Updated: 13 November 2015 Friday.
Monday 11/9: Using a column of liquid to make a barometer to measure air pressure. Switch from water to mercury changes h at 1 atm. from 10.33 m to 0.759m. The aneroid barometer. Smooth Fluid Flow: Pressure from a column of liquid looks like P.E. Create a Kinetic Pressure term which looks like K.E. and add in the base pressure for total pressure to create Bernoulli's Equation and the Continuity Equation. The Water Tower and the Faucet Problem. Why the water tower needs a vent.
Tuesday 11/10: Bernoulli's Equation and the Continuity Equation. Want Smooth Continuous Flow, not Turbulent Flow or Viscous Flow. Flow rate = Volume / time = Cross-sectional Area × Speed. The faster the fluid flow, the lower the Pressure. Example: The aspirator -- a vacuum pump with no moving parts. Example: Air flow around a wing. (Faster air over top means lower pressure on top, so net force is up -- Lift.) Spoilers -- doors open in wing to allow air to pass between upper and lower surfaces, thus "spoiling the lift" by eliminating the pressure difference. Why the Mackinac Bridge has grates on the inside north- and soundbound lanes.
Wednesday 11/11: Old wooden water towers and farm silos are wrapped in steel bands which are not evenly spaced. They are closer together at the bottom, farther away at the top. This is due to the increased pressure on the bottom. When these towers fail, it's usually on the bottom where the pressure is greatest. Q10 in-class.
Thursday 11/12: Dr. Phil out sick -- no subsitute available in time -- class cancelled.
Friday 11/13: Dr. Phil out sick -- substitute. Expected material available:
Problem: Spray can. (Inside: P_{1} = 200,000 Pa, v_{1} = 0, h_{1} = h_{2}. Outside: P_{2} = 100,000 Pa. Find v_{2}.) Problem: RMS Titanic is on the bottom of the Atlantic, about 2½ miles down. When James Cameron made his movie, he rode the Mir submersibles to the wreck. Find the speed of the water shooting into the sub if there is a leak. Find the water pressure at that depth. (P_{1} = P_{2}, v_{1} (on surface) = 0, h_{1} = 3821 m, h_{2} = 0, ρ_{seawater} = 1030 kg/m³.) Actually the water pressure is higher, due to the fact that with hundreds of atmospheres of pressure, the seawater is slightly compressed and so the mass-to-volume ratio isn't constant, but increasing.
Temperature & Heat. Heat = Energy. Two objects in thermal contact, exchange heat energy, Q. If net heat exchange is zero, the two objects are at the same temperature. Temperature Scales: °F, °C and K (Kelvins).
Monday 9/7: Labor Day <No Classes>
Tuesday 9/8: Class begins. The nature of studying Physics. Science education in the United States. The Circle of Physics. Intro Handout. (Click here for a copy if you didn't make it to the first class.)
Wednesday 9/9: Aristotle and the Greek Philosophers. Observation vs. Experiment - Dropping the book and the piece of paper (2 views). Mechanics is the study of motion. So what is motion? (Xeno) Zeno of Elea -- Zeno's Paradoxes. Speed Limit 70: What does it mean? First Equation: Speed = Distance / Time. v = d/t . Rewrite as: d = v t.
Thursday 9/10: Distribute Syllabus. v = d / t or d = v t. So d = distance. Position also has units of length, is x, y or z. Displacement is final position minus initial position, or d = x_{final} - x_{initial} = x_{f} - x_{i}. x_{i} = x_{t=0} = x_{0} (x-sub-zero). Physicists love our subscripts -- they're like name tags. Finally d = x - x_{0} = v t or x = x_{0} + v t. All these equations are rewrites of the same thing, but these rewrites will become useful soon. Q1 (Attendance -- Q1A coming if you weren't in class.)
Friday 9/11: Topic 1 assigned. (Updated Searchable booklist available online NOT UP YET -- use last years for now .) Q1½ In-Class worksheet -- "No Stress Quiz".
Monday 9/14: PTPBIP - Putting The Physics Back Into The Problem. Go over Q1½. The P-O-R (Press-On-Regardless) road rally problem. "You can't average averages." Note that Q1½ part (c) is just like the P-O-R problem. Speed. 60 m.p.h. = "A Mile A Minute". It's a nice alliterative phrase and wasn't possible for Man to move at 60 mph until 1848: The Antelope, but it really isn't a special speed, just an accident of the English system of measurement.
