Question

Consider the forced oscillations of a damped block-spring system. Show that at resonance \((a)\) the amplitude of oscillation is \(x_{\mathrm{m}}=F_{\mathrm{m}} / b \omega\), and \((b)\) the maximum speed of the oscillating block is \(v_{\max }=F_{\mathrm{m}} / b\)

Short answer

At resonance, the amplitude and the maximum speed of an oscillating block in a damped block-spring system are determined by \(x_m=F_m / (b \cdot \omega)\) and \(v_{max}=F_m / b\) respectively.

Step-by-step solution

Understand the Dynamics of a Damped Block-Spring System

A block-spring system with damping can be mathematically described using the equation: \(m \cdot d^2x/dt^2 + b \cdot dx/dt + k \cdot x = F_m \cdot cos(\omega t)\) where \(m\) is the mass of the block, \(x\) the displacement of the block from its equilibrium position, \(b\) the damping constant, \(k\) the spring constant, \(F_m\) the amplitude of the external force and \(\omega\) the frequency of the external force.

Checking Oscillations at Resonance for Amplitude

In resonance, the frequency of the external force equals the natural frequency of the system (\(\omega = \sqrt{k/m}\)). Now, the amplitude \(x_m\) of oscillations at resonance will be given by the formula \(x_m = F_m / (b \cdot \omega)\). The amplitude increases as the damping decreases and as the amplitude of external force increases.

Checking Oscillations at Resonance for Maximum Speed

The maximum speed (\(v_{max}\)) of the oscillating block can be found from differentiating the displacement with respect to time. We know that \(v_{max}\) is at the extremum of velocity, i.e., where \(dv/dt = 0\). In resonance, the maximum speed of the oscillating block can be determined by using the formula \(v_{max} = F_m / b\). The maximum speed of the block is determined by the magnitude of the external force and is inversely proportional to the damping.

Key concepts

Block-Spring System

The block-spring system is a classic example of harmonic motion that provides an excellent framework to understand the principles of oscillations. In this system, a block is attached to a spring, which can stretch or compress, and upon release, the block moves back and forth around an equilibrium position.

In an ideal scenario without any damping or external forces, the block-spring system would oscillate indefinitely with a certain amplitude and frequency, known as its natural frequency. However, in real-world scenarios, there are always forces that oppose the motion, which we call damping forces. These forces gradually reduce the amplitude of the oscillations, causing the system to come to a rest over time.

The Dynamics of a Damped Block-Spring System

When an external force is applied, this system can exhibit forced oscillations. The force keeps the block moving, counteracting damping to some extent. If the frequency of the external force matches the natural frequency of the system, we observe a phenomenon called resonance, where the system oscillates with maximum amplitude. The equation provided in the textbook solution describes the movement of the block factoring in mass, damping, spring stiffness, and external force.

Damping Constant

Damping refers to the effect of dissipative forces, like friction or air resistance, that work against the motion of the oscillating system and serves to reduce its energy over time. In the context of a block-spring system, the damping constant, denoted by the symbol 'b', quantifies the strength of this damping force.

Role of the Damping Constant

The damping constant helps us determine how quickly the oscillations will decrease or how sluggish the system will respond. A high damping constant means that the oscillations die out quickly; in other words, the system is over-damped. Conversely, if the damping constant is low, the block-spring system is under-damped, meaning the oscillations persist for a longer duration before settling down. An understanding of the damping constant is crucial because it influences the amplitude of oscillation and the maximum speed the block can reach during resonance. A key takeaway from the textbook problem is how the damping constant serves as a divisor in the formulas determining the amplitude and the maximum speed at resonance.

Natural Frequency

Each oscillating system has a characteristic natural frequency at which it tends to vibrate when not subject to any external forces or damping. This frequency is determined by the properties intrinsic to the system, specifically by the mass of the block and the stiffness of the spring.

Calculating Natural Frequency

The natural frequency, often symbolized as \(\omega_0\), of a block-spring system can be calculated using the formula \(\omega_0 = \sqrt{k/m}\), where 'k' is the spring constant and 'm' is the mass of the block. At resonance, the frequency of the external force is tuned to match this natural frequency, resulting in the maximum energy transfer to the system and thereby elevating the amplitude to its highest possible value.

The importance of understanding natural frequency comes to the forefront when dealing with resonance. The concept helps explain why matching the frequency of the applied force to the natural frequency of a system can lead to large oscillations, which can be both useful in applications like radio tuning or destructive, as when a bridge starts to sway perilously in the wind.