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Grade 12U Physics The Conical Pendulum Lab: Measuring the gravitational field strength of the earth Instructions: At home or in class, attach one end of a long string to a pivot point fixed to the ceiling or some elevated support as shown in the diagram. Attach the other end to a compact object which is much heavier than the string used. Allow the object to hang at rest so the string defines a vertical or plumb line. Using a metre stick or metric measuring tape, measure the distance from the pivot point to the centre of the suspended object, called the pendulum length L . Being mindful that only one “rightmost” digit is allowed to have some uncertainty, record the pendulum length measurement to the correct number of significant digits on the space provided below. Now pull back the mass roughly 30° to 40°. Using a metre stick or metric measuring tape, measure the perpendicular distance from the vertical or plumb line to the centre of the object, called the pendulum radius R . Being mindful that only one “rightmost” digit is allowed to have some uncertainty, record the pendulum radius measurement to the correct number of significant digits on the space provided below. Note that the string now makes an angle Θ to the vertical or plumb line. With practice, the object can now be given a push perpendicular to the radius and on the horizontal plane so the object traces out a approximate UCM (uniform circular motion) on that horizontal plane. Using a timing device such as a smart phone, stopwatch or watch, measure the time it takes the object to make ten revolutions or cycles. Being mindful that only one “rightmost” digit is allowed to have some uncertainty, record “the time to make ten revolutions” measurement to the correct number of significant digits on the space provided below. Record measurements to correct number of significant digits here: Ceiling a) L = ______________________ m State measuring device: (ex. mm or cm ruler etc. ) _____________________________ _____________________________ Pivot String L L R b) R = ______________________ m object State measuring device: (ex. mm or cm ruler etc. ) _____________________________ _____________________________ c) time to undergo ten revolutions in UCM on horizontal plane: __________________________ seconds State measuring device: (ex. Stopwatch, clock etc. ) _____________________________ _____________________________ Θ Plumb line page #2 Questions and Calculations: A formal report is not required. Hand-in a cover page with your name, due date and your instructor's name. Other than the cover page, your report must be in your own handwriting, not computer generated. Plagiarism will result in a zero for all parties concerned. For calculations, all steps must be show in a neat and logical manner. Explanations must be concise and in complete sentences. Marks will be deducted for missing steps, missing units and incorrect # of significant digits. Hand-in this question booklet with your report. Use the marking scheme on the next page as a checklist. 1. Assuming R stays constant during the ten revolutions of the conical pendulum, calculate Θ to the correct number of significant digits. Be sure to show formula and substitutions. 2. Read about “reporting data involving measurements” in the new textbook p 694 or “error analysis in experimentation” in the old textbook p755. Using your reading as a guide, state the uncertainty in L and R, then convert these uncertainties to percentage uncertainties. Be sure to show the steps in the conversions. Noting that a division is required to obtain Θ, find the percentage uncertainty in Θ. Be sure to show steps. 3. Find the period of revolution for the conical pendulum. Assuming the “ten cycles” are exact, state the uncertainty and percentage uncertainty in the period. 4. Why is it better to measure the time to complete ten cycles as opposed to measuring the time to complete one cycle? 5. Review the derivation for the formula of a car rounding a circular banked curve with negligible friction. Using this derivation as a guide, derive or create a fully-simplified formula for the period (T) of the conical pendulum in terms of Θ , L and g. Ignore any friction effects in this derivation. Set up the positive x-axis so it is aligned with the centripetal acceleration of the object. This derivation must include a properly labelled FBD, equation of motion, vector and scalar statements, and all steps in logical order. Use the marking scheme that follows as a checklist to complete your derivation correctly. Note “g” is the magnitude of the gravitational field strength of the earth in N/kg. Now solve for “g” in terms of Θ, T and L. 6. Showing all steps, use the derived formula for “g” to find the experimental measured magnitude of the earth's gravitational field strength to the correct number of significant digits. 7. Using the reading in question #2 as a guide, and showing all steps, find the percentage uncertainty in the measured value calculated. Would there be any reason why the percentage uncertainty would be higher? Explain. 8. The accepted value for the magnitude of the earth's gravitational field strength is 9.81 N/kg. Using the reading in question #2 as a guide, and showing the formula and substitutions, find the % error in the measured value of the magnitude of the gravitational field strength as determined by this experiment. 9. Use proportionality to explain how a decrease in Θ during the experiment might affect the value “g” ? (too high or too low) 10. Is the % error in the measured value within the percentage uncertainty calculated? Other than the factor examined in question #7, list at least two other significant factors as to why the experimental or measured value of “g” might not be equal to the accepted value. For each reason or factor listed, explain how this affects the measured value of “g” (too high or too low) For your experiment, how do these factors relate to the discrepancy between the % uncertainty and % error? 11. Why is the mass of the object not a factor in this experiment? How would the calculation of the measured value of g change if the string and object had similar masses? Page #3 Marking Scheme: 80 marks A. Cover page with name, due date and instructor's name: /2 B. Lab handout included with recorded measurements on p1: recorded measurement of L, R, and time/10 rev measuring instruments stated proper # significant digits /3 /3 /3 Questions and Calculations: 1. Calculate Θ... formula, sub, ans /3 2. Uncertainty in L, R % uncertainty in L, R with steps shown % uncertainty in Θ with steps shown /2 /4 /2 3. Period and calculation (correct sigs) Period uncertainty % period uncertainty and step shown /2 /1 /2 4. Explanation about time measurement in complete sentences. 5. FBD... |-----------|----------| 0 1 2 poor excellent real forces on diagram with arrows. acceleration direction and expressed as scalar XY plane, components drawn steps to find components expressed in terms of Θ Y-forces equation of motion, vector statement, scalar statement X-forces equation of motion, vector statement, scalar statement combining x and y equations to simplify for T simplified equation for g /3 /4 /1 6. Measured value of g (correct sigs)...formula, sub, ans /3 7. % uncertainty in g with steps shown Explanation about higher % uncertainty in complete sentences. 8. % error calculation with formula, sub, ans 9. proportionality determined, arrow notation steps, conclusion /7 /4 /3 /2 |-----------|----------| 0 1 2 poor excellent /3 /6 10. % error statement two factors for ex error plus explanation in complete sentences how factors are relevant to difference in %error and %uncertainty in complete sentences 11. explanation for mass irrelevance in complete sentences explanation in complete sentences if mass of string ≈ mass of object /1 |----------|-----------| 0 2 4 poor excellent |-----------|-----------| 0 2 4 |------------|-----------| 0 1 2 |------------|-----------| 0 1 2