## Friday, February 24, 2017

Executive summary:
================
E max lost to sound: 0.5% per 200 clacks (100 cycles)
E max loss to air drag: 1.7% per 100 cycles
E lost to string: 0.8% per cycle (not measured carefully)
E lost to heat in balls and strings: 3% per 1 clack

Variables: 5 cm drop (1 m/s),  44 g steel balls 1.1 cm radius.
12 cm high strings.
Observation: about 0.5 cm distance sideways after 200 clacks (100 swings)
This is 12*cos(asin(0.5/12)) = 0.01 cm height from initial 5 cm = 0.2% remaining energy
(not measured carefully)
Two balls were dropped simultaneously from both sides, so it's 0.1% of initial energy
E=mgh * 2 = 44 mJ initial energy both balls.

max Energy lost in sound, 0.5% per 200 clacks:
====== proof =======:
Assume worst case, it's like noisy office at 5 meters for 8 ms per clack.
10E-6 W/m^2 * 4pi 5^2 * 0.008 secs= 2.5 uJ per clack
For 200 clacks assume max loud for 100 clacks, 0.250 mJ
Max energy in sound, 0.25/44 = 0.5%

max Energy loss to air drag, 3.41% per 200 clacks :
======= proof  =========
F=1/2 * C * density * Area * v^2
C=0.5 sphere
air 1.2 kg/m^3
v = 0.75 m/s for 5 cm drop in first few clacks
Area= pi * 0.011^2
F = 62 uN each ball
12 cm string  height 0.68 secs per 2 clacks.
5 cm drop => ~12 cm travel per 1 clack
E=Fd = 62 uN * 0.12 * 100 clacks (100 as a high average even for 200 clacks) = 0.74 mJ per 200 clacks max
Max energy in air friction  = 0.74/44 =1.7% per 100 cycles absolute max

Can I measure the heat increase with an infrared thermometer?
Steel heat capacity: 2.2 Celsius * gr/J * 0.022 J /  44 g = 0.001 C increase if all energy lost to heat in 1 ball in 1st clack.  Can't measure it.

Energy loss in string resistance 0.8% per cycle,
====== proof =======
Measured 80% energy loss after 100 complete swings of ball on the string (no clacks).
(1-0.80)^(1/200) = 99.2% => 0.8% loss in string per cycle (2 clacks when in cradle).

Energy loss from all sources, measured 3.4% per clack
===================
Observed about 200 max clacks (100 cycles) before stopping =>
0.001^(1/200) = 0.966 energy retained after each clack. ( 0.966^200 = 0.001)

Energy lost to heat in balls (or in strings due to off-center rotational effects): 3.4%-0.8%/2 - 1.7%/200 - 0.5/200=3% per clack
=================

96% energy retained is what I measured for steel on glass block, very close to this 96.6%, but that is without a string, possibly offsetting the error of not bouncing steel on steel.

Liquid metal seemed to be 1/2 the height after 44 bounces.  0.5^(1/44) = 98.4% retained energy per bounce.