Why Does a Small Bell Sound Higher Than a Big Bell?

A small bell rings high and a big bell rings low, even when both are made of the same metal in the same shape. So why does size change the note? In 2 minutes 40 seconds, rendered brass bells, a slowed-down model and a formula for a steel bar show what pitch is (how many times each second the metal vibrates, called frequency), how that differs from loudness (how far the metal moves), and why exact copies of one bell, with the wall thickness scaled too, vibrate half as often when they are twice as big (880 versus 440 vibrations a second in the picture). A steel bar 100 mm long and 2 mm thick vibrates about 163 times a second by the Euler-Bernoulli formula; the 200 mm by 4 mm copy about 82. Real bells vibrate in many ways at once, and size alone does not decide the pitch across different shapes or instruments. No science background needed. Pause at 0:35 and make your own guess before the answer. 0:00 A small bell and a big bell 0:50 Pitch is how often 1:21 Why size matters 1:53 Real bells are complicated 2:21 A question to chase HONEST LABELS Everything in the picture is a rendered illustration or a labelled model, not footage and not an experiment. The bells are exact copies of one shape (the large one twice the small one in every direction, thickness included); the frequencies 880 and 440 per second are illustrative values for a two-to-one size ratio. The short tones in the film are computed tones (a plain decaying sine wave at the stated frequency), not recordings of any bell, and are kept quiet. Real bells ring with many partials at once. The rim motion is a simplified model of a bell's lowest vibration pattern, greatly exaggerated and slowed down 440 times; the real motion is far too fast and too small to see. The rim graph is the same closed form that moves the bell, not a measurement. The trip picture (twice the size, twice the time) is simplified reasoning, not the full bell model. The steel-bar numbers are computed from a formula for an ideal thin bar held at one end (steel: 200 GPa, 7850 kg/m3), slowed down 80 or 160 times and exaggerated; a real bar would differ a little. The partial ratios in the chart (hum 0.500, prime 1.000, tierce 1.183, quint 1.506, nominal 2.000) are typical of a tuned church bell, not universal; the rule that frequencies scale with 1/size is for exact copies in one material, and real bells and instruments differ in shape, thickness and material. The supplied thumbnail is conceptual artwork, not experimental evidence. SOURCES Sathoff and Rossing, Scaling of bells (ASA 1993): https://auditory.org/asamtgs/asa93ott... Wikipedia, Euler-Bernoulli beam theory (cantilever frequency): https://en.wikipedia.org/wiki/Euler%E... Engineering Toolbox, Young's modulus of steel: https://www.engineeringtoolbox.com/yo... Engineering Toolbox, density of steel: https://www.engineeringtoolbox.com/me... Wikipedia, Strike tone (bell partials): https://en.wikipedia.org/wiki/Strike_... Wikipedia, Bell (instrument): https://en.wikipedia.org/wiki/Bell_(i...) Wikipedia, Pitch (music): https://en.wikipedia.org/wiki/Pitch_(...) Wikipedia, Loudness: https://en.wikipedia.org/wiki/Loudness OpenStax University Physics, sound waves: https://openstax.org/books/university... Rossing, bells (ASA 133rd meeting): https://acoustics.org/pressroom/httpd... More from the series:    • Science From Start   A question to chase: why can't sound cross empty space? Narration is a synthetic voice. The Fun is in finding out. #science #physics #sound