Tuesday 9/15: English system of measurement. SI Metric System. Prefixes. What do we mean by Measurements? "Units will save your life." What is "1 m/s"? We need a few benchmark values to compare English and SI Metric quantities. NOTE: English-to-Metric conversions will NOT, with two exceptions, be tested on in this course. 60 m.p.h. = 26.8 m/s. 1.00 m/s = slow walking speed. 10.0 m/s = World Class sprint speed (The 100 meter dash -- Usain Bolt is the current Olympic (9.683 seconds) and World (9.58 seconds) record holder.) 26.8 m/s = 60 m.p.h.. 344 m/s = Speed of sound at room temperature. 8000 m/s = low Earth orbital speed. 11,300 m/s = Earth escape velocity. 300,000,000 m/s = speed of light in vacuum (maximum possible speed).
Wednesday 9/16: Delta (Δ): Δx = x_{f} - x_{i} = x - x_{0} = "the change in x". So our first equation becomes v = Δx / Δt . A simplified trip to the store -- The S-Shaped Curve. Acceleration. Physics Misconceptions: Things you think you know, are sure you know, or just assume to be true in the back of your mind... but aren't true. Aristotle was sure that heavier objects always fell faster than lighter objects, but we did a demostration on Wednesday which showed that wasn't always true. Example: You're driving a car. To speed up, you need to put your foot on the accelerator (gas pedal), so YES, you are accelerating -- True. To drive at a constant speed, you must still have your foot on the accelerator, so YES, you are accelerating -- Not True because constant v means a = 0. To slow down, you must take your foot off the accelerator and put it on the brake pedal, so NO, you are not accelerating -- Not True because v is changing, so a < 0 (negative). Just as the equation v = d / t is for constant or average speed, the equation a = Δv / Δt is for constant or average acceleration. Finding the set of Kinematic Equations for Constant Acceleration.
Thursday 9/17: Kinematic Equations for Constant Acceleration. What do we mean by a = 1 meter/sec² ? You cannot accelerate at 1 m/s² for very long. Types of Motion: No Motion (v=0, a=0), Uniform Motion (v=constant, a=0), Constant Acceleration (a=constant). We generally cannot accelerate for very long. Example: A car goes from rest to 60.0 mph (26.8 m/s) with an acceleration a = 2.00m/s². Find the distance traveled and the time. Strategy: To aid in setting up problems with the kinematic equations, you might try to list all six kinematic variables (x_{0}, x, v_{0}, v, a and t) and give the values for those you know, those you don't know and those you want to find out. This will help you choose which kinematic equation(s) you'll need. In this case: The problem tells you v_{0} = 0, v = 26.8 m/s, a = 2.00 m/s². Choose x_{0} = 0. That's 4 out of 6 of the kinematic variables. Try to find t using one of the kinematic equations. (Hint: I'd use the 2nd equation.) Then either 1st or 4th equation to find x. Both work. Dr. Phil's Simplified Significant Figures for multiplication, division and trig functions. (Click here if you need a copy.)
t (seconds) | v = a t (m/sec) | x = ½ a t² (meters) |
0 | 0 | 0 |
1.00 | 1.00 | 0.500 |
2.00 | 2.00 | 2.00 |
3.00 | 3.00 | 4.50 |
4.00 | 4.00 | 8.00 |
5.00 | 5.00 | 12.5 |
10.0 | 10.0 | 50.0 |
25.0 | 25.0 | 312.5 |
Friday 9/18: Problem: A rifle bullet is fired from rest to faster than the speed of sound, 425 m/s, in a distance of 1.00 m. Find a. Answer, a = 90,310 m/s². This is huge, which is why we don't fire people out of rifle barrels. Find t = 0.004706sec. Again, we could solve for t using two different equations, but will still get the same result because there is one Physics. To aid in setting up problems with the kinematic equations, you might try to list all six kinematic variables (x_{0}, x, v_{0}, v, a and t) and give the values for those you know, those you don't know and those you want to find out. This will help you choose which kinematic equation(s) you'll need. Free-Fall: If we ignore air resistance, all objects near the surface of the Earth fall towards the Earth at the same rate. a_{y} = -g ; g = 9.81 m/s². That's nearly ten times the acceleration a = 1 m/s² we just talked about. With these two known accelerations, we can now have something to compare our accelerations a. How much acceleration can a human take? See story of Scott Crossfield below.
