HARMONIC OCSILLATIONS M Theory 1. The period T (sec) of a simple harmonic oscillator is given by Т% 2л 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 2. f = 3. Circular frequency of oscillations wo (rad/sec) here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg) Relationship between circular frequency and period 4. 2л 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) 3 Асos(0ot + Ф), 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); 4o is initial phase (rad); wo is circular frequancy (rad/sec). Position х X- Amplitude X -X Speed v (m/sec) of a simple harmonic oscillator 6. dx v(t) = = -Awosin(@,t + Po) = -vmSin(@ot + Po), here vm = Awo is maximum speed, or speed amplitude. Acceleration a (m/sec²) of a simple harmonic oscillator d?x 7. = -Awžcos(wot + 4o) = -amcos(wot + Po), here am = Aw is maximum acceleration or amplidude of acceleration. a(t) = dt2 8. Kinetic energy E (J) of a simple harmonic oscillator kA? sin²(wot + Po). ту? 2 9. Potential energy E, (J) of a simple harmonic oscillator kx2 E, = = kA? - cos²(wot + Po). %3D 10. Full energy E (J) of a simple harmonic oscillator тозА? kA² E = Ex + Ep %3D T rad =180° or 1 rad = 180° 30° G 60° () 90° G) 45° ) 0° 1 VZ/2 V3/2 1/2 sin(0) V3/2 V2/2 cos(0) 1/2 1 V3/3 V3 tg(0) cos(0 + n) = -cos® sin(0 + n) = -sin0 3. An object having mass m = 5 gr oscillates with simple harmonic motion along the x axis. Its displacement from the origin varies with time according to the equation x = 4cos (nt +- where t is in seconds and the angles in the parentheses are in radians. (a) Determine kinetic and potential energy of the object at moment of time t = 1 sec.
HARMONIC OCSILLATIONS M Theory 1. The period T (sec) of a simple harmonic oscillator is given by Т% 2л 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 2. f = 3. Circular frequency of oscillations wo (rad/sec) here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg) Relationship between circular frequency and period 4. 2л 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) 3 Асos(0ot + Ф), 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); 4o is initial phase (rad); wo is circular frequancy (rad/sec). Position х X- Amplitude X -X Speed v (m/sec) of a simple harmonic oscillator 6. dx v(t) = = -Awosin(@,t + Po) = -vmSin(@ot + Po), here vm = Awo is maximum speed, or speed amplitude. Acceleration a (m/sec²) of a simple harmonic oscillator d?x 7. = -Awžcos(wot + 4o) = -amcos(wot + Po), here am = Aw is maximum acceleration or amplidude of acceleration. a(t) = dt2 8. Kinetic energy E (J) of a simple harmonic oscillator kA? sin²(wot + Po). ту? 2 9. Potential energy E, (J) of a simple harmonic oscillator kx2 E, = = kA? - cos²(wot + Po). %3D 10. Full energy E (J) of a simple harmonic oscillator тозА? kA² E = Ex + Ep %3D T rad =180° or 1 rad = 180° 30° G 60° () 90° G) 45° ) 0° 1 VZ/2 V3/2 1/2 sin(0) V3/2 V2/2 cos(0) 1/2 1 V3/3 V3 tg(0) cos(0 + n) = -cos® sin(0 + n) = -sin0 3. An object having mass m = 5 gr oscillates with simple harmonic motion along the x axis. Its displacement from the origin varies with time according to the equation x = 4cos (nt +- where t is in seconds and the angles in the parentheses are in radians. (a) Determine kinetic and potential energy of the object at moment of time t = 1 sec.
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Transcribed Image Text:HARMONIC OCSILLATIONS
M Theory
1.
The period T (sec) of a simple harmonic oscillator is given by
Т% 2л
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
2.
f =
3.
Circular frequency of oscillations wo (rad/sec)
here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg)
Relationship between circular frequency and period
4.
2л
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) 3 Асos(0ot + Ф),
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); 4o is
initial phase (rad); wo is circular frequancy (rad/sec).
Position
х
X-
Amplitude X
-X
Speed v (m/sec) of a simple harmonic oscillator
6.
dx
v(t) =
= -Awosin(@,t + Po) = -vmSin(@ot + Po),
here vm = Awo is maximum speed, or speed amplitude.
Acceleration a (m/sec²) of a simple harmonic oscillator
d?x
7.
= -Awžcos(wot + 4o) = -amcos(wot + Po),
here am = Aw is maximum acceleration or amplidude of acceleration.
a(t) =
dt2
8.
Kinetic energy E (J) of a simple harmonic oscillator
kA?
sin²(wot + Po).
ту?
2
9.
Potential energy E, (J) of a simple harmonic oscillator
kx2
E, = =
kA?
- cos²(wot + Po).
%3D
10. Full energy E (J) of a simple harmonic oscillator
тозА?
kA²
E = Ex + Ep
%3D
T rad =180° or 1 rad =
180°
30° G
60° ()
90° G)
45° )
0°
1
VZ/2
V3/2
1/2
sin(0)
V3/2
V2/2
cos(0)
1/2
1
V3/3
V3
tg(0)
cos(0 + n) = -cos®
sin(0 + n) = -sin0
Transcribed Image Text:3. An object having mass m = 5 gr oscillates with simple harmonic motion along the x axis.
Its displacement from the origin varies with time according to the equation x = 4cos (nt +-
where t is in seconds and the angles in the parentheses are in radians. (a) Determine kinetic and
potential energy of the object at moment of time t = 1 sec.
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