1st Pre-Lab Quiz 2nd Pre-Lab Quiz 3rd Pre-Lab Quiz 4th Pre-Lab Quiz

Today’s Lecture: Brief logistics Experiment #1 Error Propagation: Intro Procedural comments Experimental considerations Error Propagation: Recap from last lecture Intuitive view Useful hints for experiment 1

Logistics etc… Please consult course web page regularly. Important: This is where we post all relevant announcements It is our ONLY way to communicate with you! 2BL.UCSD.EDU Your lab sections and TAs schedule on web page Important: Bring printouts of guidelines to Lab Be on-time. Take quiz Go to beach

The Four Experiments Determine the average density of the earth Weigh the Earth, Measure its volume Measure simple things like lengths and times Learn to estimate and propagate errors Non-Destructive measurements of densities, inner structure of objects Absolute measurements Vs. Measurements of variability Measure moments of inertia Use repeated measurements to reduce random errors Construct, and tune a shock absorber Adjust performance of a mechanical system Demonstrate critical damping of your shock absorber Measure coulomb force and calibrate a voltmeter. Reduce systematic errors in a precise measurement.

Experiment 1: Measure Density of Earth Need to know: Mass, Volume <Density> = Mass/Volume Volume: assume earth is a sphere (99.8% accurate) Mass: acceleration of gravity on Earth’s surface (Galileo’s experiment’s) Two measurements: (a) Earth’s Radius Re - Watch a sunset (b) Local acceleration of gravity g - Swing a pendulum (a) Radius of Earth, direct measurements of: Length, Time, Angle. Deduce: Radius of Earth (b) Mass of Earth, direct measurements of: Length, Time. Deduce: Mass of earth Use Newton’s constant G = 6.67 × 10-11 N m2/kg2 ==> Calculate average density r

Experiment 1: Basics Gravitational force Local g Measuring g & Re What is the value of g? ( M ) Local g What is the radius of the earth? Measuring g & Re ==> Solve for r

What Element(s) make up the Earth Assume most of earth’s volume is one element. Assume earth is a sphere Need to know: Mass, Volume Densities (g/cm3) Which of these can be eliminated as the dominant constituent in the earth’s composition? Look it up! 10% measurement needed to determine composition.

Measure Earth’s Radius using Dt Sunset - height above the sea level From right triangle - distance to the horizon line How do we convert the distance to the horizon line, L, into the sunset time delay, t ? Sunset on earth’s surface (tangent) The length of the earth’s circumference, corresponds to a time delay of Therefore, Sunset at height h Is this time delay measurable?

Measure Earth’s Radius using Dt Sunset - height above the sea level Now, is this time delay measurable? - distance to the horizon line - our cliff Sunset on earth’s surface (tangent) Sunset at height h Have we forgotten something? Looks doable!

Correct for Latitude and Earth’s Axis winter view Set your mode to Rad as appropriate when using your calculators… La Jolla latitude Solar latitude varies. d = days since March 20 (or Sept 22). This formula accounts for our latitude and for the angle of the earth’s axis from the plane of its orbit.

“The Equation” for Experiment 1a Which are the variables that contribute to the error significantly? from previous page. Time difference between the two sunset observers. Season dependant factor slightly greater than 1. The formula for your error analysis. What other methods could we use to measure the radius of the earth?

Procedural Comments

Measuring the Height of the Cliff Work in two groups: Half on the cliff and Half on the beach The group on the cliff will measure l, using a laser range finder. The group on the beach will measure q. Note that it is measured relative to gravity! l h1 h1=l cos(q) q

Your Height Above Sea Level on Beach The formula we derived is for height above sea level. The experimenter on the beach also views the sunset from above sea level (at least her height…) Check the error propagation for the sensitivity of the measurement of the earth’s radius to the h2 measurement! h2 must be measured at the same approximate time as Dt. Remember: tides change the sea level by 1 to 2 meters! h2

Cliffs West of Muir Campus (not to scale) h1 Access is easy but wear walking shoes. It may be cold in the evening. h2

Weather is important: Need a clear day! Sunset is later than most lab sections --> you will have to return to observe and measure the sunset times. When you go to measure the time of sunset, go in groups larger than two and remember to: Synchronize your watches before you split Agree on the exact phase of sunset you are timing! Weather is important: Need a clear day! What phase of sunset would make the best reference point? Sun touching the horizon? One half below the horizon? Completely disappeared?