Monday 9/21: Prepping for 2-D Motion: We can look at motion in 1-dimension in different directions. We usually use x in the horizontal. y can either be another horizontal dimension or in the vertical. We can rewrite the Kinematic Equations for constant acceleration for x or y. It turns out that if x and y are perpendicular to each other, then they are independent, so we will be able to break down 2-D motion into two 1-D motion problems. Free-Fall: If we ignore air resistance, all objects near the surface of the Earth fall towards the Earth at the same rate. a_{y} = -g ; g = 9.81 m/s². That's nearly ten times the acceleration a = 1 m/s² we talked about last week. With these two known acceleratoins, we can now have something to compare our accelerations a. Rewriting the Kinematic Equations for motion in the y-direction, pre-loading them for free-fall. Example: Falling off a ten-foot roof (3.00 meters). The consequences of Falling Down... ...and Falling Up.
Tuesday 9/22: The consequences of Falling Down... ...and Falling Up. The Turning Point ( v_{y} = 0, but a_{y} = -g during whole flight). The illusion of "hanging up there in the air" at the turning point. Example: It is 5.50 meters to the ceiling in 1104 Rood. Consider tossing a ball up to the ceiling. If we ignore air reistance, then the time to rise is the same as the time to fall -- the latter is easier to calculate because if you start a problem at the turning point, then v_{0y} = 0.
Wednesday 9/23: Q3 in-class on Free Fall motion.
Thursday 9/24: Return Q2. Motion in Two-Dimensions: You may be able to break it down into two one-dimensional problems, connected by time, which you can already solve. Example: The guy with the fedora and the cigar. There are 6 variables from the first dimension (x_{0}, x, v_{0x}, v_{x}, a_{x}, t), but only 5 from the second (y_{0}, y, v_{0y}, v_{y}, a_{y}), because time is the same. Remarkably, with a couple of reasonable assumptions, there are only 3 unknown variables (v_{0x}, t, v_{y}). Time links the two one-dimensional problems together. We need to find v_{0x} , but we don't know the time. So we can find the time it takes to fall from the top of the building in the y-problem, then use that in the x-problem. Two kinds of numbers: Scalars (magnitude and units) and Vectors (magnitude, units and direction). Adding and subtracting vectors: Graphical method. To generate an analytical method, we first need to look at some Trigonometry. Right Triangles: Sum of the interior angles of any triangle is 180°. Examples: vector C = vector A + vector B, vector D = vector A - vector B.
Friday 9/25: To generate an analytical method, we first need to look at some Trigonometry. Right Triangles: Sum of the interior angles of any triangle is 180°, Pythagorean Theorem (a² + b² = c²). Standard Angle (start at positive x-axis and go counterclockwise). Standard Form: 5.00m @ 30°. Practical Trigonometry. SOHCAHTOA. Adding and subtracting vectors: Analytical method. (Check to make sure your calculator is set for Degrees mode. Try cos 45° = sin 45° = 0.7071) Why arctangent is a stupid function on your calculator. Sig.Figs for angles doesn't quite make sense -- both 359° and 1.00° look like 3 sig. fig. But how can we know one angle to 0.01° and the other only to a whole degree, when they are just 2° apart? Examples: vector C = vector A + vector B, vector D = vector A - vector B. Dr. Phil's Method uses a table you fill out with the x- and y-components, to allow you to easily add or subtract the columns. Then use your sketch to check your work. Q4 is a Take-Home on Vector Addition, due on Tuesday 29 September 2015 at the beginning of class (or by 12:45pm). (Click here for a copy.)
Monday 9/28: Adding and subtracting vectors: Analytical method. (Check to make sure your calculator is set for Degrees mode. Try cos 45° = sin 45° = 0.7071) Why arctangent is a stupid function on your calculator. Examples: vector C = vector A + vector B, vector D = vector A - vector B. Dr. Phil's Method uses a table you fill out with the x- and y-components, to allow you to easily add or subtract the columns. Then use your sketch to check your work. Example: A-vector = 2.00 m @ 17° and B-vector 6.00 m @ 173°.
Vector | A_{x} = A cos θ | A_{y} = A sin θ |
A | 1.913 m | 0.585 m |
B | -5.955 m | 0.731 m |
A+B | -4.042 m | 1.316 m |
C = 4.595 m , Arctan gives us -18.0°, which is a small angle, so given that C_{x} points in the -x and C_{y} in the +y, if you make a sketch you should find that θ = 162.0° or C-vector = 4.595 m @ 162.0°. Another problem solved by using two linked 1-D problems: Classic Simple Pursuit (Cop and the Speeder). Starting from rest, the contant accelerating cop ends up with a final speed twice that of the uniform motion speeder -- because they both have to have the same average speed (same place, same time). Note that (1) if you take a suqare root of t , you get two answers, +/- , (2) if you divide both sides of an equation by t then t = 0 is also a solution. In both cases, these other solutions may not be useful to our problem, because they occur at the beginning or before the problem starts -- the equations go on forever in the past or the future, but the problem is defined only over a narrow range.
