperature waveform.

The temperature varies between 20°C and 100°C, so it can be described by the equation:

T(t)=Tavg+Asin(ωt)T(t) = T_{\text{avg}} + A \cdot \sin(\omega t)

Where:

  • TavgT_{\text{avg}} is the average temperature.
  • AA is the amplitude of the sinusoidal part.
  • ω\omega is the angular frequency.

The average value TavgT_{\text{avg}} is simply the midpoint between the minimum and maximum temperatures:

Tavg=Tmax+Tmin2=100C+20C2=60CT_{\text{avg}} = \frac{T_{\text{max}} + T_{\text{min}}}{2} = \frac{100^\circ C + 20^\circ C}{2} = 60^\circ C

So, the average temperature is 60°C.

(b) Find the amplitude of the sinusoidal part of the temperature waveform.

The amplitude AA is the distance from the average value to the maximum (or minimum) value:

A=TmaxTavg1=100C60C=40CA = \frac{T_{\text{max}} - T_{\text{avg}}}{1} = 100^\circ C - 60^\circ C = 40^\circ C

So, the amplitude of the sinusoidal temperature waveform is 40°C.

(c) Find the angular frequency of the sinusoidal part of the temperature waveform.

The period TT of the sinusoidal waveform is 30 seconds. The angular frequency ω\omega is related to the period by:

ω=2πT=2π30s=π15rad/s\omega = \frac{2\pi}{T} = \frac{2\pi}{30 \, \text{s}} = \frac{\pi}{15} \, \text{rad/s}

So, the angular frequency is π15\frac{\pi}{15} rad/s.

(d) Find the average value of the system output voltage waveform.

Assume the system output voltage V(t)V(t) is directly proportional to the temperature waveform T(t)T(t), i.e.,

V(t)=kT(t)V(t) = k \cdot T(t)

Where kk is a proportionality constant. The average value of the system output voltage VavgV_{\text{avg}} would be:

Vavg=kTavg=k60CV_{\text{avg}} = k \cdot T_{\text{avg}} = k \cdot 60^\circ C

Thus, the average value of the system output voltage is 60k60k (in units consistent with the proportionality constant kk).

(e) Find the amplitude of the sinusoidal part of the system output voltage waveform.

The amplitude of the sinusoidal part of the system output voltage VampV_{\text{amp}} is directly proportional to the amplitude of the temperature waveform:

Vamp=kA=k40CV_{\text{amp}} = k \cdot A = k \cdot 40^\circ C

So, the amplitude of the sinusoidal part of the system output voltage is 40k40k.

Summary:

  • (a) Average temperature: 60°C
  • (b) Amplitude of temperature waveform: 40°C
  • (c) Angular frequency: π15\frac{\pi}{15} rad/s
  • (d) Average system output voltage: 60k60k
  • (e) Amplitude of system output voltage: 40k40k

Let me know if you need any further explanation!

Questions for Further Exploration:

  1. How would the output voltage waveform change if the temperature varied in a non-sinusoidal manner?
  2. What role does the proportionality constant kk play in the system output?
  3. If the period of the sinusoidal temperature variation were doubled, how would this affect the angular frequency?
  4. How could you determine the phase shift of the sinusoidal temperature waveform if additional information were provided?
  5. What would be the impact on the system output voltage if the amplitude of the temperature variation increased?

Tip: Understanding how sinusoidal signals are transformed when passing through a system is fundamental in control systems and signal processing. Always ensure you correctly identify the amplitude, frequency, and phase of the signal.

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Math Problem Analysis

Mathematical Concepts

Sinusoidal Functions
Average Value
Amplitude
Angular Frequency
Proportionality

Formulas

Average Value formula
Amplitude formula
Angular Frequency formula
Proportionality formula

Theorems

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Suitable Grade Level

Advanced High School / College