Oscillators An oscillator is anything whose motion repeats itself, but we are mainly interested in a particular type called a ‘Simple Harmonic Oscillator’. 1. Understanding the terminology Displacement (x): Distance from equilibrium position at any moment in time. Amplitude (A): Maximum displacement Time Period (T): Time to make one complete oscillation Frequency: Number of oscillations per second Restoring Force The force which always tends to accelerate the oscillating body back towards its equilibrium position. There are many types of oscillators: Bouncing balls, car spring systems etc, but physicists are most interested in ‘Harmonic Motion’. Harmonic Means that the time period is constant even although the amplitude may be decreasing. ‘Isochronous’ is another word meaning the same thing. This type of motion occurs when the restoring force and  acceleration is proportional to, but in the opposite direction to the displacement. So for a Harmonic Oscillator: F = - kx Simple Harmonic The same as harmonic, but applied to idealised ‘friction free’ systems, so that amplitude remains constant. Applies to many situations in Physics and is referred to as ‘Simple Harmonic Motion’ (SHM) or ‘Free Oscillations’

Phase Difference The ‘phase difference’ between two oscillators or between two waves is the amount by which one ‘has been delayed’ compared to the other. It can be expressed in radians or degrees for oscillators, and in radians, or degrees or fractions of a wavelength for waves. EG Pendulums as shown with the red one at max amplitude: Formula for calculating Phase Difference Example 2 1 2 1 2 NB. Phase has nothing to do with this angle ! PD = Phase Difference in Radians t = Difference in time at which the two oscillators reach a certain point 360 180 90 T = Time Period of both oscillators. (It must be the same) 1 lags 2 by 90 or /2 radians 1 lags 2 (or 2 lags 1)by 180 or  radians

Questions page 35. 1a. Initial downward acceleration of ‘g’. Acceleration decreases, at first due to air resistance When distance below platform = unstretched length of bungee cord, acceleration decreases more rapidly due to F=kx from cord.c At some point kx = weight so v is momentarily constant. Then kx becomes greater than his weight, so he starts to decelerate. Eventually his velocity is momentarily 0. Then he starts to accelerate upwards because kx > weight. Eventually kx = weight, so momentarily v is constant. Then weight > kx so he decelerates..... Etc. 2. a) Free oscillation : One where no friction forces are present so amplitude is constant. b) Measure amplitude and see whether it changes 3. a)

4. a) b)

Using ‘projected’ angular velocity to give a formula for SHM. Oscillator Equivalent rotating object r Displacement r Displacement = r Cos    x = r Cos   But r = Amplitude of Oscillation Equilibrium  x = A Cos  But for circular motion ,  = t  Formula for SHM x = A Cos t x = Displacement A = Amplitude  = angular velocity = 2  Frequency  = 2f t = Time from start position. NB x = A cos t applies to oscillators which start at maximum displacement. If they start at the equilibrium position, the formula is: x = A sin t

The equation for displacement of an oscillator is: This has a graph of the form : Graph and Formula for velocity of the oscillator ?? A Time (s) Displacement We know that velocity = gradient of displacement time graph so: Max -ve gradient gives max -ve velocity 0 gradient gives 0 velocity Max +ve gradient gives max +ve velocity Time (s) velocity A The equation must have something to do with : But…. Higher frequency oscillators have steeper graphs -A   affects the amplitude of the velocity graph For mathmeticians, differentiation gives this result acceleration Time (s) Graph and formula for acceleration of the oscillator ?? 2A We know that acceleration (a) = gradient of velocity time graph so using same arguments as above: -2A

Questions page 37. NB.Displacement always means from equilibrium, not from starting point. It has reached maximum displacement upwards, so +25 mm. It is instantaneously stopped while changing direction from up to down. b. Zero displacement moving downwards c. It has reached maximum displacement downwards, so -25 mm. It is instantaneously stopped while changing direction from down to up. d. Back to zero displacement moving upwards. 2. a. b. (i) -0.25ms-2 (ii) (iii) 0.25ms-2 3. a. b. Initial acceleration = Acceleration when x = Amplitude (= maximum acceleration 4. a. 1.0 s is half an oscillation. x = -32 mm, a = 0.32ms-2 b. 1.5 s is three quarters of an oscillation. x = 0 , a = 0

Questions page 39. NB Your calculator must be in ‘Radians mode’ to use the formula x = a cos (t) 1. a) b) 2. a) (i) Amplitude = 12 mm (ii) b) 3. a) b)

4. a) b) (i) This is negative, so towards maximum negative displacement. (ii) This is positive, so towards maximum positive displacement.

Oscillators An oscillator is anything whose motion repeats itself, but we are mainly interested in a particular type called a ‘Simple Harmonic Oscillator’. 1. Understanding the terminology Displacement (x): Amplitude (x0): (Time) Period: Frequency: Restoring Force There are many types of oscillators: Bouncing balls, car spring systems etc, but physicists are most interested in ‘Harmonic Motion’. Harmonic So for a Harmonic Oscillator: Simple Harmonic The same as harmonic, but applied so that amplitude remains constant. Applies to many situations in Physics and is referred to as ‘Simple Harmonic Motion’ (SHM) or ‘free oscillations’

Phase Difference and Superposition The ‘phase difference’ between two oscillators or between two It can be expressed in radians or degrees for oscillators, and in radians, or degrees or fractions of a wavelength for waves. EG Pendulums as shown with the red one at max amplitude: Example 2 1 2 1 2 NB. Phase has nothing to do with this angle !

Questions page 35.

Using ‘projected’ angular velocity to give a formula for SHM. Oscillator Equivalent rotating object r r Displacement =   x =  But r = Equilibrium  x = But for circular motion ,  Formula for SHM x = Displacement A = Amplitude  = angular velocity = 2  Frequency t = Time from start position. NB x = A cos t applies to oscillators which start at maximum displacement. If they start at the equilibrium position, the formula is:

The equation for displacement of an oscillator is: This has a graph of the form : Graph and Formula for velocity of the oscillator ?? X0 Time (s) Displacement We know that velocity = Max -ve gradient gives 0 gradient gives Max +ve gradient gives Time (s) velocity X0 The equation must have something to do with : But…. Higher frequency oscillators have steeper graphs -X0   affects the amplitude of the velocity graph For mathmeticians, differentiation gives this result acceleration Time (s) Graph and formula for acceleration of the oscillator ?? 2X0 We know that acceleration (a) = gradient of velocity time graph so using same arguments as above: -2X0

Questions page 37.

Questions page 39. NB Your calculator must be in ‘Radians mode’ to use the formula x = a cos (t)