Tuesday 9/29: Types of Motion: No Motion (v=0, a=0), Uniform Motion (v=constant, a=0), Constant Acceleration (a=constant). Ballistic or Projectile Motion 2-D problem where a_{x} = 0 and a_{y} = -g. Covers anything shot, thrown or kicked into the air which is unpowered and where we can ignore air resistance. Ancient cannons. We can always use the Kinematic Equations, but we can also derive specialized equations: Max Height, Time to Max Height, Range Equation.
Wednesday 9/30: Exam 1.
Thursday 10/1: Ballistic or Projectile Motion 2-D problem where a_{x} = 0 and a_{y} = -g. Covers anything shot, thrown or kicked into the air which is unpowered and where we can ignore air resistance. Ancient cannons. We can always use the Kinematic Equations, but we can also derive specialized equations: Max Height, Time to Max Height, Range Equation. Two Dangerous Equations. You can only use the Range Equation if the Launch Height = Landing Height. But the sin (2 θ) term in the Range Equation means that (1) 45° gives the maximum range for a given initial velocity and (2) that all other angles have a complementary angle (90° - θ) that gives the same range (but a different time and height). High and low trajectories for Range Equation.
Friday 10/2: Cannonball example: if v_{0} is 100. m/s @ 30° and lands at launch height, y = y_{0}. Find range R, height h, time of flight t -- note that the latter is for the full flight. What changes if you use the complementary launch angle of 60°? Or 45°, which should give the maximum range? Compare h, time t to max height and range R. To Jump A Gap, you MUST have a positive v_{0y}, because you are immediately in freefall. Demo: A bullet fired horizontally has the same y-motion as a bullet dropped from the muzzle height. Use two pieces of chalk, one tossed, one dropped. They hit the table at the same time.
Monday 10/5: Recap: Our studies so far have described "How" things move, and allow to say "When" and "Where" things move, but not "Why" things move. For that we have to start talking about Forces -- and that means Newton. Some stories about Sir Isaac Newton. (Reeding on the edge of the silver shilling or a U.S. dime/quarter.) (Mad as a hatter -- from mercury poisioning.) Newton's Three Laws of Motion: Zeroeth Law - There is such a thing as mass. First Law - An object in motion tends to stay in motion, or an object at rest tends to stay at rest, unless acted upon by a net external force. Second Law - F=ma.
Tuesday 10/6: Newton's Three Laws of Motion: Zeroeth Law - There is such a thing as mass. SI unit of mass = kilogram (kg). SI unit of force = Newton (N). English unit of force = pound (lb.). English unit of mass = slug (Divide pounds by 32. For English units, g = 32 ft/sec².). Force is a vector. First Law - An object in motion tends to stay in motion, or an object at rest tends to stay at rest, unless acted upon by a net external force. Second Law - F=ma. Third Law - For every action, there is an equal and opposite reaction, acting on the other body. (Forces come in pairs, not apples.) Force is a vector. Free Body Diagrams. Normal Force (Normal = Perpendicular to plane of contact). The normal force does NOT automatically point up and it is not automatically equal to the weight -- we have to solve for the normal force. "The Normal Force is NOT automatically present -- you have to be in contact with a surface. The Normal Force does NOT automatically point up -- F_{N} is perpendicular to the surface. The Normal Force is NOT automatically equal to the weight. F_{N} = mg only if there are no other forces in the y-direction." Sum of forces in x or y equations -- either will be equal to 0 (Newton's 1st Law) or ma (Newton's 2nd Law). Example of 125 kg crate being dragged/pushed around. (Near the surface of the Earth, you can use the relationship that 1 kg of mass corresponds [not "equals"] to 2.2 lbs. of weight. So multiple 125 by 2 and add 10%... 250 + 25 = 275... so a 125 kg crate has a weight of mg = 1226 N or 275 lbs.). Variations as we allow for an applied force that it at an angle. Push down and Normal Force increases; pull up and Normal Force decreases -- though it cannot go negative.
Wednesday 10/7: Example of 125 kg crate being dragged/pushed around. (Near the surface of the Earth, you can use the relationship that 1 kg of mass corresponds [not "equals"] to 2.2 lbs. of weight. So multiple 125 by 2 and add 10%... 250 + 25 = 275... so a 125 kg crate has a weight of mg = 1226 N or 275 lbs.). Variations as we allow for an applied force that it at an angle. Push down and Normal Force increases; pull up and Normal Force decreases -- though it cannot go negative. "You can't push on a rope." Since the force from a wire/string/rope/chain/thread/etc. can only be in one direction, Dr. Phil prefers to call such forces T for Tensions rather than F for Forces. Q5 in-class.
