1. The period T (sec) of a simple harmonic oscillator is given by T = 27 here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg). Frequency f (Hz) of a simple harmonic oscillator f = Circular frequency of oscillations wo (rad/sec) 2. 3. wo = m' here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg) Relationship between circular frequency and period 4. T= here T is the oscillation period (sec), wo is circular frequency (rad/ sec) 5. Law of harmonic motion In general, a particle moving along the x axis exhibits simple harmonic motion when x, the particle's displacement from equilibrium, varies in time according to the relationship x(t) = Acos(wot + Po). here x is the particles displacement (m) at moment of time t (sec); A is maximum displacement from equilibrium or amplitude (m); p = wot + po is phase of oscillatory motion (rad); 4, is initial phase (rad); wo is circular frequancy (rad/sec). Position Amplitude X 27 6. Speed v (m/sec) of a simple harmonic oscillator v(t) = = -Awosin(wgt + Po) = -tmsin(wot + Po). here vm = Aw, is maximum speed, or speed amplitude. 7. Acceleration a (m/sec") of a simple harmonic oscillator a(t) = = -Awfcos(wot + Po) = -amcos(wot + Po). here am = Awf is maximum acceleration or amplidude of acceleration. 8. Kinetic energy Ex (J) of a simple harmonic oscillator = sin° (wot + Po). mu? kA? Ex = Potential energy Ex (J) of a simple harmonic oscillator 9. kx? kA? = cos (wot + Po). Ep = 10. Full energy E (J) of a simple harmonic oscillator kA mužA? E = Eg + Ep = I rad =180" or 1 rad = 180° 90° ) 45° ) 30° 60° 0° v2/2 V3/2 1/2 sin(0) V3/2 VZ/2 cos(8) 1 1/2 V3/3 V3 tg(8) 1 cos (0 + n) = -cose sin(0 + n) = -sine

College Physics

10th Edition

ISBN:9781285737027

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter13: Vibrations And Waves

Section: Chapter Questions

Problem 32P: A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.50 N is...

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A 100 gram mass is attached to a spring and undergoes simple harmonic motion with period of T = 2 sec. If the total energy of the system is E = 5 J, find (a) force constant k , and (b) the amplitude of motion A.

1. The period T (sec) of a simple harmonic oscillator is given by
T = 27
here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg).
Frequency f (Hz) of a simple harmonic oscillator
f =
Circular frequency of oscillations wo (rad/sec)
2.
3.
wo =
m'
here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg)
Relationship between circular frequency and period
4.
T=
here T is the oscillation period (sec), wo is circular frequency (rad/ sec)
5. Law of harmonic motion
In general, a particle moving along the x axis exhibits simple harmonic motion when x, the
particle's displacement from equilibrium, varies in time according to the relationship
x(t) = Acos(wot + Po).
here x is the particles displacement (m) at moment of time t (sec); A is maximum displacement
from equilibrium or amplitude (m); p = wot + po is phase of oscillatory motion (rad); 4, is
initial phase (rad); wo is circular frequancy (rad/sec).
Position
Amplitude X
27
6. Speed v (m/sec) of a simple harmonic oscillator
v(t) = = -Awosin(wgt + Po) = -tmsin(wot + Po).
here vm = Aw, is maximum speed, or speed amplitude.
7. Acceleration a (m/sec") of a simple harmonic oscillator
a(t) = = -Awfcos(wot + Po) = -amcos(wot + Po).
here am = Awf is maximum acceleration or amplidude of acceleration.
8. Kinetic energy Ex (J) of a simple harmonic oscillator
= sin° (wot + Po).
mu?
kA?
Ex =
Potential energy Ex (J) of a simple harmonic oscillator
9.
kx?
kA?
= cos (wot + Po).
Ep =
10. Full energy E (J) of a simple harmonic oscillator
kA
mužA?
E = Eg + Ep =

I rad =180" or 1 rad =
180°
90° )
45° )
30°
60°
0°
v2/2
V3/2
1/2
sin(0)
V3/2
VZ/2
cos(8)
1
1/2
V3/3
V3
tg(8)
1
cos (0 + n) = -cose
sin(0 + n) = -sine

Definition Definition Special type of oscillation where the force of restoration is directly proportional to the displacement of the object from its mean or initial position. If an object is in motion such that the acceleration of the object is directly proportional to its displacement (which helps the moving object return to its resting position) then the object is said to undergo a simple harmonic motion. An object undergoing SHM always moves like a wave.

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