Simple Harmonic Motion

Do Now 5 min – Explain why a pendulum oscillates using words and pictures. Work INDIVIDUALLY. Share with your table partner … add/make changes to your answer if necessary.

Vocab Review! What does the word oscillation mean? back and forth movement When is oscillatory motion is called periodic motion? If the motion repeats If the motion follows the same path in the same amount of time We refer to these repeating units of periodic motion motion as… The time it takes to complete one cycle is called the … Oscillation – a back and forth movement; regular variation in magnitude or position around a central point cycles period (T) Example: Earth’s rotation has a period of 24 hours, or 86,400 s.

Simple Harmonic Motion Pendulums and springs are special examples of motion that not only oscillatory and periodic, but also simple harmonic. Simple harmonic motion is a type of periodic motion in which the force that brings the object back to equilibrium is proportional to the displacement of the object. e.g. greater displacement = greater force Oscillation – a back and forth movement; regular variation in magnitude or position around a central point

Restoring Force - CFUs In which position(s) is the restoring force Of the pendulum … … greatest? … zero? … angled downward and towards the right? In which position(s) is the restoring force of the spring … … directed upwards? A, G D G, F, E A B C D E F G G A B, C, D, E, F, G Springs can also be compressed! Any elastic (stretchable) material will act somewhat like a spring.

Calculating restoring (net) force In pendulums … Look at the diagram. What forces cancel out? What is the net force? In springs … Fspring = kx where k is spring constant, x = displacement T and mgcosθ cancel out … we know because there is no a in that direction mgsinθ

We do: Calculating restoring (net) force Force (N) Displacement (mm) 2 1.0 3 1.5 4 2.0 5 2.5 6 3.0 An engineer measured the force required to compress a spring. Based on the data, what is the spring constant? Predict the force required to compress the spring by 3.5 mm. k = 2 N/mm = 0.002 N/m F = 7 N Use the simulator! How do the spring constants of spring 1 and spring 2 compare? Calculate the spring constant for spring 1. Calculate the spring constant for spring 3. Predict how far the spring will stretch with a 250 g weight. Determine the weight of each cylinder.

Period is NOT affected by the amplitude of motion! Calculating period In pendulums … In springs … NOTE: Period is NOT affected by the amplitude of motion! Period only depends on length & gravity Longer string = longer period Weaker gravity = longer period Period only depends on mass and spring constant. Higher mass = longer period Looser spring / smaller k = longer period

Period CFUs – Turn & Talk If you stretch and release a slinky, you will notice that the amplitude of its motion decreases over time (why?). How does this decrease in amplitude affect the period of motion? Will a grandfather clock run slower or faster if placed on the moon? Why? How does doubling the mass affect the period of a pendulum? How does doubling the mass affect the period of a spring? It doesn’t! Amplitude of motion does NOT affect period. The grandfather clock will run slow (have a longer period) because as acceleration due to gravity decreases, the period increases. Doubling the mass has NO affect on the period of a pendulum. Doubling the mass of a spring increases the period by a factor of √2

Conservation of energy Ideally, pendulums and springs both conserve energy. (Realistically, they lose energy over time due to friction). In both cases, PE is maximum at maximum displacement. PE gradually converts to KE, and reaches zero at the equilibrium point. KE shows the opposite trend – it is maximum at equilibrium and reaches zero at maximum displacement. We have a simple formula for the PE in a spring. PEspring = ½ kx2 In pendulums … In springs … TE

Conservation of energy CFU A and G have equal heights. D is equilibrium position Fill in the following table: Position PE (J) KE (J) A 50 B 35 C 15 D G

You Do Problems - 1) A spring stretches by 18 cm when a bag of potatoes weighing 56 N is suspended from its end. Determine the spring constant, k How much EPE does the spring have when it is stretched this far? Hooke’s Law F = -k(x) k = 310 N/m b) EPE = .5k(x^2) EPE = 5 J

You Do Problems - 2) A pendulum swings from its release point, past equilibrium, to its highest point on the opposite side in 0.4 seconds. The highest point is 8 degrees above equilibrium. What is its frequency? Its period? Its amplitude? Frequency = 1 per 0.8 sec = 1.25 Hz Period = 1/frequency = 1.6 s Amplitude = 8o 3) You need to know the height of a tower but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower? T = 2π√ 𝐿 𝑔 𝑇√𝑔 2π = L L = 36 m 4) Billie releases the bob of his pendulum at an angle of 10° from the vertical. At the same time, Bobby releases the bob of his pendulum at an angle of 20° from the vertical. The two pendulums have the same length. Does Billie’s bob reach the vertical position before, after, or at the same time as Bobby’s bob? Explain. Same time. Amplitude does not affect period.

Damping and Resonance Damping is the decrease in amplitude of a wave. All real pendulums and springs have damping. Energy is lost due to friction Amplitude of motion becomes smaller, until it ceases Some systems are designed to heavily damped, such as shock absorbers on a car Damping mechanisms in the foundations of buildings in earthquake zones Heavy damping