Thursday 10/8: Return X1. "You can't push on a rope." Since the force from a wire/string/rope/chain/thread/etc. can only be in one direction, Dr. Phil prefers to call such forces T for Tensions rather than F for Forces. Simple pulleys (Massless, frictionless, dimensionless, only redirect the forces). "There is no free lunch." The bracket for the pulley will have to support a force greater than the weight of the hanging object. Mechanical advantage: multiple pulleys allow us to distribute the net force across multiple cables or the same cable loop around multiple times. Tension in the cable is reduced, but you have to pull more cable to move the crate.
Friday 10/9: Atwood's Machine -- two masses connected by a single cable via a simple pulley. They share a common acceleration, a, with one mass going up and the other going down. More Elevator Comments. The Normal Force represents the "apparent weight" of the person in the elevator. Like Atwood's Machine, we can hang a counterweight on a cable and a pulley and support all or some of hte weight of the elevator. The elevator will go one way and the counterweight will go the other way.
Monday 10/12: Hanging a sign with angled wires -- still the same procedure: Sketch of the problem, Free Body Diagram, Sum of Forces equations in the x- and y-directions, solve for unknowns. Discussion of guy wires to help support a very tall antenna. Two kinds of Friction: Static (stationary) and Kinetic (sliding). For any given contact surface, there are two coefficients of friction, µ, one for static (µ_{s}) and one for kinetic (µ_{k}). Coefficients of friction have no units, so they are the same for SI metric and English. They are generally less than 1. Static is always greater than kinetic. Kinetic friction opposes motion and is always F_{f,k} = µ_{k} F_{n}. Static friction opposes impending motion and can be zero to a maximum of F_{f,s,max} = µ_{s}F_{n}.
Tuesday 10/13: Two kinds of Friction: Static (stationary) and Kinetic (sliding). For any given contact surface, there are two coefficients of friction, µ, one for static (µ_{s}) and one for kinetic (µ_{k}). Coefficients of friction have no units, so they are the same for SI metric and English. They are generally less than 1. Static is always greater than kinetic. Kinetic friction opposes motion and is always F_{f,k} = µ_{k} F_{n}. Static friction opposes impending motion and can be zero to a maximum of F_{f,s,max} = µ_{s}F_{n}. Example: Using our 125 kg. crate, applied force with F_{1} = 100.N and coefficients 0.800 and 0.600. (1) If at rest, F_{1} < F_{f,s,max}, therefore F_{f,s} = 100.N and the crate does not move. (2) If moving with v_{0} = 5.00 m/s, then F_{f,k} > F_{1} , so the crate will be slowing down. (3) If at rest and apply force F_{2} = 1000. N > F_{f,s,max} then break the staic friction barrier and switch to kinetic friction, so will move, and accelerate around 5 m/s². (4) To move crate at constant speed, get it started then apply force F_{3} = F_{f,k}. Anti-Lock Brakes and Traction Control. ABS works by monitoring the rotation of all four wheels. If one wheel begins to "lose it" and slip on the road while braking, it will slow its rotation faster than the other tires, so the computer releases the brake on that wheel only until it is rolling without slipping again. This can be done many times a second, much faster than the good old "pump your brakes to stop on ice" trick older drivers are familiar with. Traction control uses the ABS sensors to monitor the wheel slip during acceleration -- keeps the wheels from spinning.
Wednesday 10/14: Friction Problems: (1) A car moving at 70 mph (31.3 m/s) on dry concrete (coefficients of friction 1.00 and 0.800) -- find the shortest stopping distance. To find the distance, we need the acceleration. To find the acceleration we need to find the net force (F = ma). To find the forces, we need the sum of forces equations in x and y, which we get from the Free Body Diagram. Shortet stopping distance requires maximum static friction -- if the car is moving to the right, friction must point to the left to stop. (2) Repeat for the car on sheer ice -- divide the coefficients of friction by 10 (0.100 and 0.0800) -- where in a panic the cars slides to a stop. Q6 in-class.
