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 O Theory 1. The period T (sec) of a simple harmonic oscillator is given by T = 2rE here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg). 2. Frequency / (Hz) of a simple harmonic oscillator , f= 3. Circular frequency of oscillations ag (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. T= here T is the oscillation period (sec), y 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(gt + Pa). here x is the particles displacement (m) at moment of time t (sec), A is maximum displacement from equilibrium or amplitude (m): = wgt + Po is phase of oscillatory motion (rad): Po is initial phase (rad); we is circular frequancy (rad/sec). Position TAmplitude X 27 Speed v (m/sec) of a simple harmonic oscillator 6. v(t) ==-Auosin(ahot + P%) = -Bmsin(upt + Po). here vm = Aag is maximum speed, or speed amplitude. Acceleration a (m/see?) of a simple harmonic oscillator 7. a(t) = - -Aajcos(at + Po) = -acos(agt + .). here am = Aj is maximum acceleration or amplidude of acceleration. Kinetie energy E, (J) of a simple harmonic oscillator mal E. -m 8. - Asin (ayt + ). Potential energy E, (J) of a simple harmonic oscillator 9. E, =-cos"(atot + wn). 10. Full energy E (J) of a simple harmonic oscillator E = E, + E, = mata E rad =180 or 1 rad = 30" 45" 60° " 90" e 0° vZ/2 v3/2 sin(0) 1/2 V3/2 vz/2 1/2 cos(0) v3/3 V3 tg(8) cos(e +*) = -cose sin(6 +) = -sine
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 O Theory 1. The period T (sec) of a simple harmonic oscillator is given by T = 2rE here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg). 2. Frequency / (Hz) of a simple harmonic oscillator , f= 3. Circular frequency of oscillations ag (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. T= here T is the oscillation period (sec), y 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(gt + Pa). here x is the particles displacement (m) at moment of time t (sec), A is maximum displacement from equilibrium or amplitude (m): = wgt + Po is phase of oscillatory motion (rad): Po is initial phase (rad); we is circular frequancy (rad/sec). Position TAmplitude X 27 Speed v (m/sec) of a simple harmonic oscillator 6. v(t) ==-Auosin(ahot + P%) = -Bmsin(upt + Po). here vm = Aag is maximum speed, or speed amplitude. Acceleration a (m/see?) of a simple harmonic oscillator 7. a(t) = - -Aajcos(at + Po) = -acos(agt + .). here am = Aj is maximum acceleration or amplidude of acceleration. Kinetie energy E, (J) of a simple harmonic oscillator mal E. -m 8. - Asin (ayt + ). Potential energy E, (J) of a simple harmonic oscillator 9. E, =-cos"(atot + wn). 10. Full energy E (J) of a simple harmonic oscillator E = E, + E, = mata E rad =180 or 1 rad = 30" 45" 60° " 90" e 0° vZ/2 v3/2 sin(0) 1/2 V3/2 vz/2 1/2 cos(0) v3/3 V3 tg(8) cos(e +*) = -cose sin(6 +) = -sine
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Chapter15: Oscillatory Motion
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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.
Transcribed Image Text:HARMONIC OCSILLATIONS
O Theory
1. The period T (sec) of a simple harmonic oscillator is given by
T = 2rE
here k is force constant or stiffness coefficient (N/m), m is mass of oscillating particle (kg).
2. Frequency / (Hz) of a simple harmonic oscillator
,
f=
3. Circular frequency of oscillations ag (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.
T=
here T is the oscillation period (sec), y 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(gt + Pa).
here x is the particles displacement (m) at moment of time t (sec), A is maximum displacement
from equilibrium or amplitude (m): = wgt + Po is phase of oscillatory motion (rad): Po is
initial phase (rad); we is circular frequancy (rad/sec).
Position
TAmplitude X
27
Speed v (m/sec) of a simple harmonic oscillator
6.
v(t) ==-Auosin(ahot + P%) = -Bmsin(upt + Po).
here vm = Aag is maximum speed, or speed amplitude.
Acceleration a (m/see?) of a simple harmonic oscillator
7.
a(t) = - -Aajcos(at + Po) = -acos(agt + .).
here am = Aj is maximum acceleration or amplidude of acceleration.
Kinetie energy E, (J) of a simple harmonic oscillator
mal
E. -m
8.
- Asin (ayt + ).
Potential energy E, (J) of a simple harmonic oscillator
9.
E, =-cos"(atot + wn).
10. Full energy E (J) of a simple harmonic oscillator
E = E, + E, = mata
E rad =180 or 1 rad =
30" 45"
60°
"
90" e
0°
vZ/2
v3/2
sin(0)
1/2
V3/2
vz/2
1/2
cos(0)
v3/3
V3
tg(8)
cos(e +*) = -cose
sin(6 +) = -sine
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