Hooke's law is an empirical physical law describing the linear relationship between the restorative force exerted by a spring and the distance by which the spring is displaced from its equilibrium length. A spring which obeys Hooke's law is said to be Hookean. In addition to springs, Hooke's law is often a good model for arbitrary physical systems that exhibit a tendency to return to a state of equilibrium quickly after perturbation.
If a Hookean spring is compressed or extended by some displacement from equilibrium, the spring will exert a force proportional to this displacement in the opposite direction:
The proportionality constant , called the spring constant, is dependent on the stiffness of the spring, which in turn depends on its shape and the material from which the spring is made.
How can Hooke's law be used to determine the mass of an object?
Given a Hookean spring of spring constant , fix one end of the spring to the ceiling and the other end to the object. In equilibrium, the spring force will balance the downward force of gravity on the object, which allows computation of the mass from the displacement of the spring.
Gravity exerts a force downward proportional to the mass of the object, which is perfectly matched by the spring force in equilibrium, where the negative sign indicates that the spring force acts in the opposite direction. The mass is obtained by setting these forces equal:
The linear relationship of Hooke's law empirically holds only for small displacements . For large deformations, a spring or other Hookean material can be permanently distorted and exhibit a nonlinear restorative force.
In one dimension, the potential energy of a spring can be obtained from Hooke's law by integration:
If a spring is compressed or extended and then released, it will oscillate (indefinitely if friction is neglected; otherwise, the spring will eventually return to equilibrium). The oscillation of a mass on a spring can be derived directly from Newton's second law of motion, . Since acceleration is the second derivative of velocity, setting the force equal to the spring force yields
So the equation of motion of the mass on the spring is
where dots indicate time derivatives. The coefficient is often denoted as , the square of the natural frequency of oscillation of the spring. This is because the general solution to this equation of motion is
for some constants and depending on initial conditions.
The equation of motion above is often called the equation of motion of the simple harmonic oscillator, and systems obeying a similar equation of motion to the above are therefore said to exhibit simple harmonic motion.
A mass suspended by two springs each of spring constant as in the diagram, is compressed upwards a displacement from the equilibrium length of the springs and allowed to fall under the influence of gravity. Find the subsequent displacement as a function of time.
The mass obeys the equation of motion of the simple harmonic oscillator where the coefficient is replaced by since there are two spring forces acting directly on the mass. The displacement therefore obeys the equation
for constants and to be fixed by initial conditions. These initial conditions are
Plugging into the general solution above, these initial conditions yield
The solution is thus uniquely fixed to be
Show that a circuit with an inductor and capacitor, called an LC circuit, obeys the simple harmonic oscillator equation.
According to Kirchoff's laws, the voltage around a closed loop must sum to zero. The voltage across the capacitor is where is the capacitance and is the charge stored on the capacitor. The voltage across the inductor is where is the current in the circuit and is the inductance. Since the current is the time rate of change of the charge on the capacitor, Kirchhoff's loop rule reduces to
This is the simple harmonic oscillator equation where the inductance plays the role of the mass and the inverse of the capacitance plays the role of the spring constant. The linear voltage response with increasing charge stored on the capacitor is analogous to Hooke's law, with the rate of change of current through the circuit analogous to the acceleration.
The solution to the equations of motion above, in terms of cosines and sines, will continue to oscillate forever. However, real oscillators eventually return to a stable equilibrium. This is because real oscillators feel damping forces which remove energy from the system.
Almost any physical system with a stable equilibrium state is described well by Hooke's law when it is displaced slightly from equilibrium. To see why, it is useful to consider potential energy diagrams, which graphically display the potential energy of a system in different states.
The above diagram plots in blue the potential energy of some system as a function of location . A state of sufficiently low total energy in this system will not have enough kinetic energy to escape the potential well around the minimum of . The minimum of is thus an equilibrium state. Futhermore, it is stable: if the system starts at and is perturbed slightly, it will tend to exert a restorative force towards .
As a result, Taylor expanding the potential energy about the minimum yields
The first term, , is just a constant energy shift. The quadratic term is the lowest-order term that describes how changes with . This quadratic potential exactly matches the form of the spring potential . Therefore, near the equilibrium point , the behavior of physical systems can be well approximated by simple harmonic motion, i.e. a force obeying Hooke's law. Above, the quadratic approximation to a potential at its minimum is plotted in red.
For a system with multiple interacting masses, it is useful to define the reduced mass by
The frequency of oscillation about the minimum of a potential is then, in analogy to the Hookean spring,
The Lennard-Jones 6,12 potential is used to model the interactions between two neutral atoms. The potential contains an attractive term for the van der Waals interaction and a repulsive term that models the exchange force due to the Pauli exclusion principle:
where controls the depth of the potential well and is the minimum of the potential well. Find the frequency of oscillation about equilibrium for two atoms of mass whose interactions are modeled by this potential.
The Taylor expansion of about to second order is
The reduced mass of the system is . The frequency of oscillation about equilibrium is therefore
 D. Klepper and R. Kolenkow, An Introduction to Mechanics. McGraw-Hill, 1973.
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