Thursday 10/15: Inclined plane problems: Change the co-ordinate system, change the rules. In the tilted x'-y' coordinates, this is a one-dimensional problem, not two-dimensional. Inclined plane with and without friction. Finding the coefficient of static friction by tilting: µ_{s} = tan(θ_{max}). Similar for kinetic friction, except one has to tap the board to "break the static friction barrier". Rubber on concrete. Can µ_{ }be greater than 1? Means θ_{max} greater than 45° -- rare, but yes. Driving up and down mountains. Truck Escape Ramps.
Friday 10/16: Air Resistance. Low speed (F_{drag} = -bv ) and high speed (F_{drag} = -cv²) air resistance. The minus sign is there to remind us that air resistance, like friction, opposes motion. If allowed to drop from rest, then a real object may not free fall continuously, but may reach a Terminal Velocity (Force of gravity down canceled by Drag force up) and doesn't accelerate any more. Ping-pong balls in free-fall, vs. being hit with a paddle in a world-class table tennis match. What is the terminal velocity of a falling person? It depnds on clothing and orientation -- aerodynamics, streamlining, cross-sectional area, composition of the air are all part of the drag coefficients b and c. World's Record Free-Fall (old). (NEW Sunday 10/14/2012)
Monday 10/19: We are not done with Forces, but some problems cannot easily be solved by using forces. Collisions, for example, are very complex if we have to put in all the forces of bending and breaking and mashing things. Need a simpler way of looking at the problem. "Inertia" is a word which isn't used much today, but it is the same as "momentum" -- represents some kind of relentless quality of movement. It takes a force to change the momentum, otherwise it just continues on, i.e., Newton's 1st Law. Linear Momentum ( p = mv ) is a vector quantity. Newton's form of the 2nd Law: F = Δp / Δt = change in momentum / change in time instead of F=ma, but really the same thing. Impulse Equation: Δp = F Δt .Two extremes in collisions: Totally Elastic Collision (perfect rebound, no damage) and Totally Inelastic Collision (stick together, take damage). Linear momentum is conserved in all types of collisions . Totally Inelastic Collisions. Example: The Yugo and the Cement Truck with numbers. Real Head-On collisions. Three example collisions: Head-on Collisions. Rear-end Collisions. (The Non-Collision -- if the car following is going slower, it isn't going to run into the car ahead. PTPBIP.)
Tuesday 10/20: Three example collisions: Head-on Collisions. Rear-end Collisions. (The Non-Collision -- if the car following is going slower, it isn't going to run into the car ahead. PTPBIP.) 2-D Side Impact (vector) collision. Real crashes. Interactions of safety systems: Seat belts, shoulder belts, steel beams in doors and crumple zones. The myth of it being better to be "thrown clear from the wreck". What happens in a wreck.
Wednesday 10/21: Work: A Physics Definition (Work = Force times distance in the same direction). Work = Energy. Q7 in-class. Q8 Take-Home on Inclined Planes, due Tuesday 27 October 2015. (Click here for a copy.)
Thursday 10/22: Work: A Physics Definition (Work = Force times distance in the same direction). Work = Energy. Power = Work / time. Kinetic Energy -- an energy of motion, always positive, scalar, no direction information. Work-Energy Theorem (net Work = Change in K.E.). Using the Work-Energy Theorem to find a final speed. Potential Energy: Storing energy from applied work for later. Gravitational P.E. = mgh. Location of h=0 is arbitrary choice. Conservation Laws are very important in Physics. Conservation of Total Mechanical Energy (T.M.E. = K.E. + P.E.). Lose angle and directional information because energy is a scalar, not a vector. We can change height for speed and vice versa.
Example: Consider: a 1.00 kg mass undergoing a net force of 1.00 N for an acceleration of 1.00 m/s². (Please note I chose simple numbers for this illustration. The mass isn't usually 1.00 kg, so the actual numbers aren't going to be the same between columns, but the ratios mentioned below still work..)
t | v = at | p = mv | d = ½at² | KE = ½mv² |
0 | 0 | 0 | 0 | 0 |
1.00 sec | 1.00 m/s | 1.00 kg·m/s | 0.500 m | 0.500 J |
2.00 sec | 2.00 m/s | 2.00 kg·m/s | 2.00 m | 2.00 J |
3.00 sec | 3.00 m/s | 3.00 kg·m/s | 4.50 m | 4.50 J |
4.00 sec | 4.00 m/s | 4.00 kg·m/s | 8.00 m | 8.00 J |
- From 1.00 sec to 2.00 sec, doubling the time gives double the speed, double the momentum, but 4x the distance and 4x the KE.
- From 1.00 sec to 4.00 sec, quadrupling (4x) the time gives 4x the speed, 4x the momentum, but 16x the distance and 16x the KE.
