simple pendulum

Examples of simple pendulum in the following topics:

• The Simple Pendulum

  • For small displacements, a pendulum is a simple harmonic oscillator.
  • For angles less than about 15º, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.
  • For the simple pendulum:
  • or the period of a simple pendulum.
  • Even simple pendulum clocks can be finely adjusted and accurate.

• The Physical Pendulum

  • Recall that a simple pendulum consists of a mass suspended from a massless string or rod on a frictionless pivot.
  • This is of the same form as the conventional simple pendulum and this gives a period of:
  • As with a simple pendulum, a physical pendulum can be used to measure g.
  • A brief introduction to pendulums (both ideal and physical) for calculus-based physics students from the standpoint of simple harmonic motion.
  • This is another example of a physical pendulum.

• Second-Order Linear Equations

  • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.

• Projecting Vectors Onto Other Vectors

  • We did this, in effect, when we computed the tangential force of gravity on a simple pendulum.

• Energy in a Simple Harmonic Oscillator

  • The total energy in a simple harmonic oscillator is the constant sum of the potential and kinetic energies.
  • To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have.
  • For example, for a simple pendulum we replace the velocity with v=Lω, the spring constant with k=mg/L, and the displacement term with x=Lθ.
  • If we start our simple harmonic motion with zero velocity and maximum displacement (x=X), then the total energy is:
  • A similar calculation for the simple pendulum produces a similar result, namely:

• Simple Harmonic Motion

  • Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.
  • In addition, other phenomena can be approximated by simple harmonic motion, such as the motion of a simple pendulum, or molecular vibration.
  • A system that follows simple harmonic motion is known as a simple harmonic oscillator.
  • A brief introduction to simple harmonic motion for calculus-based physics students.
  • Relate the restoring force and the displacement during the simple harmonic motion

• Introduction to Simple Harmonic Motion

  • You will remember from your elementary physics courses that if you want to know the electric field produced by a collection of point charges, you can figure this out by adding the field produced by each charge individually (my treatment of elementary simple harmonic motion is standard in most introductory physics textbooks.
  • For instance, the motion of a plane pendulum of length $\ell$ (Figure 1.1) is governed by
  • So for small displacements, the equation for the pendulum is:
  • Thus the equation of motion for the pendulum is linear in $\theta$ when $\theta$ is small.

• Energy Transformations

  • For example, imagine a pendulum in a vacuum.
  • However, when the pendulum is at its lowest point, all of its energy exists in the form of kinetic energy.
  • This animation shows the velocity and acceleration vectors for a pendulum.
  • One may note that at the maximum height of the pendulum's mass, the velocity is zero.
  • This corresponds to zero kinetic energy and thus all of the energy of the pendulum is in the form of potential energy.

• Applications of Second-Order Differential Equations

  • The system under consideration could be an object attached to a spring, a pendulum, etc.
  • If $F$ is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.

• Time

  • For example, the movement of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart have all been used as a standard for time keeping.