# Central Force 1 Waves and things that repeat. Repeaters 2 Hypothesize how 1. Circles 2. Oscillators (mass on a spring) 3. Waves are related.

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Central Force 1 Waves and things that repeat

Repeaters 2 Hypothesize how 1. Circles 2. Oscillators (mass on a spring) 3. Waves are related.

Hypothesize a quantitative model for the velocity of an object moving in a circle if you only have a meter stick and a timer. What other physical quantities about the circle could you measure? Circular Motion 3

Hypothesize a quantitative model for the force which moves objects in circles using a ball on a string and measuring devices. Central Force 4

central force Central Force & Circular Motion 5 What equations can you derive?

central force Central Force & Circular Motion 6

Central Force & Circular Motion Problems 7 Mr. Mayes is swinging a bucket of water vertically on Earth. The bucket and the water is 10kg. Mayes’ total wingspan (fingertip to fingertip) is 79 inches and it takes him 1 second to spin the bucket in a full circle. How much force does he have to exert at the top of the bucket’s motion? 300N How much force does he have to exert at the bottom of the bucket’s motion? 493N Mayes is tired of this insipid, loathsome, cantankerous water bucket talking smack about his students. He decides he is going to throw it into orbit at a height of his outstretched arm. If the radius of Earth is 6,371 km, his height is 2 m, and his arm length is 1 m, what is the orbital velocity of the water bucket? 7907m/s

You are spinning a sandbag on a string horizontally above your head 1. What would happen to the period, frequency, angular frequency, and period if you keep the force you exert the same but make the radius 5 times larger? 2. What would happen to the force and tangential velocity if you doubled the frequency? Central Force & Circular Motion Problems 8

You want to be a cowboy so you buy a lasso and practice. You get bored of spinning it the same old way so you make the following adjustments from the original way you were spinning it. -If you double the radius but keep period the same, what happens to the force? -If you double the frequency but keep the radius the same what happens to the force? -If you triple the velocity but keep the radius the same what happens to the period? -If you keep the force the same but make the radius five times larger, what happens to the period, frequency, and angular frequency? -If you triple the force and double the period, what happens to the radius? Central Force & Circular Motion Problems 9

Hypothesize the relationship between the quantities which determine the gravitational force between two objects. Materials: phet.colorado.edu ‘Gravity Force Lab’ Gravitational Force 10

Gravitational Force 11

Kepler's 3 rd Law 12 Derive an expression to relate the period and frequency of one planet orbiting the sun to the period and frequency of another planet orbiting the sun.

What is the gravitational force between Earth and Sol (the sun)? A: 3.5 x 10 22 N Gravitational Force 13

Predict the tension in the string which supports a flying pig. You must have 2 observation experiments using only a meter stick, timer, and mass scale to come up with a mathematical hypothesis and a quantitative prediction. You will be graded on how well your predicted value matches the value as measured by a spring scale. Central Force 14

Linear Motion (translational) Circular Motion (translational and maybe rotational) Spin Motion (rotational) Quantity & description Time [∆t or t] The time, clock reading, or time interval of an occurrence. 1.Time [∆t or t] The time, clock reading, or time interval of an occurrence. 2. Period [T] The time for one revolution around a circular path. 1.Time [∆t or t] The time, clock reading, or time interval of an occurrence. 2. Period [T] The time for one revolution around a circular path. Create a chart relating the physical quantities we use in linear force and motion to the quantities we invented for circular motion and rotations. Circles and Rotations 15

By substituting the physical quantities we have for rotations into the physical quantities for linear motion, derive kinematic equations for rotational motion. Circles and Rotations 16

When you pick up a plank or pole, where do you choose to pick it up from? Why do you choose this spot? If you were to try to open a book cover with a piece of pencil lead, where do you choose to place the lead? Create experiments to hypothesize why you choose these spots. Create an index to explain why. Torque 17

Design an experiment to test the hypothesis that the force which exerts a torque and the displacement between the force and the pivot point must be perpendicular to each other. Predict the outcome of this experiment then compare to the actual outcome to test your hypothesis. Torque 18

A painter stands 4/5 of the way up a long ladder. To reach the height needed the angle the ladder makes with the ground must be at minimum 60 o. How can you find the friction force necessary to support the painter and ladder? Torque 19

Create a mobile using meter sticks, strings, random objects, and tape. You will be graded on how well your mobile stays together for 10s. Rules: The mobile may not… 1.be symmetrical or repetitive in any dimension. 2.use any more or less than 3 meter sticks. 3.be constructed with trial and error in any way. You must follow a design. Torque 20

If mass represents the resistance of an object to accelerate when there is a net force exerted on it, what similar physical quantity could we come up with for spins? Moment of Inertia 21

1.A long, massive rod is supported by a string at one end and held horizontally. How can you find the angular acceleration of the rod be if it is released? α= ¾ g/r 2.A long, uniform wrench is not touching anything on the space station. A force is exerted to one end and you can measure the angular acceleration. How can you find the mass? m= 3F/(αr) Torque 22

Torque with Parallel Axis Theorem 24 A massive, uniform rod on Earth is suspended by a string which is 75% of from one end of the rod. How could you find the angular acceleration in terms of the radius of the rod and acceleration due to gravity? α= 6 / 7 (g/r)

Torque: Wheel and Axle 25 Arguably one of the greatest inventions ever made is the wheel & axle. It moves us… literally. A wheel & axle is a simple machine. Create an engineering schematic of a wheel & axle.

