What is periodic motion

The running of the batsman between the wickets. The swinging of the branches of the tree. The periodic motion and non-periodic motion differ from each other in their way of existence in nature, that are as follows:.

Periodic Motion. Non-Periodic Motion. Repeated motion. Non-repetitive motion. The occurrence of periodic motion is totally related to the time interval. Time period.

It has a time period. It has a time of motion. Type of motion. Vibratory or oscillatory motion. Displacement of an object. The motion of a girl sitting in the swing. 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. Because a simple harmonic oscillator has no dissipative forces , the other important form of energy is kinetic energy KE.

Conservation of energy for these two forms is:. This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role.

In the case of undamped, simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates.

So for the simple example of an object on a frictionless surface attached to a spring, as shown again see , the motion starts with all of the energy stored in the spring.

As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.

Energy in a Simple Harmonic Oscillator : The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface.

All energy is potential energy. The conservation of energy principle can be used to derive an expression for velocity v. This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each.

The conservation of energy for this system in equation form is thus:. Notice that the maximum velocity depends on three factors. It is directly proportional to amplitude. As you might guess, the greater the maximum displacement, the greater the maximum velocity.

It is also greater for stiffer systems because they exert greater force for the same displacement. This observation is seen in the expression for v max ; it is proportional to the square root of the force constant k. Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of m.

For a given force, objects that have large masses accelerate more slowly. Experience with a simple harmonic oscillator : A known mass is hung from a spring of known spring constant and allowed to oscillate. The time for one oscillation period is measured. This value is compared to a predicted value, based on the mass and spring constant. The solutions to the equations of motion of simple harmonic oscillators are always sinusoidal, i.

If the mass -on-a-spring system discussed in previous sections were to be constructed and its motion were measured accurately, its x — t graph would be a near-perfect sine-wave shape, as shown in. It may not be surprising that it is a wiggle of this general sort, but why is it a specific mathematically perfect shape?

Why is it not a sawtooth shape, like in 2 ; or some other shape, like in 3? It is notable that a vast number of apparently unrelated vibrating systems show the same mathematical feature. A tuning fork, a sapling pulled to one side and released, a car bouncing on its shock absorbers, all these systems will exhibit sine-wave motion under one condition: the amplitude of the motion must be small.

Sinusoidal and Non-Sinusoidal Vibrations : Only the top graph is sinusoidal. The others vary with constant amplitude and period, but do no describe simple harmonic motion. As touched on in previous sections, there exists a second order differential equation that relates acceleration and displacement.

When this general equation is solved for the position, velocity and acceleration as a function of time:. These are all sinusoidal solutions. Consider a mass on a spring that has a small pen inside running across a moving strip of paper as it bounces, recording its movements. Mass on Spring Producing Sine Wave : The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.

The above equations can be rewritten in a form applicable to the variables for the mass on spring system in the figure. Recall that the projection of uniform circular motion can be described in terms of a simple harmonic oscillator.

Uniform circular motion is therefore also sinusoidal, as you can see from. Sinusoidal Nature of Uniform Circular Motion : The position of the projection of uniform circular motion performs simple harmonic motion, as this wavelike graph of x versus t indicates.

The equations discussed for the components of the total energy of simple harmonic oscillators may be combined with the sinusoidal solutions for x t , v t , and a t to model the changes in kinetic and potential energy in simple harmonic motion. Privacy Policy. Skip to main content. Waves and Vibrations. Search for:. Periodic Motion. Period and Frequency The period is the duration of one cycle in a repeating event, while the frequency is the number of cycles per unit time.

Learning Objectives Practice converting between frequency and period. Key Takeaways Key Points Motion that repeats itself regularly is called periodic motion. The duration of each cycle is the period. The frequency refers to the number of cycles completed in an interval of time.

Key Terms period : The duration of one cycle in a repeating event. Learning Objectives Identify parameters necessary to calculate the period and frequency of an oscillating mass on the end of an ideal spring. Key Takeaways Key Points If an object is vibrating to the right and left, then it must have a leftward force on it when it is on the right side, and a rightward force when it is on the left side. The restoring force causes an oscillating object to move back toward its stable equilibrium position, where the net force on it is zero.

If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system amplitude : The maximum absolute value of some quantity that varies.

Simple Harmonic Motion Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. Learning Objectives Relate the restoring force and the displacement during the simple harmonic motion. Any system that obeys simple harmonic motion is known as a simple harmonic oscillator.

Simple Harmonic Motion and Uniform Circular Motion Simple harmonic motion is produced by the projection of uniform circular motion onto one of the axes in the x-y plane.

Learning Objectives Describe relationship between the simple harmonic motion and uniform circular motion. Key Takeaways Key Points Uniform circular motion describes the movement of an object traveling a circular path with constant speed. The one-dimensional projection of this motion can be described as simple harmonic motion.

In uniform circular motion, the velocity vector v is always tangent to the circular path and constant in magnitude. The acceleration is constant in magnitude and points to the center of the circular path, perpendicular to the velocity vector at every instant.

Key Terms centripetal acceleration : Acceleration that makes a body follow a curved path: it is always perpendicular to the velocity of a body and directed towards the center of curvature of the path.

The Simple Pendulum A simple pendulum acts like a harmonic oscillator with a period dependent only on L and g for sufficiently small amplitudes. Learning Objectives Identify parameters that affect the period of a simple pendulum. Key Takeaways Key Points A simple pendulum is defined as an object that has a small mass, also known as the pendulum bob, which is suspended from a wire or string of negligible mass.

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