Friday 10/23: See chart problem above. This helps explain why it takes longer and longer to speed up -- because the Work needed to increase the K.E. keeps getting much bigger and bigger because of the v^{2}. Work = Energy. Power = Work / time. You are limited in how much work you can do in a given time by the maximum power available. NOTE: for a car, the upper speed limit is generally where all the available work from the max power goes into opposing the work done by friction and air resistance -- leaving nothing for the Work-Energy Theorem to increase the speed. Conservation of T.M.E. (P.E. + K.E.) on a roller coaster. Total energy limits maximum height. If speed at top of the first hill is about zero, then this P.E. is all we have. Cannot get higher, but we can change height for speed. Example: Rollercoaster with h_{1} = 50.0 m, v_{1} = 0, h_{2} = 0 (bottom of loop-the-loop), h_{3} = 12.0 m (top of loop-the-loop, making D = 12.0 m and r = D/2 = 6.00 m). Results: v_{2} = 31.32 m/s, v_{3} = 27.30 m/s. v_{3} is well above the minimum speed to safely do the loop-the-loop (7.672 m/s from F_{N} = 0 and mv²/r = mg ) NOTE: We usually have covered Uniform Circular Motion by now, but that's our next topic...
Monday 10/26: Types of Motion: No Motion (v=0, a=0), Uniform Motion (v=constant, a=0), Constant Acceleration (a=constant). Uniform Circular Motion (UCM): speed is constant, but vector velocity is not; magnitude of the acceleration is constant, but the vector acceleration is not. Velocity is tangent to circle, Centripetal Acceleration is perpendicular to velocity and points radial INWARD. a_{c} = v²/r. You can generate very large centripetal acclerations very quickly. Space Shuttle in Low-Earth Orbit. (There's still gravity up there!) Comments on Free Fall vs. "zero gravity" in space. The Centripetal Force, F_{c} = mv²/r = ma_{c} . The Centripetal Force is "the answer" on the right side of the Sum of Forces equation. Something has to cause the centripetal force -- it does not show up on the F.B.D. as F_{c}.
Tuesday 10/27: Uniform Circular Motion (UCM): speed is constant, but vector velocity is not; magnitude of the acceleration is constant, but the vector acceleration is not. Velocity is tangent to circle, Centripetal Acceleration is perpendicular to velocity and points radial INWARD. Uniform Circular Motion (UCM): speed is constant, but vector velocity is not; magnitude of the acceleration is constant, but the vector acceleration is not. Velocity is tangent to circle, Centripetal Acceleration is perpendicular to velocity and points radial INWARD. a_{c} = v²/r. You can generate very large centripetal acclerations very quickly. Demo: Rodney Reindeer does UCM. Examples: A hard disk drive spinning at 3600 rpm (60 times a second, time for one revolution = 1/60th of a second). The guard around a circular saw blade takes the sawdust and broken bits which shoot out tangentially from the blade and redirects them to a bucket -- improves safety and makes less of a mess. Review for X2.
Wednesday 10/28: Exam 2.
Thursday 10/29: Return Q7. UCM Revisited: Centripetal Force, F_{c} = ma_{c} = mv²/r. No such thing as Centrifugal Force. Only the Centrepital Force, which points radial inward, just like the centripetal acceleration. Note that the Centripetal Force is an ANSWER to the sum of forces equation -- it does not show up in the F.B.D. directly -- something has to CAUSE the Centripetal Force, such as a Normal Force (or component), tension, friction, or a combination of forces, etc. Making "artificial gravity" for long-duration space flight by living in a rotating object. Test tube example. The story of the 50,000 rpm Ultra-Centrifuge and the Fresh Rat's Liver. Newton's Universal Law of Gravity (or Newton's Law of Universal Gravity). Use Universal Gravity to check "g". The value we calculate is close, 9.83m/s², which turns out to be only off by 0.2%. (9.83/9.81 = 1.002) Why is it off? Because using Univeral Gravity in this manner makes the assumption that the entire Earth is uniform and homogenous from the surface to the core -- which it is not. We would need calculus to integrate over layers to get the observed value of 9.81m/s².
Friday 10/30: UCM Revisited -- A Problem to do this weekend: The Shuttle in Low Earth Orbit (Revisited). Calculating g(r) for r = 6,770,000 m (the radius of the Earth plus the height of 400 km for Low Earth Orbit), we get a value somewhat different than we found for the centripetal acceleration a few weeks ago. Working backwards, discover for this radius that the period T = 5542 sec and NOT the estimated 5400 sec (90 minutes) we had started with before. Newton's Law of Universal Gravity + U.C.M: Each radius of circular orbit has a different value of g(r). As r increases, v decreases and T increases. Newton's Law of Universal Gravity and Tides (high/low, spring/neap). Water is more flexible than land, so it can be influenced by the weak gravitational forces from the Moon (a quarter million miles away) and the Sun (93 million miles away).
