A 200 g block attached to a horizontal spring is oscillating...

A $200 \mathrm{g}$ block attached to a horizontal spring is oscillating with an amplitude of $2.0 \mathrm{cm}$ and a frequency of $2.0 \mathrm{Hz}$. Just as it passes through the equilibrium point, moving to the right, a sharp blow directed to the left exerts a $20 \mathrm{N}$ force for $1.0 \mathrm{ms}$. What are the new (a) frequency and (b) amplitude?

Key Concepts

Amplitude in Oscillatory Motion

Amplitude is the maximum displacement from the equilibrium position in an oscillatory system and is closely related to the energy contained in the motion. When an external impulse alters the momentum of the system, it can change the amplitude of the oscillation, reflecting a change in the total energy of the oscillating mass. The relationship between the kinetic energy at equilibrium and the potential energy at maximum displacement helps determine the new amplitude after such perturbations.

Mass-Spring System Frequency

The frequency of oscillation in a mass-spring system is determined by the physical properties of the system—namely, the mass attached to the spring and the spring’s stiffness (spring constant). The mathematical relationship is given by the formula f = (1/2?)?(k/m), indicating that unless there is a change in these intrinsic properties, the frequency remains constant regardless of external impulses.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes the type of periodic oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. It is characterized by parameters such as amplitude, frequency, and phase, and is commonly observed in systems like mass-spring oscillators and pendulums. The analysis of SHM often involves understanding how these parameters interact without external dissipation.

Impulse and Momentum Change

Impulse refers to the effect of applying a force over a short period of time and is defined as the product of force and the duration of its application. It results in a change in momentum of the object. In oscillatory systems, an impulse can alter the velocity of the oscillating mass, thereby affecting the energy of the system, though not necessarily its frequency if the system’s inherent mechanical properties remain unchanged.

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