Energy of the Simple Harmonic Oscillator

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1 Energy of the Simple Harmonic Oscillator
Kinetic energy of a simple harmonic oscillator The elastic potential energy stored in the spring The total mechanical energy of the simple harmonic oscillator

2 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. Not that U is zero at x zero and K is zero at x max where v=0 We can use the principle of conservation of energy to obtain the velocity for an arbitrary position by expressing the total energy at some arbitrary position x as

3 Example A kg cart connected to a light spring for which the force constant is 20.0 N/m oscillates on a horizontal, frictionless air track. A) Calculate the total energy of the system and the maximum speed of the cart if the amplitude of the motion is 3.00 cm.

4 B) What is the velocity of the cart when the position is 2. 00 cm
B) What is the velocity of the cart when the position is 2.00 cm? The positive and negative signs indicate that the cart could be moving to either the right or the left at this instant. (C) Compute the kinetic and potential energies of the system when the position is 2.00 cm.

5 Example: A block–spring system oscillates with amplitude of 3.50 cm. If the spring constant is 250 N/m and the mass of the block is kg, determine (a) the mechanical energy of the system, (b) the maximum speed of the block and (c) the maximum acceleration. Solution:

6 Example: A 50.0-g object connected to a spring with a force constant of 35.0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.00 cm. Find (a) the total energy of the system and (b) the speed of the object when the position is 1.00 cm. Find (c) the kinetic energy and (d) the potential energy when the position is 3.00 cm. Solution:

7 Wave Motion Their are two main types of waves mechanical waves and electromagnetic waves. Mechanical waves require a source of disturbance and a medium that can be disturbed. Electromagnetic waves do not require a medium to propagate. All waves carry energy, but the amount of energy transmitted through a medium and the mechanism responsible for that transport of energy differ from case to case.

8 Their are two main types of electromagnetic waves Transverse wave and Longitudinal wave
A travelling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave. Longitudinal wave A travelling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called a longitudinal wave.

9 Sinusoidal Waves The wave represented by this curve is called a sinusoidal wave (sine curve) because the curve is the same as that of the function sin θ plotted against θ The sinusoidal wave is the simplest example of a periodic continuous wave and can be used to build more complex waves.

10 The motion of the wave can be represented by the motion of two curves toward the right
The point at which the displacement of the element from its normal position is highest is called the crest of the wave. The wavelength λ of a wave is the distance between adjacent crests or adjacent troughs. The wavelength is the minimum distance between any two identical points (such as the crests) on adjacent waves, as shown in Figure

11 The period of the wave is the same as the period of the simple harmonic oscillation of one element of the medium. The frequency of the wave is the same as the frequency of the simple harmonic oscillation of one element of the medium. Any one-dimensional wave travelling with a speed v in the x direction can be represented by a wave function of the form Where the positive sign applies to a wave travelling in the negative x direction and the negative sign applies to a wave travelling in the positive x direction.

12 The wave function for a one-dimensional sinusoidal wave travelling to the right can be expressed as
Where A is the amplitude, λ is the wavelength, k is the angular wave number, and ω is the angular frequency. If T is the period and f the frequency, v, k, and ω can be written

13 Example For a certain transverse wave, the distance between two successive crests is 1.20 m, and eight crests pass a given point along the direction of travel every 12.0 s. Calculate the wave speed. Solution

14 Example A sinusoidal wave on a string is described by where k = 3
Example A sinusoidal wave on a string is described by where k = 3.10 rad/cm and ω=9.30 rad/s. How far does a wave crest move in 10.0 s? Does it move in the positive or negative x direction? Solution

15 Example A sinusoidal wave is travelling along a rope. The oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along the rope in 10.0 s. What is the wavelength? Solution

16 Example Write the expression for y as a function of x and t for a sinusoidal wave travelling along a rope in the negative x direction with the following characteristics: A=8.00 cm, λ= 80.0 cm, f =3.00 Hz, and y(0, t) = 0 at t =0. Solution

17 Example A wave on a string is described by the wave function y = (0.100 m) sin (0.50x- 20t). (a) Show that a particle in the string at x =2.00 m executes simple harmonic motion.(b) Determine the frequency of oscillation of this particular point. Solution