Starting from Eq. \(17-43\), find the velocity \(v_{x}(=d x / d t)\) in forced oscillatory motion. Show that the velocity amplitude is $$ v_{\mathrm{m}}=F_{\mathrm{m}} /\left[\left(m \omega^{\prime \prime}-k / \omega^{\prime}\right)^{2}+b^{2}\right]^{1 / 2} $$ The equations of Section \(17-8\) are identical in form with those representing an electrical circuit containing a resistance \(R\), and inductance \(L\), and a capacitance \(C\) in series with an alternating emf \(V=V_{\mathrm{m}} \cos \omega^{\prime \prime} t\). Hence \(b, m, k\), and \(F_{\mathrm{m}}\), are analogous to \(R, L, 1 / C\), and \(V_{\mathrm{m}}\), respectively, and \(x\) and \(v\) are analogous to electric charge \(q\) and current \(i\), respectively. In the electrical case the current amplitude \(i_{\mathrm{m}}\), analogous to the velocity amplitude \(v_{\mathrm{m}}\) above, is used to describe the quality of the resonance.

Step by step solution

01

- Mechanical System Equations

First identify the equation describing the forced harmonic oscillator. This was given like Eq. 17-43. If the motion is forced by a force \( F(t) = F_m \cos \omega'' t \), then the equation of motion can be represented as \( m \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + kx = F_m \cos \omega'' t \). Here \( x \) is the displacement, \( v_x = \frac{dx}{dt} \) is the velocity, \( m \) is the mass, \( b \) is the damping coefficient, \( k \) is the spring constant, \( F_m \) is the amplitude of the forcing function, and \( \omega'' \) is the angular frequency of the forcing function.

02

- Electrical System Equations

Next, identify the similarly formed equation for an electrical circuit containing resistance \( R \), inductance \( L \), and capacitance \( C \) in series, being powered by an alternating emf \( V = V_m \cos \omega'' t \). This equation is represented as \( L \frac{d^2 q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = V_m \cos \omega'' t \). In this case, \( q \) is the electric charge, \( i = \frac{dq}{dt} \) is the current, \( L \) is the inductance, \( R \) is the resistance, \( 1/C \) stands for \( k \) in the mechanical system, and \( V_m \cos \omega'' t \) represents the forcing function.

03

- Analogy of Variables

Now, observe that by analogy, \( b, m, k, F_m \) are analogous to \( R, L, 1/C, V_m \) respectively. Therefore, \( x, v_x \) are analogous to \( q, i \) respectively.

04

- Derive the Expression for Velocity Amplitude

Using the expression of current amplitude for the electrical system, derive the expression for the velocity amplitude in the mechanical system. This can be done by replacing the electrical quantities by the corresponding mechanical quantities in the expression for \( i_m \). This gives \( v_{m}=F_{m} / \left[\left(m \omega''^2-k / \omega''\right)^{2}+b^{2}\right]^{1 / 2} \).

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