Wheel and Axle Problem 26 You have two cylinders with ropes wrapped around them oppositely. The smaller cylinder has a diameter of 10cm, a mass of 2.5kg, and a 4kg mass is attached to the rope which hangs from the right side of the cylinder. The larger cylinder has a diameter of 20 cm, a mass of 11kg, and a 2.5kg mass is attached to the rope which hangs from the cylinder’s left side. The two cylinders are attached to each other by a frictionless axle which goes through both of their centers. What will the angular acceleration of both be?

Hypothesize… …what happens to people when they are spinning like a fan with their arms out, then bring their arms in close. Relate what we know about linear momentum to this scenario. This is an observation experiment. Angular Momentum 27

Predict… …what happens to a person and a wheel who sits in a frictionless spinny chair with their feet up (isolated frictionless system) and no spin. This person then spins a wheel with a specific mass at a specific counterclockwise angular frequency while holding the face of the wheel parallel to the ground. This is a testing experiment. Angular Momentum 28

Predict… …why a person who spins on a frictionless stool very slowly holding a wheel spinning very fast in the opposite direction will stop when this person stops the wheel. This is a application experiment. Angular Momentum 29

Want to Ride Bikes? Observe your memories in this thought experiment. How did you learn to ride a bike? Why is it easier to balance on a moving bike as opposed to stationary one? How do you turn on a bike that is moving very fast and why does this action make you turn? Angular Momentum 30

Create 3 problems about angular momentum. These problems should be different types and should have NO NUMBERS. You are trying to predict what types of angular momentum problems you may encounter. Angular Momentum 31

Attach a set of keys to a very light box with a 1m string. Drape the string over a pencil about 1/3 of the length of the string away from the keys. Let the keys hang vertically but hold the box horizontally. Predict what will happen when you let go of the very small box. Angular Momentum 32

Derive an expression for rotational kinetic energy and rotational work by relating linear physical quantities to rotational physical quantities. Rotational Kinetic Energy 33

Create 3 problems about rotational kinetic energy. These problems should be different types and should have NO NUMBERS. You are trying to predict what types of rotational kinetic energy problems you may encounter. Rotational Kinetic Energy34

Four people of the same mass are competing in a tight rope walking competition across the Grand Canyon. Each competitor may bring their own balancing poles, but the balance poles must be of the same length and mass. This has received a lot of media attention so you and your friends decide to place bets as you watch on TV. Rank the performance from best to worst of the following competitors. 1.Tall with a stiff balancing pole. 2.Short with a stiff balancing pole. 3.Tall with a droopy balancing pole. 4.Short with a droopy balancing pole. Do this for two sets of assumptions. In one case assume that the best competitor is the one which has the least torque exerted on her. In another assume the best competitor which has the easiest time correcting herself after starting to fall. Defend your argument with physical principles in paragraph format. Rotation Problem 35

Five objects roll down the same hill, have the same radius and mass, and start from rest at the same time. The objects are a hoop, a sphere, a wheel with thick spokes, a cylinder, and a shell. Rank the order in which the objects will reach the bottom of the hill starting with the first. Explain your answer in detail. Rotation Problem 36

Circles and Sines 37 I have a physics riddle for you to explain: Circles and squares, sines and lines, It takes two lines two times to square. It takes two sines in time to circle. Squares and lines, circles and sines.

Hypothesize an equation for the position of a mass on a mass-spring system at any time. Use the physical quantities we came up with for circles and spins. Analyze the poem on the previous page to help you. Simple Harmonic Motion: Spring Oscillation 38

Derive equations for an oscillating mass on a spring, knowing the position at any time during oscillation, to describe: Velocity Acceleration Spring force (ignore others) Kinetic energy Spring potential energy Simple Harmonic Motion: Spring Oscillation 39

Derive an expression for the total energy (U elastic and K) of an oscillator (spring and mass system) at any time. Simple Harmonic Motion: Spring Oscillation 40

Derive an expression for the angular frequency of a simple harmonic oscillator if you know the spring constant and the mass. Assume the only force acting on your system is the spring force. Simple Harmonic Motion: Spring Oscillation 41

Simple Harmonic Motion: Spring Oscillation 42 Image courtesy of Hyperphysics Maximum Energy = ½ kA 2

Simple Harmonic Motion: Simple Pendulum 43 A pendulum is like a spring & mass in that is repeats its motion due to a restoring force. In this case, though, the restoring force is some component of the gravitational force. Draw a force diagram of a simple pendulum swinging to its highest height to hypothesize an equation which describes this restoring force.

