Rotation, Gravity, Oscillation Monday, November 23, 2009 Torque Torque and the see-saw A see-saw is an example of a device that twists. A force that causes a twisting

motion, multiplied by its distance from the point of rotation, is called a torque. Torque is what makes a see saw fun. Torque If we know the angle between F and r, we can calculate torque!

= r F sin is torque r is moment arm F is force is angle between F and r The SI unit of torque is the Nm. You cannot substitute Joule for Nm

in the case of torque. Hinge (rotates) r Direction of rotation F

Sample Problem Consider the door to the classroom. We use torque to open it. Identify the following: A. The point of rotation. B. The point of application of force. C. The moment arm (r). D. The angle between r and F

(best guess). Sample Problem A crane lifts a load. If the mass of the load is 500 kg, and the cranes 22-m long arm is at a 75o angle relative to the horizontal, calculate the torque exerted about the point of rotation at the base of the crane arm. Torque simplified Usually, will be

Hinge: rotates o 90 , and r =rF is torque r is moment arm F is force

Direction of rotation F Problem A standard door is 36 inches wide, with the doorknob located at 32 inches from the hinge. Calculate the torque a person applies when he pushes on the doorknob at right angles to the door with a force of 110 N. (Use 1 inch = 2.54 cm

to calculate the torque in SI units). Problem A double pulley has two weights hanging from it as shown. A) What is the net torque? B) In what direction will the pulley rotate? 3 cm

2 cm 10 kg 2 kg Now consider a balanced situation 40 kg 40 kg

ccw = cw This is called rotational equilibrium! Sample Problem A 5.0-meter long see saw is balanced on a fulcrum at the middle. A 45-kg child sits all the way on one end. Where must a 60-kg child sit if the see-saw is to be balanced?

Sample Problem A 5.0-meter long see saw is balanced on a fulcrum at the middle. A 45-kg child sits all the way on one end. And a 60-kg child sits all the way on the other end. If the see saw has a mass of 100 kg, where must the fulcrum be placed to attain a balanced situation? Check against notes; mass is different Sample Problem

A 10-meter long wooden plank of mass 209 kg rests on a flat roof with 2.5 meters extended out beyond the roofs edge. How far out on the plank can an 80-kg man walk before he is in danger of falling? Tuesday, November 24, 2009 Torque Lab Torque Lab data collection

Create a torque balance with the meter stick, two known masses and one unknown mass. Rules All masses, known and unknown, must be attached to clips. The meter stick cannot be balanced at the 50 cm point Data collected

Positions on meter stick of all hanging masses, and position of fulcrum. Masses of all known components. DO NOT MASS THE UNKNOWN! DRAW A DIAGRAM THAT IS CLEARLY LABELED! Calculate your unknown.

We will complete the lab on the computers. Torque Lab II Use Excel to determine if your unknown calculation was OK. Turn in:

Hand calculation of torque lab. This will include your diagram, your data, and your calculation of your unknown mass. Torque lab spreadsheet. Submit in one of the following ways: Drag and drop into the Prinkey class temp folder. Print and submit to me a hard-copy.

Email the spreadsheet to me. Tuesday, December 1, 2009 Universal Law of Gravity Torque lab tables Lets take a minute to review the torque lab, and entry of the data into a spreadsheet and calculation with Excel. ccw 2

82 cm 23 g clip 85 g unknown ccw 1 cw 2 cw 1 50 cm 35 cm

145 g meter stick 19 cm 8 cm 20 g clip 19 g clip 150 g weight 150 g weight

The Universal Law of Gravity Newtons famous apple fell on Newtons famous head, and lead to this law. It tells us that the force of gravity objects exert on each other depends on their masses and the distance they are separated from each other. The Force of Gravity

Remember Fg = mg? Weve use this to approximate the force of gravity on an object near the earths surface. This formula wont work for planets and space travel. It wont work for objects that are far from the earth. For space travel, we need a better formula.

The Force of Gravity Fg = -Gm1m2/r2 F : Force due to gravity (N) G: Universal gravitational constant g 6.67 m

x 10-11 N m2/kg2 and m2: the two masses (kg) r: the distance between the centers of the masses (m) 1 The Universal Law of Gravity ALWAYS works, whereas F = mg only works sometimes.

