Simple Harmonic Motion

simple harmonic motion

• it is the simplest kind of oscillatory motion. It is simple harmonic if:
• - the acceleration of the object is directly proportional to its displacement from its equilibrium position
• -" it is a system which acceleration is towards a fixed point and is at maximum acceleration at that fixed point".
• MASS SUSPENDED VERTICALLY ON A SPRING
• -the displacement can be negative or positive with respect to the equilibrium position
• -the resultant force acting on the mass is composed of its weight and tension in the spring= it will always be "trying" to pull the mass back to the equilibrium position.

Diapositiva 2

• 
• -he restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'.
• -this doesn't happen due to DAMPING.

Diapositiva 3

• springs

displacement

• 
• -displacement is 0 at equilibrium point if it is measured from there
• -The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
• IT CAN BE AFFECTED BECAUSE IF THE DISPLACEMENT IS AT MAX WHEN T=0 THEN THE COSINE GRAPH APPLIES
• IF THE DISPLACEMENT IS AT ZERO WHEN T=0 THEN THE SINE GRAPH APPLIES (the one on the right)

displacment

Diapositiva 6

• 
• -when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle.
• -At maximum displacement the object has to come to rest, and then start moving in the opposite direction. I

Diapositiva 7

• T=1 t=seconds per cycle
• F
• (eg) if there are 4 oscillations per second. whats T
• T=1/4 = 0.25s
• from circular motion. one complete oscillation is equivalent to one revolution for circular motion
• w = 2π f
• therefore T= 2π/ w
• 

Acceleration

• calculate acceleration of an object at any point during its oscillation
• a= -(2πf)2 X x
• a= -(w)2 X x
• x= the displacement from the equilibrium in metres
• negative value of acceleration shows it is opposite direction to the displacement.
• 

Diapositiva 9

• 
• - WHEN DISPLACEMENT IS ZERO, VELOCITY IS AT MAX
• -WHEN DISPLACEMENT IS ZERO, ACCELERATION IS AT ZERO
• -WHEN DISPLACEMENT IS AT MAX, VELOCITY IS ZERO
• -WHEN DISPLACEMENT IS AT MAX, ACCELERATION IS AT MINIMUM

Ke and GPE energy

• for an undamped SHM, the total energy of the system remains constant.
• -however, the potential and kinetic energy are inversley related
• -as the kinetic energy of the mass increases, the potential energy decreases

damping

• -there is friction causing energy to be lost from the system
• -if energy is removed form a system, amplitude of a vibration is reduced
• -some systems have so much damping that no oscillations occur, critical damping.
• 

examples of damping

• grandfather clock
• - lightly damped by friction at the pivot and via air resitance. damping is not wanted as it will cause the clock to cease
• car suspension systems
• -fitted with springs and shock absorbers to damp out the vibrations in a moving car. without this the car would bounce up and down on the springs. they need to be slightly less than critically damped. if there is too much damping, then the system does not return back to equilibrium in time for the next bump in the road.

forced osculations and resonance

• natural frequency = the frequency at which a system oscillates when not subjected to a continuous or repeated external force.
• - if the frequency of the applied force is applied the same as the natural frequcy it will cause the amplitude vibrations to build up.
• -when the driving force is applied at same time, we say resonance occurs
• -eg) if people walk the same footsteps over a bridge it causes resonance and can sometimes lead to collapse.
• increased damping= the system responds to a wider range of frequencies.
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