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Chapter 3: Problem 13

Output Power from a Motor. The output power produced by a rotating motor is given by the equation $$ P=\tau_{\text {IND }} \omega_{m} $$ where \(\tau_{\text {IND }}\) is the induced torque on the shaft in newton- meters, \(\omega_{m}\) is the rotational speed of the shaft in radians per second, and \(P\) is in watts. Assume that the rotational speed of a particular motor shaft is given by the equation $$ \omega_{m}=188.5\left(1-e^{-02 t}\right) \mathrm{rad} / \mathrm{s} $$ and the induced torque on the shaft is given by $$ T_{\mathrm{IND}}=10 e^{-0.2 t} \mathrm{~N} \cdot \mathrm{m} $$ Plot the torque, speed, and power supplied by this shaft versus time for \(0 \leq t \leq 10 \mathrm{~s}\). Be sure to label your plot properly with the symbols \(\tau_{I N D}\) and \(\omega_{n}\) where appropriate. Create two plots, one with the power displayed on a linear scale, and one with the output power displayed on a logarithmic scale. Time should always be displayed on a linear scale.

Short Answer

Expert verified
The output power P(t) of the rotating motor can be calculated by substituting the given values of induced torque and rotational speed into the equation: \(P(t) = \left(10e^{-0.2t}\right)\left(188.5\left(1-e^{-0.2t}\right)\right)\). To analyze the behavior of this motor, create two separate plots. The first plot displays the induced torque, rotational speed, and output power with respect to time for \(0 \leq t \leq 10\) seconds on a linear scale. The second plot displays the output power with respect to time on a logarithmic scale while keeping the time scale linear.

Step by step solution

01

Calculate Output Power

Recall the equation given for the output power of a rotating motor: \( P = \tau_{IND} \omega_m, \) where, \( \tau_{IND} = 10 e^{-0.2 t} \mathrm{~N} \cdot \mathrm{m} \) \( \omega_m = 188.5\left(1-e^{-0.2 t}\right) \mathrm{rad} / \mathrm{s} \) Now, substitute the formula for the torque and rotational speed into the output power equation. \(P(t) = \left(10e^{-0.2t}\right)\left(188.5\left(1-e^{-0.2t}\right)\right)\)
02

Create Plots of Torque, Speed, and Power with respect to time

We will now proceed to plot the following functions with respect to time for 0 ≤ t ≤ 10 seconds: - Induced torque: \( \tau_{IND}(t) = 10 e^{-0.2 t} \mathrm{~N} \cdot \mathrm{m} \) - Rotational speed: \( \omega_m(t) = 188.5\left(1-e^{-0.2 t}\right) \mathrm{rad} / \mathrm{s} \) - Output power: \( P(t) = \left(10e^{-0.2t}\right)\left(188.5\left(1-e^{-0.2t}\right)\right) \) Please use graphing software or tools like Desmos, MATLAB, or Python with Matplotlib to create the plots. You should label the axes properly with correct symbols: "τ_IND" for induced torque, "ω_m" for rotational speed, and "P" for output power.
03

Plot Output Power on a Linear and Logarithmic Scale

For the first plot, display the output power (P) with respect to time (t) on a linear scale. For the second plot, display the output power (P) with respect to time (t) on a logarithmic scale while keeping the time scale linear. Use the graphing software again, and make sure both plots have the appropriate labels for the axes. After completing these steps, the student should have two separate plots, one with the output power on a linear scale and one with the output power on a logarithmic scale, each with proper labeling and correct functions plotted.

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