Sample Problem A. B. How much force does the earth exert on the moon? How much force does the moon exert on the earth? Sample Problem What would be your weight if you were orbiting the earth in a satellite at an altitude of 3,000,000 m above the

earths surface? (Note that even though you are apparently weightless, gravity is still exerting a force on your body, and this is your actual weight.) Sample Problem Sally, an astrology buff, claims that the position of the planet Jupiter influences events in her life. She surmises this is due to its gravitational pull. Joe scoffs at Sally and says your Labrador Retriever exerts more gravitational pull on your body than the planet Jupiter does. Is Joe correct? (Assume a 100-lb Lab 1.0 meter away, and Jupiter at its farthest distance from Earth).

Wednesday, December 2, 2009 Gravitational Acceleration and Orbit Acceleration due to gravity Remember g = 9.8 m/s2? This works find when we are near the surface of the earth. For space travel, we

need a better formula! What would that formula be? Acceleration due to gravity g = GM/r2 This formula lets you calculate g anywhere if you know the distance a body is from the center of a planet. We can calculate the acceleration

due to gravity anywhere! Sample Problem What is the acceleration due to gravity at an altitude equal to the earths radius? What about an altitude equal to twice the earths radius? Acceleration and distance Surface gravitational

acceleration depends on mass and radius. Planet Radius(m Mass (kg) g (m/s2) Mercury 2.43 x 106 3.2 x 1023

3.61 Venus 6.073 x 106 4.88 x1024 8.83 Mars

3.38 x 106 6.42 x 1023 3.75 Jupiter 6.98 x 106

1.901 x 1027 26.0 Saturn 5.82 x 107 5.68 x 1026 11.2

Uranus 2.35 x 107 8.68 x 1025 10.5 Neptune

2.27 x 107 1.03 x 1026 13.3 Pluto 1.15 x 106 1.2 x 1022

0.61 Sample Problem What is the acceleration due to gravity at the surface of the moon? Johannes Kepler (1571-1630) Kepler developed some extremely important laws about planetary

motion. Kepler based his laws on massive amounts of data collected by Tyco Brahe. Keplers laws were used by Newton in the development of his own laws. Keplers Laws 1. 2.

3. Planets orbit the sun in elliptical orbits, with the sun at a focus. Planets orbiting the sun carve out equal area triangles in equal times. The planets year is related to its distance from the sun in a predictable way.

Keplers Laws Lets look at a simulation of planetary motion at http://surendranath .tripod.com/Applets .html Sample Problem

(not in packet) Using Newtons Law of Universal Gravitation, derive a formula to show how the period of a planets orbit varies with the radius of that orbit. Assume a nearly circular orbit. Satellites Orbital speed

At any given altitude, there is only one speed for a stable circular orbit. From geometry, we can calculate what this orbital speed must be. At the earths surface, if an object moves 8000 meters horizontally, the surface of the earth will

drop by 5 meters vertically. That is how far the object will fall vertically in one second (use the 1st kinematic equation to show this). Therefore, an object moving at 8000 m/s will never reach the earths surface.

Some orbits are nearly circular. Some orbits are highly elliptical. Centripetal force and gravity The orbits we analyze mathematically

will be nearly circular. Fg = Fc (centripetal force is provided by gravity) GMm/r2 = mv2/r The mass of the orbiting body cancels out in the expression above. One of the rs cancels as well

GM/r = v2 Sample Problem A. B. What velocity does a satellite in orbit about the earth at an altitude of 25,000 km have? What is the period of this satellite?

Sample Problem A geosynchronous satellite is one which remains above the same point on the earth. Such a satellite orbits the earth in 24 hours, thus matching the earth's rotation. How high must must a geosynchronous satellite be above the surface to maintain a geosynchronous orbit? Thursday, December 3, 2009 Gravitational Potential Energy and Escape Velocity

Gravitational Potential Energy Remember Ug = mgh? This is also an approximation we use when an object is near the earth. This formula wont work when we are very far from the surface of the earth. For space travel, we need another formula.

