Question

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.

Short answer

The velocity amplitude for forced oscillatory motion is given by \( v_{m}=F_{m} / \left[\left(m \omega''^2-k / \omega''\right)^{2}+b^{2}\right]^{1 / 2} \).

Step-by-step solution

- 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.

- 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.

- 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.

- 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 \( v_{m}=F_{m} / \left[\left(m \omega''^2-k / \omega''\right)^{2}+b^{2}\right]^{1 / 2} \).

Key concepts

Velocity Amplitude

In the study of forced oscillations, velocity amplitude is an essential concept used to describe the maximum speed reached by a mass on a spring during harmonic motion. In the equation for forced oscillations, we commonly see a term such as the velocity amplitude, typically denoted as \( v_m \).
To find the velocity amplitude in a forced harmonic oscillator, we can begin with the equation of motion for a damped and driven oscillator:

• Force: \( F(t) = F_m \cos \omega'' t \)
• Mass: \( m \)
• Damping: \( b \)
• Spring constant: \( k \)
• Angular frequency: \( \omega'' \)

Understanding and calculating the velocity amplitude is important because it affects how energy is transferred in mechanical and analogous electrical systems.
The velocity amplitude \( v_m \) can be expressed in terms of these parameters and the forcing frequency as:\[v_{m}= \frac{F_{m}}{\sqrt{\left(m \omega''^2-\frac{k}{\omega''}\right)^{2}+b^{2}}}\]
This expression helps measure how the system's energy, delivered via force \( F_m \), gets translated into kinetic energy.

Mechanical System

A mechanical system in the context of forced oscillations refers typically to a spring-mass-damper setup. This is used to model phenomena where a mass is subjected to restoring, damping, and forcing forces.
The governing equation for such a system when dealing with forced oscillations is given by:\[m \frac{d^2 x}{dt^2} + b\frac{dx}{dt} + kx = F_m \cos \omega'' t\]
This is a second-order linear differential equation where:

• \( m \) is the mass.
• \( b \) is the damping coefficient, reflecting energy loss often through friction or resistance.
• \( k \) is the spring constant associated with the elasticity of the system.
• \( F_m \) is the amplitude of the external force driving the oscillations.

This equation essentially states how these forces balance each other as the system moves. The solution to this equation, often involving calculus, will provide the system's response in terms of displacement and velocity, crucial for designing stable mechanical systems.

Electrical Circuit Analogy

Electrical circuits can act just like mechanical systems under forced oscillations due to analogous components and forces. In an electrical context, components like resistors, inductors, and capacitors parallel the functioning of mechanical dampers, masses, and springs respectively.
The equation governing the behavior of such an RLC circuit when connected to an alternating voltage supply is:\[L \frac{d^2 q}{dt^2} + R\frac{dq}{dt} + \frac{q}{C} = V_m \cos \omega'' t\]
Here:

• \( R \) is the electrical resistance, analogous to mechanical damping \( b \).
• \( L \) is the inductance, corresponding to the mass \( m \) in the mechanical system.
• \( \frac{1}{C} \) serves as the mechanical stiffness \( k \).
• \( V_m \) equates to the force amplitude \( F_m \) in terms of pressure applied to the system.
• \( q \) (charge) and \( i \) (current) are analogous to displacement \( x \) and velocity \( v \) respectively.

Recognizing these analogies allows for solutions to complex mechanical problems by examining their electrical counterparts and vice versa.

Resonance Quality

Resonance quality, often talked about in systems dealing with oscillations, describes how sharply defined the resonance frequency of a system is. It is denoted by the term quality factor, or \( Q \)-factor, indicating how underdamped a resonant system is.
For mechanical systems, resonance quality is influenced by factors such as:

• The mass \( m \) and its ability to store kinetic energy.
• The damping coefficient \( b \) that causes energy to dissipate.
• Spring constant \( k \) aiding in periodic recovery of energy.

In electrical systems, it's measured similarly but uses the term current amplitude to express the quality of resonance, analogous to velocity amplitude in mechanical systems.
Generally, higher \( Q \) values mean lower energy loss and narrower resonance peaks. This allows for clear distinctions between resonant and non-resonant frequencies, assisting in fine-tuning applications like radio receivers or optical equipment.