C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Investigation: Testing Conservation of Energy

C2.4 C2.3 A1.1 A1.5 A1.6 A1.8 A1.10 A1.13About 75 minutes

Learning goals

I can plan an investigation to test the law of conservation of energy, identifying variables and a hypothesis.
I can use measurements from a bouncing ball, a pendulum or a simulation to calculate how much mechanical energy is kept or transformed.
I can calculate percent difference and percent uncertainty and use them to judge my results.
I can write a conclusion that answers the question, uses evidence and discusses sources of error.

Words to know

hypothesis
A testable explanation or prediction, based on theory, made before an investigation.

independent variable
The variable the investigator deliberately changes.

dependent variable
The variable that is measured to see how it responds.

percent difference
|measured − predicted| ÷ predicted × 100%; compares a measurement with a prediction.

percent uncertainty
(uncertainty ÷ measured value) × 100%; shows how precise a measurement is.

Putting the law to the test

So far you have used the law of conservation of energy to make predictions. Scientists do not just trust a law; they test it, measure where energy goes, and decide whether the evidence supports it. This lesson walks you through three investigations the curriculum suggests: a bouncing ball, a simple pendulum and a bungee simulation. Along the way you will practise the inquiry skills of strand A: asking a testable question, controlling variables, recording data with units and uncertainty, analysing graphs and writing a conclusion.

Remember from Grade 11: in SPH3U you tested conservation of energy with a ramp and a pendulum. Here you will go further: you will measure how much energy is transformed at each step and compare your measurements with predictions.

Planning an energy investigation

Question: for example, "What fraction of a ball's mechanical energy is kept after each bounce, and does it depend on the drop height?"
Hypothesis and prediction: based on theory, such as "Mechanical energy is not conserved in a bounce, so the bounce height will be less than the drop height; if a constant fraction is lost, bounce height will be proportional to drop height."
Variables: independent (what you change: drop height), dependent (what you measure: bounce height), controlled (same ball, same floor, same release method, same temperature).
Method: repeat each measurement (at least three trials) and average.
Analysis: calculate energies, percentages and differences; draw a graph; look for a pattern.
Conclusion: answer the question, refer to the evidence, discuss sources of error and suggest improvements.

Useful calculations

Percentage of energy kept: (final mechanical energy ÷ initial mechanical energy) × 100%. For a bouncing ball, Eg is proportional to height, so this is just (bounce height ÷ drop height) × 100%.
Percent difference between a prediction and a measurement: |measured − predicted| ÷ predicted × 100%.
Percent uncertainty of a measurement: (uncertainty ÷ measured value) × 100%.

A ball dropped from 100 cm bounces to 72, 52, 37 and 27 cm. Each bounce height is about 72% of the one before, so about 28% of the mechanical energy becomes thermal energy and sound at each bounce.

Worked example

Example 1: Analysing bounce data

A tennis ball is dropped from 100 cm and its peaks are filmed: 72, 52, 37 and 27 cm. What fraction of its energy does it keep per bounce? Is that fraction constant?

At each peak the ball is momentarily at rest, so its mechanical energy is just Eg = mgh, proportional to h.
Ratios: 72 ÷ 100 = 0.72; 52 ÷ 72 = 0.72; 37 ÷ 52 = 0.71; 27 ÷ 37 = 0.73.
The ratios agree within the precision of the measurements (about ±1 cm), so the fraction is close to constant.

The ball keeps about 72% of its mechanical energy on each bounce; about 28% becomes thermal energy (the ball and floor warm very slightly) and sound.

Bounce height against drop height for several drops (sample data). The points lie close to a straight line through the origin, so the fraction of energy kept does not depend on the drop height in this range.

Worked example

Example 2: Testing a pendulum

A pendulum 1.00 m long is released from rest at 40°. Slow-motion video shows a speed of 2.05 m/s at the bottom. Compare with the prediction from conservation of energy.

Prediction: h = L(1 − cos 40°) = 1.00(1 − 0.7660) = 0.234 m. v = √(2gh) = √(2 × 9.8 × 0.234) = 2.14 m/s.
Percent difference: |2.05 − 2.14| ÷ 2.14 × 100% = 4.2%.
Energy kept: Ek ÷ Eg = (v_measured ÷ v_predicted)² = (2.05 ÷ 2.14)² = 0.92, so about 92% (the mass cancels).

The measured speed is about 4% lower than predicted, so about 8% of the energy was transformed by air resistance and friction at the pivot, or the measurement has error. A 4% difference is small enough that the evidence supports the conservation-of-energy model for a pendulum over one swing.

Worked example

Example 3: Uncertainty

A student measures a drop height of 100.0 cm with a metre stick read to ±0.5 cm, and a bounce height of 72 cm read from video to ±2 cm. Which measurement limits the result most?

Drop height: 0.5 ÷ 100.0 × 100% = 0.5%.
Bounce height: 2 ÷ 72 × 100% = 2.8%.
The percent uncertainty of the ratio is roughly the sum, about 3%, so the fraction kept is 0.72 ± 0.02.

The bounce height (read from video) limits the precision. Improvement: film from farther away, level with the peak, with a metre stick in view, to reduce parallax.