Monday 11/2: Four Fundamental Forces in Nature: Gravity (weakest), E & M, Weak Nuclear Force, Strong Nuclear Force (strongest). We've asked: How do things move? (kinematics) Why do things move? (forces) What effort does it take to move? (work and energy) Now we ask -- What moves? Extended Objects: Mass occupies a volume and shape. Mass-to-Volume Ratio (Density). NOTE: Do not confuse the Density of the Materials with the Mass-to-Volume Ratio of the OBJECT. Density of Water built into the SI metric system (1 gram/cm³ = 1000 kg/m³). Sea water is 1030 kg/m³ ; sugar water is 1060 kg/m³. Floating on the Surface: Mass-to-Volume Ratio of the boat < Mass-to-Volume Ratio of the Liquid. Why Boats Float. Density of Water built into the SI metric system (1 gram/cm³ = 1000 kg/m³). Sea water is 1030 kg/m³ ; sugar water is 1060 kg/m³. Example: Front lab table as a 250 kg boat with 4.00 m³ volume.
Tuesday 11/3: Archimedes and Eureka! (I found it!) Using mass-to-volume ratio and water displacement to determine if gold crown was solid gold or not. Example: Front lab table as a 250 kg boat with 4.00 m³ volume. Buoyant Force = Weight of the Boat = Weight of the Water Displaced by the Submerged Part of the Boat. Calculating the amount of the boat submerged, by using the fact that the mass of the boat and the displaced water are the same, but the Total Volume of the Boat and Submerged Volume of the Boat ( = Volume of Displaced Water) are different. Large Ships sre described by how many tons of water (1 ton = 2000 lb.) they displaced. I believe that empty, RMS Titanic displaced 69,000 tons of water. It is importatant gto know the draft or how far down the keel or bottom of the boat is -- heavily laden ships always leave or arrive at high tide, when there is more water in the harbor and therefore more water under the keel. Water is unusual: (2) The mass-to-volume ratio of ice (solid) is LESS than liquid water, so ice floats. Ice which floats doesn't add to volume of water when it melts, but grounded ice (non-floating) does. This is one of the reasons why people worry about what global warming might do to the great ice sheets around the world.
Wednesday 11/4: Newton's Law of Universal Gravity and Tides: We don't see tides in the Great Lakes (or your bathtub), because they are isolated and cannot draw water from a quarter of the way around the Earth. The oceans, however, are all connected, so we do get ocean tides. Q9 in-class.
Thursday 11/5: Return X2. Pressure = Force / Area. SI unit: Pascal (Pa). Example: Squeezing a thumbtack between thumb and forefinger. 1 Pa = 1 N/m², but Pascals are very small, so we get a lot of them. One Atmosphere standard air pressure = 1 atm. = 14.7 psi = 101,300 Pa. Water is unusual in two ways: (1) Water is relatively incompressible. If the depth h isn't too deep, then the Mass-to-Volume ratio for water is constant. For great depths, such as the bottom of the oceans, we can't use our simple equation because rho is not constant. Air and gasses are compressible, so we can't use our pressure from a column of fluid equation either, though the air pressure here on the surface of the Earth is based on supporting the weight of the column of air above us. (2) The mass-to-volume ratio of ice (solid) is LESS than liquid water, so ice floats. Ice which floats doesn't add to volume of water when it melts, but grounded ice (non-floating) does. This is one of the reasons why people worry about what global warming might do to the great ice sheets around the world. Reset: One Atmosphere standard air pressure = 1 atm. = 14.7 psi = 101,300 Pa. Pressure at a depth due to supporting the column of liquid above: P = ρgh.
Friday 11/6: Reset: One Atmosphere standard air pressure = 1 atm. = 14.7 psi = 101,300 Pa. Pressure at a depth due to supporting the column of liquid above: P = ρgh. Pressure due to a column of water = 1 atm 101,300 Pa. at h = 10.33m = 33.86 feet. How to get liquid out of a cup using a straw -- or why Physics does not "suck", but pushes using a pressure difference. Absolute (total) Pressure vs. Gauge Pressure (difference between two readings). Possible to have a "negative gauge pressure", but absolute pressure is always zero or positive.