Презентация на тему Oscillatory motion. The simple pendulum. (Lecture 1)

pendulum.
Damped harmonic oscillations.
Driven harmonic oscillations.

to a spring moving on a frictionless surface.
(a)

When the block is displaced to the right of equilibrium (x > 0), the force exerted by the spring acts to the left.
(b) When the block is at its equilibrium position (x = 0), the force exerted by the spring is zero.
(c) When the block is displaced to the left of equilibrium (x < 0), the force exerted by the spring acts to the right.
So the force acts opposite to displacement.

given by Hook’s law:

Then we can obtain the acceleration:




That

is, the acceleration is proportional to the position of the block, and its direction is opposite the direction of the displacement from equilibrium.

motion whenever its acceleration is proportional to its position

and is oppositely directed to the displacement from equilibrium.

for harmonic motion is:


We can denote angular frequency as:


Then:

Solution for this equation is:

ω=const is the angular frequency of the motion


φ=const

is the phase constant
ωt+φ is the phase of the motion
T=const is the period of oscillations:

f of the oscillations:

body in simple harmonic motion are:

the velocity is 90° out of phase with the

position.

Acceleration vs time
At any specified time the acceleration is 180° out of phase with the position.

spring is massless
Then the kinetic energy of system spring-body

corresponds only to that of the body:

The potential energy in the spring is:

is:





That is, the total mechanical energy of a simple

harmonic oscillator is a constant of the motion and is proportional to the square of the amplitude.

of mass m suspended by a light string of

length L that is fixed at the upper end.
The motion occurs in the vertical plane and is driven by the gravitational force.
When Θ is small, a simple pendulum oscillates in simple harmonic motion about the equilibrium position Θ = 0. The restoring force is -mgsinΘ, the component of the gravitational force tangent to the arc.

depend only on the length of the string and

the acceleration due to gravity.
The simple pendulum can be used as a timekeeper because its period depends only on its length and the local value of g.

fixed axis that does not pass through its center

of mass and the object cannot be approximated as a point mass, we cannot treat the system as a simple pendulum. In this case the system is called a physical pendulum.

law:




The solution is:



The period is

such as friction, retard the motion. Consequently, the mechanical

energy of the system diminishes in time, and the motion is damped. The retarding force can be expressed as R=-bv (b=const is the damping coefficient) and the restoring force of the system is -kx then:

force is small, the oscillatory character of the motion

is preserved but the amplitude decreases in time, with the result that the motion ultimately ceases.

– the natural frequency of the system (the undamped

oscillator):

damped oscillator: when b has critical value bc= 2mω0

. System does not oscillate, just returns to the equilibrium position.
overdamped oscillator: Rmax=bVmax>kA and b/(2m)>ω0 . System does not oscillate, just returns to the equilibrium position.

a damped oscillator under the influence of an external

periodical force F(t)=F0sin(ωt). The second Newton’s law for forced oscillator is:


The solution of this equation is:

the driving force
The amplitude of the oscillator is constant

for a given driving force.
For small damping, the amplitude is large when the frequency of the driving force is near the natural frequency of oscillation, or when ω≈ω0.
The dramatic increase in amplitude near the natural frequency is called resonance, and the natural frequency ω0 is also called the resonance frequency of the system.

is close to the natural frequency of the system:

ω≈ω0. At resonance the amplitude of the driven oscillations is the largest.
In fact, if there were no damping (b = 0), the amplitude would become infinite when ω=ω0. This is not a realistic physical situation, because it corresponds to the spring being stretched to infinite length. A real spring will snap rather than accept an infinite stretch; in other words, some for of damping will ultimately occur, But it does illustrate that, at resonance, the response of a harmonic system to a driving force can be catastrophically large.