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Physics Practicum: Measurement, Error Bounds and Review

In a practicum we measure some quantities directly (length, time) and get others indirectly from a formula (g = 4π²L/T²). Every reading has an error bound, set by the instrument and by repeating. Errors combine: relative errors add for products and quotients, and powers multiply the error (T² counts twice). We test a hypothesis by plotting a straight-line graph (T² against L) and checking if the result agrees with the expected value within the error. The same habits help to review the whole physics course.

🎬 Step-by-step story

  1. We want g, the pull of gravity, but no instrument shows g. We can measure the string length L with a ruler. This is a direct measurement.
  2. The bob swings. We time 10 swings with a stopwatch and divide by 10 to get one period T. Timing 10 swings makes the small start and stop errors smaller.
  3. We repeat the timing 5 times. The blue dots are the g values from each trial. They are not equal. This spread is the random error.
  4. Now the indirect measurement. We put L and the mean T into g = 4π²L/T². The green line is our mean g.
  5. The green band is the error bound. The red line is the accepted value 9.81. If red sits inside the band, our result agrees with it.
  6. Free play. Now we test the idea T² is proportional to L. Move L: the purple points lie on a straight line. Press New trial set to go back to the g dots.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why can we not measure g directly with a ruler?

g is a rate, not a length. It must be found from other quantities that we can read, like L and T.

Why divide the time of 10 swings by 10?

The start and stop error of the stopwatch is about the same for 1 or 10 swings. Spread over 10 swings it is 10 times smaller in each T.

Why are the repeated g values different?

Reaction time and small reading differences change each trial a little. This random error is the spread of the dots.

Which measurement matters more for g, L or T?

T, because it is squared and so counts twice in the error. Improve T first.

What does "agrees within error" mean?

The accepted value falls inside the green band, i.e. the difference is smaller than the error bound.

Why use a graph instead of one reading?

Many points on one line show the rule T² ∝ L is true and average out the errors. The slope gives g.

Direct and indirect measurement

A direct measurement reads the quantity on an instrument: length with a ruler, time with a stopwatch, mass on a balance. An indirect measurement finds a quantity from a formula using direct readings. g is found from L and T, density from mass and volume, resistance from V and I.

The least count is the smallest division of an instrument (a ruler: 1 mm; a stopwatch: 0.01 s or 0.1 s). Choose an instrument with a small least count for the quantity you need.

Planning and doing a practicum

Plan like this: (1) write the aim and the formula, (2) list the direct readings, (3) choose instruments, (4) keep other things the same, (5) take readings in a table, (6) repeat at least 3 to 5 times, (7) calculate and add the error, (8) write the result with its error and a conclusion.

For the pendulum: use a thin thread, a small heavy bob, a small swing angle (under 10°), measure L from the pivot to the centre of the bob, time 10 to 20 swings, and start counting when the bob passes the middle.

Error bounds

No reading is exact. The absolute error Δx is the likely size of the doubt. It comes from the instrument (least count), from your reaction time, and from the spread of repeated readings (use half of max − min, or the standard deviation). The relative error is Δx/x and the percentage error is Δx/x × 100.

Combining: for sums and differences add the absolute errors. For products and quotients add the relative errors. A power counts as many times as the power: for g = 4π²L/T², Δg/g = ΔL/L + 2ΔT/T. The biggest term decides the quality, so improve that measurement first.

Graphs and hypothesis testing

A hypothesis is a clear guess that can fail. Here: T² ∝ L. Take readings at several L, plot T² (y) against L (x). If the points lie close to a straight line through the origin, the hypothesis is supported. The slope is 4π²/g, so g = 4π² ÷ slope.

To check an accepted value, compare the difference with the error bound. If |your value − accepted| is smaller than your error bound, the result agrees. If it is much larger, look for a mistake or a hidden effect (like a heavy thread or a big swing angle) before blaming the theory.

Reviewing the whole physics course

Review physics as a map, not a list. Mechanics (motion, force, energy), heat (molecules and temperature), electricity and magnetism (charges, fields, circuits, waves), waves and optics, atoms and quanta, and the Universe. Three threads join them: conservation laws (energy, momentum, charge), models and where they fail, and measurement with units and errors.

Good revision habit: for every chapter write the one main law, its units, one experiment, and one real-life use. Check that units match on both sides of each formula. Practise with mixed questions, not chapter by chapter.

Try it

Hang a small weight on a thread of length 1.00 m. Time 10 swings with a phone stopwatch, 5 times. Find the mean T and g. Change the length to 0.50 m and repeat. Does T² fall to half? In the 3D, press New trial set a few times and watch how the dots move but the mean stays close to 9.8.

Key formulas and definitions

Worked examples

1. The mean time for 10 swings is 17.9 s and L = 0.800 m. Find T and g.

T = 17.9 ÷ 10 = 1.79 s. g = 4π²L/T² = 39.48 × 0.800 / 3.204 = 9.86 m/s².

2. Five values of T are 1.80, 1.78, 1.82, 1.79, 1.81 s. Find the mean and the error from the spread.

Mean = 9.00 ÷ 5 = 1.80 s. Spread: (1.82 − 1.78) ÷ 2 = 0.02 s. So T = 1.80 ± 0.02 s.

3. A cube has mass 54.0 ± 0.1 g and side 3.00 ± 0.01 cm. Find its density with error.

ρ = 54.0 / 27.0 = 2.00 g/cm³. Relative error = 0.1/54.0 + 3 × 0.01/3.00 = 0.19% + 1.0% = 1.2%. Δρ = 0.024, so ρ = 2.00 ± 0.02 g/cm³.

4. In the pendulum, L = 0.800 ± 0.002 m, T = 1.79 ± 0.02 s. Find the percentage error in g.

ΔL/L = 0.25%; ΔT/T = 1.1%, counted twice = 2.2%. Δg/g = 0.25 + 2.2 = 2.5%. With g = 9.86, Δg = 0.25.

5. V = 6.0 ± 0.1 V and I = 0.50 ± 0.01 A. Find R = V/I with its error.

R = 12 Ω. Relative error = 0.1/6.0 + 0.01/0.50 = 1.7% + 2.0% = 3.7%. ΔR = 0.44, so R = 12.0 ± 0.4 Ω.

6. The T²–L graph has slope 4.00 s²/m. Find g, and say if it agrees with 9.81 ± 0.2.

g = 4π²/slope = 39.48 / 4.00 = 9.87 m/s². The difference from 9.81 is 0.06, smaller than 0.2, so it agrees.

Common mistakes

Practice quiz

1. Which is an indirect measurement?
2. For g = 4π²L/T², the percentage error in g is:
3. Why time 10 swings instead of 1?
4. T² plotted against L should give:
5. A result agrees with the accepted value when:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

How do I find g with a simple pendulum?

Measure L and the time for 10 swings, get T, and use g = 4π²L/T². Repeat for several lengths and use the slope of the T²–L graph for the best result.

What is the difference between error and mistake?

An error is the unavoidable doubt in a reading. A mistake is a wrong action, like reading the scale wrongly. Errors are reported; mistakes are removed and the measurement is repeated.

How do I review the whole physics course quickly?

Make a one-page map: each chapter gets one law, its units, one experiment and one real use. Then practise mixed questions and check units in every answer.

Learn first

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