Virtual laboratory

On measurement

Measurement is the act of comparing a quantity with a standard unit. Every instrument has a smallest readable division, and a careful reading includes one estimated digit beyond that division.

A meter rule marked in millimeters can be read reliably to about 0.5 mm, with the final digit estimated by eye. A stopwatch made for laboratory use may show hundredths of a second, but human reaction time of around 0.2 seconds often limits the true precision of a human-timed interval more than the display does. For this reason we often time many oscillations and divide by the count, rather than timing a single swing.

A measurement is meaningless without its unit and its uncertainty. A result reported as 0.92 seconds carries an implied precision, but a proper record adds an uncertainty, such as 0.92 plus or minus 0.02 seconds. Only then can two results be compared fairly.

Precision and accuracy

Precision and accuracy describe different things, and confusing them is a common beginner error.

Precision is how closely repeated measurements agree with one another. Accuracy is how close a measurement comes to the true value. A set of readings can be very precise and still be inaccurate if the instrument is mis-calibrated, or accurate on average while scattered widely. Good laboratory practice aims for both: check the instrument, take several readings, and report the spread.

Systematic errors shift all readings the same way and do not shrink when you repeat a measurement. Random errors vary unpredictably and can be reduced by averaging more trials. A good experiment write-up names which is which.

Projectile trajectory plot

Launching a projectile at 40 degrees with an initial speed of 20 m/s on level ground. The launch angle is measured from the horizontal. The horizontal component stays constant while the vertical component is accelerated downward by g.

Series circuit with a voltmeter

A battery drives current through two resistors in series. A voltmeter placed across R2 reads the potential difference V2 = I*R2, which always adds with V1 to give the battery voltage.

Experiment: measuring g with a pendulum

A small bob on a light string makes an almost ideal simple pendulum for small angles. Its period depends on the length L and the local gravitational acceleration g.

Method

Suspend a small heavy bob on a string from a clamp. Measure the length L from the point of support to the center of the bob using a meter rule. Displace the bob to a small angle, no more than about 15 degrees, release it, and time 20 complete oscillations. Divide the total time by 20 to get the period T. Repeat for several lengths.

Theory

For small angles the period of a simple pendulum is T = 2*pi*sqrt(L/g), where pi is the ratio of a circle's circumference to its diameter. Squaring both sides gives T^2 = 4*pi^2*L/g, so a graph of T^2 against L is a straight line through the origin with slope 4*pi^2/g. From the slope you can recover g.

Sample data

Using the longest row, T = 2.01 s at L = 1.00 m, the formula gives g = 4*pi^2*L/T^2 = 4*(3.1416)^2*(1.00)/(4.04), which is about 9.77 m/s^2. That value sits close to the standard 9.81 m/s^2 and differs because of the small measurement errors and the finite angle used.

Sources of error

• Reaction time when starting and stopping the stopwatch, reduced by timing many swings.
• Difficulty measuring length exactly to the center of the bob.
• The pendulum is not perfectly simple: the string has mass and the bob is not a point mass.
• Dragging the string through air and friction at the pivot slightly slow the motion.
• Using a large release angle makes the small-angle approximation less accurate.