Assuming both these system are in frictionless environments, what do they have in common? Energy is conserved Speed is constantly changing Movement would continue indefinitely Simple Harmonic Motion 1

Consider a spring/block system where the spring is stretched a displacement x The spring’s restoring force returns block to its natural position, but the block continues to move to a compressed displacement x because of its inertia If the system is frictionless, the block would continue to cycle back and forth (oscillate) indefinitely. This phenomenon is called simple harmonic motion (SHM). 2

SHM Terms and Concepts Equilibrium – the point where the spring is in its restored position and there is no net force acting on the spring in the direction of motion Amplitude (A) – the maximum displacement away from the equilibrium point Cycle – one complete motion forward and back Period (T) – the time it takes to complete one complete cycle Frequency (f) – the number of cycles per second (Hz, s -1 ). The reciprocal of the period Maximum velocity (v max ) – the velocity at the equilibrium position Maximum acceleration (a max ) – the acceleration at the amplitude where the restoring force is greatest 3

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5 In a vertical mass/spring system, the equilibrium positon is still the position where the net force is zero, BUT that position is not its restored, natural position. Equilibrium is where the restoring force from the spring is equal and opposite to the gravitational force acting on the mass. FsFgFsFg When the mass is below the equilibrium position, the net force is upward because the spring force exceeds the gravitational force. When the mass is above the equilibrium position, the net force is downward because the gravitational force exceeds the spring force. Obviously, the downward force of gravity doesn’t change…it is the upward force of the spring that changes in magnitude. The spring is always under tension while the mass is attached.

A vertical spring stretches m when a kg mass is gently lowered on it. If the system is then placed in a horizontal orientation on a frictionless table and the mass is pulled m from its equilibrium position, determine: (a)the spring constant (b)the maximum speed attained during an oscillation (c)the speed when the mass is m from equilibrium (d)the maximum acceleration of the mass N/m 0.81 m/s 0.70 m/s 6.53 m/s 2 6

Conclusion: The amplitude does NOT affect the period An object in SHM is analogous to an object in uniform circular motion (UCM) when observed from the plane of motion. The circle has radius A and tangential speed v max. You, however, would only observe the x-component of the velocity. Oscillating Systems What variables affect the period of an oscillation?

8 This character is a reason why pendulums are used as timekeeping devices. To adjust the time of a swing, the length of the pendulum simply needs to be adjusted.

A spider of mass 0.30 g waits in its web of negligible mass. A slight movement causes the web to vibrate with a frequency of 15 Hz. (a)What is the value of k for the elastic web? (b)At what frequency would you expect the web to vibrate if an insect of mass 0.10 g were trapped in addition to the spider? The length of a simple pendulum is 0.50 m, the pendulum bob has a mass of 25 g, and it is released so that the bob is 10 cm above its equilibrium position. (a)With what frequency does it oscillate? (b)What is the pendulum’s speed when it passes through the lowest point of the swing? 2.7 N/m 13 Hz Hz 1.4 m/s 9

m same period m

11 SHM Periodicity Because of the periodicity (repeated pattern) of an oscillating particle, the moving object’s displacement can be expressed as a sine function of time. Since the motions are analogous, we can analyze an object in SHM as if it was in UCM. To do that, we need to find common ground between the two models. x x

12 Calculators have to be in radians for this equation!

13 The corresponding function for the velocity of an object in SHM displays: a zero velocity when the displacement is a maximum a maximum velocity when the displacement is zero The corresponding function for the acceleration of an object in SHM displays: a maximum acceleration when the displacement is a maximum a negative acceleration when the displacement is positive a zero acceleration when the velocity is a maximum

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N/m 2.62 s m/s m/s 2 x = m v = m/s a = m/s 2

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