2. The following initial value problem describes oscillatory motion in a viscous medium: dy dy + 26- + w?y = 0; y(0) = 0, y'(0) = vo dt (2) dt2 where b, w, and vo are real, positive, constants. a) If the dimensions of y and t are length (L) and time (T), respectively, what are the dimensions of w, b, and vo? Express your answer in terms of products LªTº, where a and b are rational numbers. b) Show that y(t) = A cos(wt)+B sin(wt) is a solution to (2) when b = 0. What are the values of A and B?
2. The following initial value problem describes oscillatory motion in a viscous medium: dy dy + 26- + w?y = 0; y(0) = 0, y'(0) = vo dt (2) dt2 where b, w, and vo are real, positive, constants. a) If the dimensions of y and t are length (L) and time (T), respectively, what are the dimensions of w, b, and vo? Express your answer in terms of products LªTº, where a and b are rational numbers. b) Show that y(t) = A cos(wt)+B sin(wt) is a solution to (2) when b = 0. What are the values of A and B?
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![### Problem 2
The following initial value problem describes oscillatory motion in a viscous medium:
\[
\frac{d^2y}{dt^2} + 2b \frac{dy}{dt} + \omega^2 y = 0; \quad y(0) = 0, \quad y'(0) = v_0
\]
where \(b\), \(\omega\), and \(v_0\) are real, positive, constants.
### Part (a)
If the dimensions of \(y\) and \(t\) are length (\(L\)) and time (\(T\)), respectively, what are the dimensions of \(\omega\), \(b\), and \(v_0\)? Express your answer in terms of products \(L^a T^b\), where \(a\) and \(b\) are rational numbers.
### Part (b)
Show that \(y(t) = A \cos(\omega t) + B \sin(\omega t)\) is a solution to (2) when \(b = 0\). What are the values of \(A\) and \(B\)?
### Part (c)
Starting with \(y(t) = e^{\lambda t}\) (where \(\lambda\) is, in general, a complex number), find the solution to (2). You must show all the steps leading to the solution.
### Part (d)
What are the values of \(\omega\) and \(b\), if the period of oscillation in the viscous medium is \(5 \, s\) and in the absence of the viscous medium (\(b = 0\)) is \(3 \, s\)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f6272ad-1a25-4a67-b389-4d6bf92e8dd2%2F3507f6da-1d09-47f9-9925-825e710f16be%2Fcnjdvfb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2
The following initial value problem describes oscillatory motion in a viscous medium:
\[
\frac{d^2y}{dt^2} + 2b \frac{dy}{dt} + \omega^2 y = 0; \quad y(0) = 0, \quad y'(0) = v_0
\]
where \(b\), \(\omega\), and \(v_0\) are real, positive, constants.
### Part (a)
If the dimensions of \(y\) and \(t\) are length (\(L\)) and time (\(T\)), respectively, what are the dimensions of \(\omega\), \(b\), and \(v_0\)? Express your answer in terms of products \(L^a T^b\), where \(a\) and \(b\) are rational numbers.
### Part (b)
Show that \(y(t) = A \cos(\omega t) + B \sin(\omega t)\) is a solution to (2) when \(b = 0\). What are the values of \(A\) and \(B\)?
### Part (c)
Starting with \(y(t) = e^{\lambda t}\) (where \(\lambda\) is, in general, a complex number), find the solution to (2). You must show all the steps leading to the solution.
### Part (d)
What are the values of \(\omega\) and \(b\), if the period of oscillation in the viscous medium is \(5 \, s\) and in the absence of the viscous medium (\(b = 0\)) is \(3 \, s\)?
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