Simulations count as evidence too

The curriculum lets you test conservation of energy with a computer simulation, such as a bungee jump. Free simulations such as PhET's Masses and Springs and Energy Skate Park (University of Colorado Boulder) show bar graphs of kinetic, gravitational, elastic and thermal energy as the motion happens. To use one well: turn friction off first and check that the total stays constant, then turn it on and record how the thermal energy grows. Remember that a simulation tests the model built into it; only real measurements test nature.

Just for fun

Lab rule of thumb: if your data show a ball bouncing higher than the height you dropped it from, do not announce the end of the law of conservation of energy. First check whether someone threw it.

Hands-on activity

Investigation: an action-figure bungee jump

You need: a small action figure or doll (about 10 to 20 cm tall), 10 to 12 identical rubber bands, a tape measure, masking tape, a kitchen scale, a phone for slow-motion video, and a safe drop location such as a staircase railing or a table edge with a clear floor below (stay on the safe side of any railing; a grown-up should supervise any drop more than a metre high).

Question: how far does the figure fall for each number of rubber bands, and can you predict a safe number of bands for a bigger drop?

Weigh the figure. Loop the rubber bands into a chain (each new band through the last one). Tie one end around the figure's feet.
Hold the free end of the chain at the edge of a table, with a tape measure hanging down beside it.
Drop the figure from beside your hand with 2 bands. Film it and find the lowest point (the largest fall distance). Repeat 3 times.
Repeat with 3, 4, 5 and 6 bands.
Plot fall distance against number of bands. It should be close to a straight line.
Use your graph to predict how many bands give a fall that just misses the floor from a higher drop (for example 2.0 m), then test it carefully.
Energy check: hang the figure on 2 bands and measure the stretch to estimate k (k = mg ÷ x). Use mg(L + x) = ½kx² to predict one drop, and compare with your measurement.

What to look for: a straight-line graph of fall distance against number of bands, and predictions within about 10% of the test. Your energy check will usually predict a slightly larger fall than you measure, because rubber bands lose some energy to thermal energy and do not obey Hooke's law exactly. Writing that down, with reasons, is a good conclusion.

Your turn

A ball dropped from 1.20 m bounces to 0.84 m. What percentage of its mechanical energy did it keep? What percentage was transformed?

If the ball keeps the same fraction, predict the height of its next bounce.

A predicted speed is 2.14 m/s and the measured speed is 2.05 m/s. Find the percent difference.

In a bouncing-ball investigation, name the independent, dependent and two controlled variables.

Why should you repeat each measurement several times?

Predict the speed at the bottom for a pendulum 0.80 m long released from rest at 30°.

More practice: warm-up

What is a hypothesis?

What is a controlled variable?

Write the formula for percent difference between a measured and a predicted value.

At the top of a bounce, what kind of mechanical energy does the ball have?

Name a free simulation that can be used to test conservation of energy.

More practice: on your own

Drop heights of 50, 100, 150 and 200 cm give bounce heights of 36, 71, 108 and 142 cm. Find the fraction of energy kept and say whether it depends on drop height.

A 0.50 kg cart rolls down a ramp of height 0.30 m from rest and is measured at 2.2 m/s at the bottom. What percentage of its Eg became Ek? How much energy was transformed?

A 0.20 kg mass hangs from a spring (k = 40 N/m) at its unstretched length and is released from rest. It falls until the spring is stretched 9.8 cm, then rises. Does this support conservation of energy?

A height is measured as 100.0 cm ± 0.5 cm. What is the percent uncertainty in Eg calculated from it?

A pendulum keeps 92% of its energy in one swing in your test. Write a one-sentence conclusion.

Challenge

A ball keeps 72% of its energy at each bounce. If it is dropped from 1.00 m, how high is the fifth bounce? After how many bounces is the peak below 10 cm?

Show that the ratio of the speed just after a bounce to the speed just before is √(h₂/h₁). Find it for h₂/h₁ = 0.72.

Plan a simulation investigation to test conservation of energy for a bungee jump. Which energies would you record, at which moments, and what result would support the law?

Check your understanding

Quiz

A ball dropped from 2.0 m bounces to 1.5 m. The percentage of mechanical energy transformed in the bounce is:

In a pendulum investigation, the bob's speed at the bottom is less than predicted. The most likely reason is:

When testing whether bounce height depends on drop height, which is a controlled variable?

A predicted value is 4.00 m/s and the measured value is 3.80 m/s. The percent difference is:

Why do we use bounce height to find the energy kept, without needing the ball's mass or speed?

A computer simulation of a bungee jump with friction turned off shows a constant total energy. This result:

For parents and teachers

Big idea (C2.4, strand A): the law of conservation of energy is tested by measuring where energy goes. In a bouncing ball or a swinging pendulum, some mechanical energy becomes thermal energy and sound, but careful accounting shows it is not lost.

How to help: help with the action-figure bungee investigation (and supervise any high drop). Ask your teen to explain their graph, their prediction and why it was or was not exact.

Common difficulties:

Confusing percent kept with percent lost.
Not repeating trials, or changing two variables at once.
Writing a conclusion that just repeats the data instead of answering the question and discussing errors.
Reading heights from a video at an angle (parallax).