A pendulum is the cleanest place to watch energy change form. Take three moments in one swing:
Top: the bob is held still at height h₁. It has PE = mgh₁ and no KE.
Bottom: the height is zero, so PE is zero. If nothing were lost, all of that energy would now be KE = ½mv².
Far side: the bob climbs, slows, and stops for an instant at height h₂. It’s at rest again, so KE is zero and all of its energy is PE.
🔑In a perfect pendulum, h₂ would equal h₁ and the bob would swing forever. A real one comes back a little short every time. That shortfall is the energy you’ll be tracking.
BConcept
Why the String Does No Work
Two forces act on the bob: gravity pulls it down, and the string’s tension pulls it toward the pivot. In Activity 4-1-1 you learned that a force at 90° to the motion does no work. The bob moves along a circular arc, and the string always points along the radius of that circle, which is at 90° to the arc.
W_tension = T × d × cos 90° = 0
That leaves gravity as the only force doing work. By the work-energy theorem, gravity’s work equals the change in KE, and from Activity 4-1-3 gravity’s work is the negative of the change in PE. Put those together and PE lost equals KE gained. Everything you track in this project comes from that one result.
CMath
Finding the Release Height
For PE you need the height above the lowest point. That’s the zero level idea from Activity 4-1-3, with zero placed at the bottom of the swing. You can measure this directly with a meter stick, which is what the project asks. You can also get it from the string length L and the release angle θ:
h = L × (1 − cos θ)
When the string hangs straight down, the bob is L below the pivot. When it’s pulled to angle θ, the bob is only L × cos θ below the pivot. The difference is how far it was lifted.
Release angle θ
Height h for L = 1.00 m
20°
0.060 m
30°
0.134 m
37°
0.201 m
45°
0.293 m
60°
0.500 m
💡Notice that tripling the angle from 20° to 60° raises the bob by more than eight times (0.060 m to 0.500 m). Height isn’t proportional to angle. Your 0.20 m release is about 37°.
DExample
Pendulum Explorer
Drag the bob through the swing and watch the cyan PE and orange KE trade places. Then change the mass. Do the speeds change?
This is a perfect pendulum with no energy loss. Set the release angle, then slide the bob through the swing and watch the energy trade places. Total energy never changes.
Release height h₁ = 1.00 × (1 − cos 37°) = 0.201 m Total energy mgh₁ = 0.493 J (constant)
PE 0.493 JKE 0.000 J
Speed here0.00 m/s
EMath
Measuring Speed from Slow-Motion Video
To get the bob’s speed at the bottom you need a distance and a time. The distance is the 0.20 m gate. The time comes from counting video frames. A video records a fixed number of frames per second (fps), so the time between two frames is 1 ÷ fps.
t = frames ÷ fps v = 0.20 m ÷ t
The catch is that you can only count whole frames. If you’re off by one frame, the time is off by 1 ÷ (number of frames). A faster camera gives more frames in the same gate, so that one-frame error shrinks. For a bob moving at about 2 m/s, the gate takes 0.10 s:
Camera
Frames in the gate
Error from ±1 frame
30 fps
3 frames
±33%
60 fps
6 frames
±17%
120 fps
12 frames
±8%
240 fps
24 frames
±4%
⚠️KE depends on v², so a speed error shows up doubled in your energy. A 4% speed error is about an 8% error in KE. This is why regular 30 fps video isn’t good enough for this project.
Four habits keep the measurement honest:
Film straight on at the bob’s height, from far away and zoomed in. Filming from above or below makes the bob appear to cross the marks early or late.
Put the gate marks in the same plane as the bob’s path, not on the wall far behind it.
Always track the same point on the bob (its center or its leading edge) at both marks.
Do three runs and average the frame counts.
FExample
Frame Speed Calculator
Try 30 fps first and see how wide the range gets. Then switch to 240 fps.
A 0.250 kg bob is released from 0.20 m and crosses a 0.20 m gate. Choose a camera speed and a frame count. A perfect, loss-free swing would take about 24 frames at 240 fps. The calculator shows what a one-frame counting error does to your answer.
