Restoring force

Examples of Restoring force in the following topics:

• Period of a Mass on a Spring

  • The deformation of the ruler creates a force in the opposite direction, known as a restoring force.
  • Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero.
  • The simplest oscillations occur when the restoring force is directly proportional to displacement.
  • The force constant k is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system.
  • (c) The restoring force is in the opposite direction.

• The Simple Pendulum

  • This leaves a net restoring force drawing the pendulum back toward the equilibrium position at θ = 0.
  • Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator.
  • Thus, for angles less than about 15º, the restoring force F is
  • For small angles, then, the expression for the restoring force is:
  • Also shown are the forces on the bob, which result in a net force of −mgsinθ toward the equilibrium position—that is, a restoring force.

• Simple Harmonic Motion

  • Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.
  • Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement (i.e., it follows Hooke's Law) .
  • Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law.
  • The net force on the object can be described by Hooke's law, and so the object undergoes simple harmonic motion.
  • Relate the restoring force and the displacement during the simple harmonic motion

• Hooke's Law

  • F is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s2); and
  • As with any other set of forces, the forces of many springs can be combined into one resultant force.
  • There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right).
  • The dotted line shows what the actual (experimental) plot of force might look like.
  • The extension of the spring is linearly proportional to the force.

• Another velocity-dependent force: the Zeeman effect

  • As a classical model for the radiation of light from excited atoms we can consider the electrons executing simple harmonic oscillations about their equilibrium positions under the influence of a restoring force .
  • Remember, the restoring force is just a linear approximation to the Coulomb force and therefore , the "spring constant'', is the first derivative of the Coulomb force evaluated at the equilibrium radius of the electron.
  • Now let's suppose we apply a force that is not spherically symmetric.
  • This results in another force on the electrons of the form $q \dot{\mathbf{r}} \times B\hat{\mathbf{z}}$ (from Lorentz's force law).
  • Adding this force to the harmonic ( $-k \mathbf{r}$ ) force gives

• Elastic Potential Energy

  • That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring.
  • This work is performed by an applied force Fapp.
  • The applied force is exactly opposite to the restoring force (action-reaction), and so $F_{app}=kx$.
  • A graph shows the applied force versus deformation x for a system that can be described by Hooke's law .
  • Another way to determine the work is to note that the force increases linearly from 0 to kx, so that the average force is $\frac{1}{2}kx$, the distance moved is x, and thus

• What is Potential Energy?

  • Potential energy is often associated with restoring forces such as a spring or the force of gravity.
  • The action of stretching the spring or lifting the mass of an object is performed by an external force that works against the force field of the potential.
  • This work is stored in the force field as potential energy.
  • If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application.
  • For example, the work of an elastic force is called elastic potential energy ; work done by the gravitational force is called gravitational potential energy; and work done by the Coulomb force is called electric potential energy.

• Driven Oscillations and Resonance

  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force.
  • In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: $\vec F = -k \vec x$ where k is a positive constant.
  • If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator.
  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
  • In the case of a sinusoidal driving force: $\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t)$, where $\!

• Work Done by a Variable Force

  • Integration is used to calculate the work done by a variable force.
  • A force is said to do work when it acts on a body so that there is a displacement of the point of application in the direction of the force.
  • Thus, a force does work when it results in movement.
  • The work done by a constant force of magnitude F on a point that moves a displacement $\Delta x$ in the direction of the force is simply the product
  • According to the Hooke's law the restoring force (or spring force) of a perfectly elastic spring is proportional to its extension (or compression), but opposite to the direction of extension (or compression).

• Linear Vector Spaces

  • We can add any two, say, force vectors and get another force vector.
  • The total force on the mass is the sum of the gravitational force (which is pointed down) and the restoring force of the spring (which can be either up or down depending on where the mass is relative to its equilibrium position).
  • We can add the individual forces to get the total force because they are vectors.
  • We will get a different 3-tuple depending on whether we use Cartesian or spherical coordinates, for example, but the force vector itself is independent of these considerations.