A shock absorber is designed to quickly damp out the...

A shock absorber is designed to quickly damp out the oscillations that a car would otherwise make because it is suspended on springs. (a) Find the period of oscillation of a 1610 -kg car that is suspended by springs that make an effective force constant of $5.75 \times 10^{4} \mathrm{N} / \mathrm{m} .$ (b) Find the damping constant $b$ that will reduce the amplitude of oscillations of this car by a factor of 5.00 within a time equal to half the period of oscillation.

A shock absorber is designed to quickly damp out the oscillations that a car would otherwise make because it is suspended
on springs. (a) Find the period of oscillation of a 1610 -kg car that
is suspended by springs that make an effective force constant of
$5.75 \times 10^{4} \mathrm{N} / \mathrm{m} .$ (b) Find the damping constant $b$ that will reduce
the amplitude of oscillations of this car by a factor of 5.00 within a
time equal to half the period of oscillation.

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion describes a type of periodic oscillation where the restoring force on an object is directly proportional to its displacement from an equilibrium position. The period of such oscillations can be calculated for a mass-spring system using the formula T = 2??(m/k), where m is the mass and k is the effective spring constant.

Damped Oscillations

Damped oscillations occur when a resistive force, often proportional to the velocity of the oscillating object, acts to gradually reduce the amplitude of the oscillation over time. This damping force introduces energy loss into the system, causing the oscillations to decay exponentially rather than persisting indefinitely.

Exponential Decay in Oscillatory Systems

The exponential decay of amplitude in a damped oscillatory system is characterized by a decay rate that depends on the damping constant and the mass of the system. This decay can be modeled by an exponential function, where the amplitude decreases by a constant factor over a given time interval. Understanding this decay is crucial for designing systems like shock absorbers, where rapid reduction of oscillatory amplitude is desired.

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