Camera speed:
t = 25 ÷ 240 = 0.104 s v = 0.20 ÷ 0.104 = 1.92 m/s
PE at top = 0.490 JKE at bottom = 0.461 J94% became KE
One frame is 4% of this time. Counting one frame too many or too few moves the answer between 87% and 102%.
This range includes 100%, so your measurement can’t tell whether any energy was lost. Use a faster camera or a wider gate.
GConcept
Energy Bar Charts and Thermal Energy
An energy bar chart shows how the energy is split at one moment. Every bar chart in this project uses percent of the starting energy, so the total at each moment is 100%. There are three kinds of energy:
■ Potential■ Kinetic■ Thermal
Potential (PE): energy stored in height.
Kinetic (KE): energy of the bob’s motion.
Thermal: energy of the random jiggling of atoms and molecules in the air, string, bob, and pivot. When some of the swing’s energy gets spread into this motion, it’s still energy, but it can’t be gathered back into the swing.
Here is the pattern for a real pendulum. Look at how the thermal slice only grows:
Top of the swingPE 100% · KE 0% · thermal 0%
Bottom of the swingPE 0% · KE 94% · thermal 6%
Far side, top of the first swingPE 92.5% · KE 0% · thermal 7.5%
🔑Three rules for a correct chart. Each moment adds to 100%. PE is 0 at the bottom (zero height). KE is 0 at the far-side peak (the bob is at rest for an instant). And the thermal slice can only grow, because heat doesn’t turn back into swinging by itself.
HConcept
Where the Missing Energy Goes
The missing energy isn’t destroyed. It’s moved into thermal energy in places you can’t easily see. Here are the usual suspects:
Where
What happens
How you could test it
Air drag
The bob pushes air aside, and that air ends up swirling and slightly warmer.
Tape a card to the bob: flat to the air vs. edge-on. A flat card should return less height.
Pivot friction
The string rubs where it’s attached, warming the pivot a tiny bit on every swing.
Compare a loose, rubbing pivot with a smoother one. The smoother pivot should return more height.
Bob and string flexing
The string stretches slightly and the bob wobbles, which warms both.
Watch the video. A bob that twists or bounces is wasting energy as it moves.
Sound
Tiny clicks and air noise carry off a little energy.
Too small to measure here, but it’s part of the full ledger.
💡A heavier bob usually returns closer to its starting height. Pushing the air aside takes roughly the same energy either way, but a heavier bob started with more energy, so the share it loses is smaller.
In Activity 4-2-2 you’ll study this thermal slice head-on. Here, your job is to show that it exists and to find at least one place it goes.
IMath
Worked Example: A Full Energy Ledger
One release from 0.20 m
Given
Bob mass m = 0.250 kg
Release height h₁ = 0.20 m
Camera at 240 fps; average of 25 frames in the 0.20 m gate
Far-side height h₂ = 0.185 m
Solve
PE_top = m g h₁ = 0.250 × 9.8 × 0.20 = 0.490 J
t = frames ÷ fps = 25 ÷ 240 = 0.104 s
v = 0.20 ÷ 0.104 = 1.92 m/s
KE_bottom = ½ m v² = ½ (0.250)(1.92)² = 0.461 J
% KE = 0.461 ÷ 0.490 = 94% thermal so far = 6%
PE_far = m g h₂ = 0.250 × 9.8 × 0.185 = 0.453 J
% returned = h₂ ÷ h₁ = 0.185 ÷ 0.20 = 92.5% thermal so far = 7.5%
✓ Energy ledger: top 100% PE. Bottom 94% KE and 6% thermal. Far side 92.5% PE and 7.5% thermal.
JConcept
Quick Check
Four questions. They are not graded. Answer them to find out what still needs work.
Q1How much work does the string’s tension do on a swinging bob?
Q2A camera at 30 fps records a bob crossing a 0.20 m gate in 3 frames. What’s the problem with this measurement?
Q3A bob released from 0.20 m reaches 0.19 m on the far side. What fraction of its PE came back as height?
Q4A group’s table says the KE at the bottom is 108% of the PE at the top. What’s the best conclusion?
Score: 0 of 4
KConcept
Looking Ahead
You now have the tools to run the project and make sense of the numbers. After it, the unit shifts from how much energy to how fast it moves, and then to a closer look at friction.