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Spring Force (Hook's Law)

In physics, Hooke's law is an empirical law which states that the force, \(F_{s}\) needed to extend a spring by some distance \(d\) scales linearly with respect to that distance, $$ \vec{F_{s}} = -k\vec{d} $$ where \(k\) is a constant factor characteristic of the spring ((a measure of the spring's stiffness), and \(\vec{d}\) is the displacement of the spring's free end from its position when the spring is in its relaxed state (neither compressed nor extended) also \(d\) is small compared to the total possible deformation of the spring.

    ➤ Hooke's law: the force is proportional to the extension

    ➤ The direction of the restoring force is opposite to that of the displacement.

    ➤ A spring force is thus a variable force: It varies with the displacement of the spring's free end.

Work done by the Spring Force

If an object is attached to the spring's free end, the work \(W_{s}\) done on the object by the spring force when the object is moved from an initial position to a final position \(x\) is given by \[W_{s}= -\frac{1}{2}kx^{2}\]

Work Done by an Applied Force

If a weight block that is attached to a spring is stationary before and after a displacement, then the work done on it by the applied force displacing it is the negative of the work done on it by the spring force. \[W_{a}= -W_{s} \]

Spring Energy

The potential energy stored in the spring, \[U_{s}= \frac{1}{2}kx^{2}\]

    ➤ Work done on the spring is stored in the spring as the Spring Potenial Energy.

SIMPLE HARMONIC MOTION

Understanding simple harmonic motion (SHM)

Any motion that repeats at regular intervals is called periodic motion or harmonic motion. Here we are interested in a periodic motion of simple harmonic motion (SHM) type which can be represented using a sinusoidal function of time \(t\). In simple harmonic motion (SHM), the displacement \(x(t)\) of a particle from its equilibrium position is described by the equation $$ x = x_{m} \cos(\omega t + \phi) $$ in which \(x_{m}\) is the amplitude of the displacement, \((\omega t + \phi)\) is the phase of the motion, and \(\phi\) is the phase constant. The angular frequency \(\omega\) is related to the period and frequency of the motion by $$\omega = \frac{2\pi}{T} = 2\pi f$$ Differentiating \(x(t)\) leads to equations for the particle's SHM velocity and acceleration as functions of time:

velocity $$v = - \omega x_{m} \sin(\omega t + \phi) $$

acceleration $$ a = -\omega^{2} x_{m} \cos(\omega t + \phi) $$

In the velocity function, the positive quantity \(\omega x_{m} \) is the velocity amplitude \(v_{m} \). In the acceleration function, the positive quantity \(\omega^{2} x_{m} \) is the acceleration amplitude \(a_{m} \).

SHM in a Spring

    ➤ In SHM, the net force of the system, is proportional to the displacement and acts in the opposite direction of the displacement.

A particle with mass \(m\) that moves under the influence of a Hooke's law restoring force given by \(F = -kx\) is a linear simple harmonic oscillator with

Angular frequency, $$\omega = \sqrt \frac{k}{m} $$

and Time Period, $$ T = \frac{2\pi}{\omega} = 2\pi\sqrt \frac{m}{k} $$

The mechanical energy of a linear oscillating spring is given by the sum of potential energy and kinetic energy. \[E = U(t) + K (t)\] \[U(t) = \frac{1}{2}kx^{2} = \frac{1}{2}kx_{m}^{2} \cos^{2}(\omega t+\phi) \] \[K(t) = \frac{1}{2}mv^{2} = \frac{1}{2}m\omega^{2}x_{m}^{2}\sin^{2}(\omega t +\phi)\] where the potential energy and kinetic energy of a linear oscillator are functions of time \(t\)

    ➤ Angular frequency \(\omega = \sqrt \frac{k}{m}\) gives spring constant \(k=\omega^{2}m\)

Thus, the mechanical energy in an oscillating spring is \[E = \frac{1}{2} kx_{m}^{2} \]

    ➤ The mechanical energy of a linear oscillator is constant and independent of time.