Next need to measure g

Simple Pendulum - Reminder Measure g by constructing a simple pendulum and measure its period, T. A simple pendulum is a mass on the end of a mass-less, perfectly flexible string, suspended from a rigid support.

Measuring g with a Pendulum Can we solve this differential equation? Looks rather tough… Let’s simplify. Since: when is small Harmonic oscillation with frequency w, and period

Experimental Considerations What assumptions have been made? • m is a point mass, so that l is measured from the point of suspension to the point mass. (The center of mass will be a good approximation to a point mass.) • The mass of the “string” is negligible relative to m. (We can ensure this by choosing the correct material for the string and sufficient mass for m). The angle  is small • Rigid, stable support. Your TAs will point these out to you during Lab Important: Start calculations at home, before next week’s lab!

Quick recap from last lecture…

Error Propagation - Sum What is the perimeter of this figure? y w z x p = w + x + y + z You measure x, y, z, w and compute p. How would you calculate the error on p? Estimate errors from x, y, z, w. - They all are likely to be on the order of precision of the ruler, 1/32” or ~0.75 mm. Propagate these to compute the error on p.

Error Propagation - Sum (cont) y w z x p = w + x + y + z We estimated the errors on x, y, z, and w as: What is our estimate of error on p? Since Consider: But, since dx, dy, dz and dw are all error estimates ==> we do not know their signs! Therefore, we could do: Worst case, if all have same sign! We would normally use the rule of addition in quadrature: Independent, random errors.

Another Example of Error Propagation - Product Take x, y and q as xbest, ybest, qbest Measure (x+dx) and (y+dy) Compute (q+dq) (neglect dxdy) q Subtract q from both sides of the above equation. (Notice partial derivatives)

Another Example of Error Propagation - Product Here again, we don’t know the signs. Sometimes these contributions will cancel, sometimes add up. We can compute errors two ways: 1) Maximum possible error 2) If the uncertainties x, y are independent & random:

Fractional Errors For products like q=xy, we can add the fractional errors on the measurements (x/x) to get the fractional error on the result (q/q) : Simple Derivation This also works for ratios like .

Example

For independent random errors Summary Uncertainties in Sums and Differences: Uncertainties in Products and Ratios: General Rule: For independent random errors

Back to Error Propagation…

Error Propagation q(xbest + dx) q(xbest – dx) xbest - dx xbest qbest = q(xbest) q(xbest – dx) xbest - dx xbest xbest + dx

Error Propagation q(xbest + dx) q(xbest – dx) xbest - dx xbest dq = q(xbest+dx) – q(xbest) dq depends on: 1) dx 2) q(x) q(xbest + dx) qbest = q(xbest) q(xbest – dx) xbest - dx xbest xbest + dx

Error Propagation - Why Partial Derivatives? dq = q(xbest+dx) – q(x)

Error propagation For any function:

Propagating Errors for Experiment 1 Formula for density. Take partial derivatives and add errors in quadrature Or, in terms of relative uncertainties: shorthand notation for quadratic sum:

Propagating Errors for Re basic formula Propagate errors (use shorthand for addition in quadrature) Note that the error blows up at h1=h2 and at h2=0.

Example a Given that: h ==> Find h. l always use radians when calculating the errors on trig functions

Next Lecture Issues from last week Statistical Analysis: Histograms and Distributions The Limiting Distribution Introducing the Normal Distribution Read Taylor Chap 5.