Simple Harmonic Motion: Simple Pendulum 44 - Using your restoring force expression from the previous example, lets assume that the angle is small. What effect does this assumption have? - Substitute arc length into the equation for the restoring force of a pendulum using s=rθ. You now have quite a few constants. - Relate the equation for the restoring force of a pendulum to the equation for the restoring force of a spring. How do the constants and variables correlate? - Using this correlation, what is the period, frequency or angular frequency solely in terms of acceleration due to gravity and the length of the pendulum?

Simple Harmonic Motion: Spring & Mass and Pendulum 45 Predict the angular frequency of a simple pendulum and a mass and spring system using the hypotheses we have derived. You must have at least 3 different systems for both springs and pendulums (6 total) which have the SAME frequency.

Waves 46 How many types of waves can you make using a long slinky? Hypothesize the aspects of the two simplest types of waves that you can find.

Waves 47 Hypothesize characteristics of waves using ‘Wave on a String’ on phet.colorado.edu. How does a wave behave after it reaches a fixed end? A loose end? How can you change the speed of the wave?

Waves 48 V wave =λf

Superposition of Waves 49 Hypothesize how waves interact with other waves using ‘Wave Interference’ on phet.colorado.edu.

Superposition - Interference of Waves The amplitudes of waves combine. Superposition of Waves 50

Hypothesize how damping affects a wave? How does damping affect real world scenarios in your daily life and how is it useful? What does this type of wave motion correlate to for springs and pendulums? Damping 51 Use the following PHET simulations on phet.colorado.edu 1.Wave on a String 2.Masses and Springs 3.Pendulum Lab

Damped Oscillations Damping 52

Hypothesize how a wave which is ‘driven’ or oscillated change over time? How does this type of wave motion correlate to springs and pendulums? Use the following PHET simulations: 1. Resonance 2.Waves on a String 3. Masses and Springs 4. Pendulum Lab Resonance & Driven Oscillations 53

Just like pushing someone on a string, some oscillations have a force driving their motion. When an object is ‘driven’ or excited, it releases the energy given to it by vibrating at it’s resonance frequency. Resonance & Driven Oscillations 54

Sound 55 Hypothesize the atomic nature of sound using ‘Wave Interference’ on phet.colorado.edu. Compare this to a type of wave you can create on a slinky.

Sound 56 Common perception of… SoundReal life example Wavelength Frequency Amplitude

Sound 57 http://www.artinaid.com/2013/04/sound/

Hypothesize how sound behaves when it encounters walls or other large objects. -Thought: What does it sound like when you yell into a canyon or talk loudly in a museum? -Virtual: ‘Sound’ & ‘Wave Interference’ phet.colorado.edu -Real: Design your own! Reflection 58

Reflection 59 Reflection of Sound Waves

Hypothesize how sound behaves when it encounters obstacles or objects. -Thought: Can you hear someone when you can’t see them? Can you hear at a concert if you are the unfortunate person with seats behind a support pillar? -Virtual: ‘Sound’ & ‘Wave Interference’ phet.colorado.edu -Real: Design your own! Diffraction 60

Diffraction 61 Diffraction of Sound Waves

I went fishing on a lake very early in the morning. It was dark and the only thing to keep me warm was a steaming cup of coffee and the thought of having a beast of a fish at the end of my hook. My solitude was suddenly interrupted by the phantom voice of someone saying ‘hello’ over my left shoulder. I quickly turned to find that I was apparently still alone… then I heard it again… the phantom voice… with no source… and I was still alone. What happened? Refraction 62

Refraction 63 Refraction of Sound Waves

Standing Waves 64 Standing Waves – Superposition of Sound Waves Standing waves involve the repeated interference of two IDENTICAL waves moving in opposite directions, which results in an apparently stationary wave.

Standing Waves – Superposition of Sound Waves Create the fundamental and at least two harmonics using… -’Waves on a String’ PHET -Rope Slinky Standing Waves 65

Beats – Superposition of Sound Waves Consider pure tones with very similar frequencies, but not the same. Draw a diagram representing the superposition of the two waves when they interfere with each other. Beats 66

Beats 67 Beats – Superposition of Sound Waves Predict the frequencies you will hear on http://www.school-for- champions.com/science/sound_beat_fre quencies.htm#.VRK8NvnF_fc Frequency of the Beat Wave= Frequency of one wave – Frequency of the other wave. f B = f 1 -f 2

Doppler Shift 68 Hypothesize how the relative motion of a wave source and the observer may affect the perceived wave. 1.What happens if you are moving relative to the source? 2.What happens if the source is moving relative to you? Materials: http://highered.mheducation.com/olcweb/cgi/pluginpo p.cgi?it=swf::800::600::/sites/dl/free/0072482621/787 78/Doppler_Nav.swf::Doppler+Shift+Interactive

Doppler Shift 69 Toward is positive. Away is negative. You could redefine your reference frame relative to the observer to make life easier.

Intensity of Waves 70 How does the sound made by a person yelling close to you compare to the sound made by a person yelling far away? *HINT - Think geometrically and look at sound as a certain number of waves emanating in all dimensions from a source.*

Intensity of Waves 71

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