MeasurementMeasurement

UncertaintyUncertainty

A student measures a length of 50.0 cm with a A student measures a length of 50.0 cm with a meterstick divided with marks at each millimeter. The meterstick divided with marks at each millimeter. The uncertainty is aboutuncertainty is about

A) 1 cm.A) 1 cm. B) 5 mm.B) 5 mm. C) 0.5 %.C) 0.5 %. D) 0.2 %.D) 0.2 %. E) 0.02.E) 0.02.

How to MeasureHow to Measure

Measuring instruments are common.Measuring instruments are common.• RulerRuler• ClockClock• SpeedometerSpeedometer• ThermometerThermometer• Bathroom scaleBathroom scale

All instruments have a scale.All instruments have a scale.• Scale can be analog or digitalScale can be analog or digital• Instruments can have multiple scalesInstruments can have multiple scales

Analog ScalesAnalog Scales

Analog scales require Analog scales require interpolation and interpolation and rounding.rounding.

• Rounding when a value Rounding when a value is taken at the nearest is taken at the nearest tick marktick mark

• Interpolation when a Interpolation when a value is estimated value is estimated between two adjacent between two adjacent marksmarks

Apparent ShiftApparent Shift

A measurement device may not be at the location of A measurement device may not be at the location of the quantity being measured.the quantity being measured.• Change in observation pointChange in observation point• Change in resultsChange in results

This can be used to determine the position of the This can be used to determine the position of the observer relative to the observed point.observer relative to the observed point.

base

angle A

angle B

Observing ParallaxObserving Parallax

Observe an object against the background.Observe an object against the background. Shift one seat left and observe again.Shift one seat left and observe again. Subtract to get the parallax shift.Subtract to get the parallax shift.

0 5 10 15 20 25 30 35 40 45 50

GraphsGraphs

Both a recording tool Both a recording tool and measuring deviceand measuring device

• Keep track of Keep track of measurements as they measurements as they are recordedare recorded

• Estimate Estimate measurements from measurements from data on the graphdata on the graph

Graphs have two scalesGraphs have two scales

AccuracyAccuracy

The smallest unit on a measuring device sets the The smallest unit on a measuring device sets the accuracy.accuracy.

In general, a measurement is only as accurate as the In general, a measurement is only as accurate as the smallest unit.smallest unit.

Significant figures are a guide to the accuracy of a Significant figures are a guide to the accuracy of a measurement.measurement.

Significant FiguresSignificant Figures

Any value is expressed in some number of digits.Any value is expressed in some number of digits. The number of digits (without left side zeroes) is the The number of digits (without left side zeroes) is the

number of number of significant figuressignificant figures.. With no decimal point, skip right side zeroes. With no decimal point, skip right side zeroes.

• 3838 2 digits, 2 significant figures2 digits, 2 significant figures• 5.065.06 3 digits, 3 significant figures3 digits, 3 significant figures• 0.00410.0041 5 digits, 2 significant figures5 digits, 2 significant figures• 7,000.7,000. 4 digits, 4 significant figures4 digits, 4 significant figures• 2,0002,000 4 digits, 1 significant figure4 digits, 1 significant figure

Using Significant FiguresUsing Significant Figures

Add or Subtract Add or Subtract

Keep the significant figures Keep the significant figures to decimal place of the to decimal place of the leastleast accurate value, rounding as accurate value, rounding as needed.needed.

• 4.361 + 14.2 = 18.64.361 + 14.2 = 18.6

• 12000 + 364 = 1200012000 + 364 = 12000

Multiply or DivideMultiply or Divide

Keep the same number of Keep the same number of significant figures as the significant figures as the value with the fewest, value with the fewest, rounding as nedeed.rounding as nedeed.

• 4.361 4.361 14.2 = 61.9 14.2 = 61.9

• 12000 12000 364 = 4.4 364 = 4.4 10 1066

Absolute UncertaintyAbsolute Uncertainty

Measure 50.0 cm. Measure 50.0 cm.

There are three significant figures.There are three significant figures. The smallest figure suggests an The smallest figure suggests an

accuracy of 0.1 cm.accuracy of 0.1 cm. This is also equal to 1 mm.This is also equal to 1 mm.

The absolute uncertainty has the same type of units as the measurement.

Percent UncertaintyPercent Uncertainty

Measure 50.0 cm. Measure 50.0 cm.

Compare 0.1 cm to 50.0 cm.Compare 0.1 cm to 50.0 cm. The ratio is 0.1/50.0 = 0.002.The ratio is 0.1/50.0 = 0.002. Multiply by 100 % to get 0.2 %.Multiply by 100 % to get 0.2 %.

The percent uncertainty has no units, and is either a pure number or a percent.

next