Gravitational Potential Energy Ug = -Gm1m2/r Ug: Gravitational potential energy (J) G: Universal gravitational constant 6.67 x 10-11N m2/kg2

m1 and m2: the two masses (kg) r: the distance between the centers of the masses (m) Notice that the theoretical value of Ug is always negative. This formula always works for two or more objects.

Sample Problem What is the gravitational potential energy of a satellite that is in orbit about the Earth at an altitude equal to the earths radius? Assume the satellite has a mass of 10,000 kg. Sample Problem not in packet What is the gravitational potential energy of the following configuration of objects?

2,000 kg 1,500 kg 10 meters 10 meters 3,000 kg Escape Velocity Calculation of miniumum escape velocity

from a planets surface can be done by using energy conservation. Assume the object gains potential energy and loses kinetic energy, and assume the final potential energy and final kinetic energy are both zero. U 1 + K 1 = U 2 + K2 -GMm/r + mv2 = 0 v = (2GM/r)1/2 Sample Problem

What is the velocity necessary for a rocket to escape the gravitational field of the earth? Assume the rocket is near the earths surface. Sample Problem Suppose a 2500-kg space probe accelerates on blast-off until it reaches a speed of 15,000 m/s. What is the rockets kinetic energy when it has effectively escaped the earths gravitational field?

Friday, December 4, 2009 Periodic Motion Periodic Motion Motion that repeats itself over a fixed and reproducible period of time is called periodic motion. The revolution of a planet about its sun is an example of periodic motion. The highly reproducible period (T) of a planet

is also called its year. Mechanical devices on earth can be designed to have periodic motion. These devices are useful timers. They are called oscillators. Oscillator Demo Lets see demo of an oscillating spring using LoggerPro and a motion sensor. Simple Harmonic Motion

You attach a weight to a spring, stretch the spring past its equilibrium point and release it. The weight bobs up and down with a reproducible period, T. Plot position vs time to get a graph that resembles a sine or cosine function. The graph is sinusoidal, so the motion is referred to as simple harmonic motion. Springs and pendulums undergo simple harmonic motion and are referred to as simple harmonic oscillators.

Analysis of graph Equilibrium is where kinetic energy is maximum and potential energy is zero. 3 equilibrium 2 -3

x(m) 4 6 t(s) Analysis of graph Maximum and minimum positions

3 2 -3 x(m) 4 6 t(s)

Maximum and minimum positions have maximum potential energy and zero kinetic Oscillator Definitions Amplitude Maximum displacement from equilibrium. Related to energy. Period

Length of time required for one oscillation. Frequency How fast the oscillator is oscillating. f = 1/T Unit: Hz or s-1 Sample Problem

Determine the amplitude, period, and frequency of an oscillating spring using the CBLs and the motion sensors. See how this varies with the force constant of the spring and the mass attached to the spring. Monday, December 7, 2009 Springs

Springs A very common type of Simple Harmonic Oscillator. Our springs are ideal springs. They are massless. They are both compressible and extensible.

They will follow Hookes Law. F = -kx Review of Hookes Law m Fs mg

Fs = -kx The force constant of a spring can be determined by attaching a weight and seeing how far it stretches. Period of a spring m T 2

k T: period (s) m: mass (kg) k: force constant (N/m) Sample Problem Calculate the period of a 200-g mass attached to an ideal spring with a force constant of 1,000 N/ m.

Sample Problem A 300-g mass attached to a spring undergoes simple harmonic motion with a frequency of 25 Hz. What is the force constant of the spring? Sample Problem An 80-g mass attached to a spring hung vertically causes it to stretch 30 cm from its unstretched position. If the mass is set into oscillation on the end of the spring, what will be the period?

Sample Problem You wish to double the force constant of a spring. You A. B. C. D. Double its length by connecting it to

another one just like it. Cut it in half. Add twice as much mass. Take half of the mass off. Sample Problem You wish to double the force constant of a spring. You A. B.

C. D. Double its length by connecting it to another one just like it. Cut it in half. Add twice as much mass. Take half of the mass off. Conservation of Energy Springs and pendulums obey conservation

of energy. The equilibrium position has high kinetic energy and low potential energy. The positions of maximum displacement have high potential energy and low kinetic energy. Total energy of the oscillating system is constant. Sample problem. A spring of force constant k = 200 N/m is attached to a

700-g mass oscillating between x = 1.2 and x = 2.4 meters. Where is the mass moving fastest, and how fast is it moving at that location? Sample problem. A spring of force constant k = 200 N/m is attached to a 700-g mass oscillating between x = 1.2 and x = 2.4 meters. What is the speed of the mass when it is at the 1.5 meter point? Sample problem.

