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Simple harmonic motion
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Presented By :Presented By :
Himanshu K. KhannaHimanshu K. KhannaFAITH ACADEMYFAITH ACADEMY
• Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped.
• The motion is periodic and sinusoidal.• With constant amplitude; the acceleration of a
body executing SHM is directly proportional to the displacement of the body from the equilibrium position and is always directed towards the equilibrium position.
where, x Displacement A Amplitude of the oscillation f Frequency t Elapsed time ф Phase of oscillation
If there is no displacement at time t = 0, the phase is ф = π/2
If the spring is unstretched, there is no net force on the mass - in other words, the system is in equilibrium. However, if the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore it to the equilibrium position. In the case of the spring-mass system, this force is the elastic force, which is given by Hooke's Law,
F = -kxF = -kx
where,
F Elastic forcek Spring constantx Displacement
Any system that undergoes simple harmonic motion exhibits two key features:
• When the system is displaced from equilibrium there must exist a restoring force that tends to restore it to equilibrium.
• The restoring force must be proportional to the displacement.
• Once the mass is displaced it experiences a restoring force, accelerating it, causing it to start going back to the equilibrium position.
• As it gets closer to equilibrium the restoring force decreases; at the equilibrium position the restoring force is 0.
• However, at x=0, the mass has some momentum due to the impulse of the force that has acted on it; this causes the mass to shoot past the equilibrium position, in this case, compressing the spring.
• The restoring force then tends to slow it down, until the velocity reaches 0, whereby it will attempt to reach equilibrium position again.
Displacement x is given by:
Differentiating once gives an expression for the velocity at any time:
And once again to get the acceleration at a given time:
Simplifying acceleration in terms of displacement:
Acceleration can also be expressed as:
• Mass on a spring• Uniform circular motion• Mass on a simple pendulum
A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with
Alternately, if the other factors are known and the period is to be found, this equation can be used:
The total energy, E is constant, and given by
A mass is oscillating on a spring
Position inequal time intervals:
In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length with gravitational acceleration g is given by
Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular frequency ω around a circle of radius R centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude R and angular speed ω.
Model: oscillation coupled to a wheel spinning at constant rate
Vertical position versus time:
Period T
Period T
Sinusoidal motion
Time (s)
Displacement (cm)
Period T
Sine function: mathematically
x
y
2π
π/2 π 3π/2 2π 5π/2 3π 7π/2 4π 9π/2 5π
-1
1
y=sin(x)y=cos(x)
Sine function: employed for oscillations
x
y
π/2 π 3π/2 2π 5π/2 3π 7π/2 4π 9π/2 5π
-1
1y=sin(x)
Time t (s)
Displacement y (m)
T/2 T 2T
-A
Ay= A sin(ωt)
Sine function: employed for oscillations
Time t (s)
Displacement y (m)
T/2 T 2T
-A
A y= A sin(ωt)
What do we need ? 1. Maximum displacement A
2. ωT = 2π
3. Initial condition