A 2.0-kg mass attached to a spring oscillates with an amplitude of 12.0 cm and a frequency of 3.0 Hz. What is its total energy? Mini-Lab Estimate the force constant of the spring in the plunger cart using conservation of energy. Equipment:

Plunger cart (mass 500 g) Ramp Meter Stick Hint: consider turning spring potential energy into another form of potential energy. Turn in one paper per person with your groups data, calculations, and results (that is, the value you think k has).

Tuesday, December 8, 2009 Pendulums Pendulums The pendulum can be thought of as a simple harmonic oscillator. The displacement needs to be small for it to work properly. Pendulum Forces

T mg sin mg Period of a pendulum l T 2 g

T: period (s) l: length of string (m) g: gravitational acceleration (m/s 2) Sample problem Predict the period of a pendulum consisting of a 500 gram mass attached to a 2.5-m long string. Sample problem Suppose you notice that a 5-kg weight

tied to a string swings back and forth 5 times in 20 seconds. How long is the string? Sample problem The period of a pendulum is observed to be T. Suppose you want to make the period 2T. What do you do to the pendulum? Pendulum Lab Determine period, T, and length, l, of your groups

pendulum. For accuracy, time multiple oscillations. Write your groups data on the board. Report, due Friday: A table and graph constructed from this data. The graph must be LINEAR such that the slope can be used to obtain g. In other words, you cant just simply graph T versus l. Think of what you must do to produce a linear graph from the data. Axes must be clearly labeled. The graph may be done by hand or in Excel. Show clearly how you get g, and indicate its value. Perform a percent error calculation. Hint: Consider the formula for the period of a pendulum

to decide what to graph. 1st Period Group Number of oscillation s

Elapsed time (s) Period (s) Length (m) 2nd Period Group

Number of oscillation s Elapsed time (s) Period (s)

Length (m) 7th Period Group Number of oscillation s

Elapsed time (s) Period (s) Length (m) Tuesday, January 9, 2007

Spring Lab Announcements Rotation, Gravity, Oscillation #9 will be checked tomorrow, which is when you have your next Homework Quiz. Lunch Bunch Photoelectric Effect lab due tomorrow. US Physics Team exam: Do you

have your $5.00? Exam is Friday. Spring lab Use Hookes Law to determine the force constant of your spring. Do at least 5 trials. The report will include a graph of the data such that the slope yields k. Determine the force constant of your spring from its period of an oscillation with various attached masses. The report will

include a graph of the data such that the slope yields k. Compare the force constants obtained by these two methods. Full lab report due next Tuesday, January 16. Wednesday, January 10, 2007 Review Announcements

Homework: Due today is #9 (folder), pendulum lab (pass up), photoelectric lab (pass up). Your homework folders will be collected after homework quiz. Due tomorrow at beginning of class -- FR #2 and #3 in your packet.

US Physics Team Exam $5 is due. Review today and tomorrow. Exam Friday. Makeup lab Friday Jeopardy Thursday, January 11, 2007 Review Announcements US Physics Team Exam $5 is due.

Get out Clickers. Exam Friday. Makeup Lab Friday morning before school. Review: Torque Torque causes a twist or rotation. = r F sin is torque F is force r is moment arm

is angle between F and r Torque units: Nm Review: Keplers Laws 1. 2. 3.

Planets orbit the sun in elliptical orbits. Planets orbiting the sun carve out equal area triangles in equal times. The planets year is related to its distance from the sun in a predictable way -- derivable Review: Gravitation Fg = Gm1m2/r2 (Magnitude of Force)

Ug = -Gm1m2/r (Potential Energy) Relationships for derivations Acceleration due to gravity Fg = mg Orbital parameters (period, radius,

velocity) Fg = mv2/r Energy Conservation (escape velocity) Ug1 + K1 = Ug2 + K2 Friday, January 12, 2007 Exam

Friday, January 12, 2007 Exam