Measurement & regression

Physical pendulum

Time the plate alone and with a roller and measure its dimensions — then work out the moment of inertia in three ways.

Example: A 600 × 104 mm steel plate: 25T and 25T′, ten readings of each dimension → J₀ three ways.

Změřte periodu ocelové desky na excentrické ose, pak s válcem položeným na desce, a změřte rozměry. Simulace provede celé měření, moment setrvačnosti J třemi způsoby spočítáte v kalkulačkách níže.Time a steel plate on an off-centre axis, then with a roller on it, and measure the dimensions. The simulator runs the whole measurement; you work out the moment of inertia J in three ways in the calculators below.

Apparatusnot to scale

Table · successive differenceslap every 5 oscillations

0.00s

Plate alone · T

n	n·T / s	n+25	(n+25)·T / s	25T / s

Plate with roller · T′

n	n·T′ / s	n+25	(n+25)·T′ / s	25T′ / s

Angleθ against time, dots are the counted passes through zero

How J₀ from method 2 depends on T′the other quantities held at their measured values

Table · dimensions10 readings each, in mm

n	a / mm	b / mm	c / mm	d / mm	L / mm

Průměry a chyby spočítá kalkulačka Více veličin (a, b, c).The means and errors are worked out by the Several quantities calculator.

Presets

Standard valuesdensity and moment of inertia

Materiál desky a válcePlate and roller material	ρ / kg·m^-3	ρ / g·cm^-3
Ocel (konstrukční)Steel (structural)	78507850	7,857.85
Nerezová ocelStainless steel	79007900	7,97.9
LitinaCast iron	72007200	7,27.2
MěďCopper	89608960	8,968.96
MosazBrass	8400 – 87008400 – 8700	8,4 – 8,78.4 – 8.7
BronzBronze	88008800	8,88.8
HliníkAluminium	27002700	2,72.7
TitanTitanium	45004500	4,54.5
OlovoLead	1134011340	11,3411.34

Orientační tabulkové hustoty. Hmotnost desky M = ρ·b·c·t (t je tloušťka), díry ji snižují.Typical handbook densities. The mass of a plate is M = ρ·b·c·t (t is the thickness); holes lower it.

Momenty setrvačnosti a vzorceMoments of inertia and formulas

Těleso / situaceBody / situation	VzorecFormula
Obdélníková deska b × c, osa kolmá k desce středemRectangular plate b × c, axis perpendicular through the centre	J_E = M·(b² + c²) / 12
Tenká tyč délky l, osa kolmá středemThin rod of length l, axis perpendicular through the centre	J_E = M·l² / 12
Válec nebo kotouč poloměru r, osa válceCylinder or disc of radius r, about its own axis	J_E = M·r² / 2
Koule poloměru r, osa středemSphere of radius r, axis through the centre	J_E = 2·M·r² / 5
Steinerova věta, osa ve vzdálenosti a od těžištěParallel-axis theorem, axis at distance a from the centre of mass	J = J_E + M·a²
Fyzické kyvadlo, malé kmityPhysical pendulum, small swings	T = 2π√(J / (M·g·a))
Tíhové zrychlení použité v simulaciGravitational acceleration used in the simulation	g = 9,81 m·s^-2g = 9.81 m·s^-2

Calculate your result

Rozměry a, b, c, d, L zpracujte v kalkulačce Více veličin, rozdíly 25T a 25T′ v Pravděpodobné chybě. Z period a rozměrů spočítejte J₀ třemi způsoby podle vzorců nahoře a porovnejte je.Work up the dimensions a, b, c, d, L in the Several quantities calculator, and the 25T and 25T′ differences in Probable error. Then calculate J₀ in the three ways from the formulas above and compare them.

Several quantities Probable error

Key equations

(141)

T = 2 π \sqrt{\frac{J}{M g a}} \Rightarrow J = \frac{g}{4 π^{2}} a M T^{2}

Method 1 — plate alone

(142)

J_{p} = \frac{1}{8} m d^{2}

added cylinder about its own axis

(143)

J_{op} = J_{p} + m L^{2}

Steiner’s theorem

(145)

J = \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} m [\frac{g L {\overline{T}}^{2}}{4 π^{2}} - \frac{d^{2}}{8} - L^{2}]

Method 2 — main result, with the added body

(146)

J = \frac{1}{12} M (b^{2} + c^{2}) + M a^{2}

Method 3 — direct calculation for the rectangular plate

Parameters

SymbolSymbol	VýznamMeaning	JednotkaUnit
$J$	moment setrvačnosti desky kolem osy O (J₀)moment of inertia of the plate about pivot O (J₀)	kg·m²
$M$	hmotnost deskymass of the plate	kg
$m$	hmotnost přidaného válcemass of the added cylinder	kg
$a$	vzdálenost osy O od těžištědistance pivot O – centre of mass	m
$b, c$	délka a šířka deskyplate length and width	m
$d$	průměr přidaného válcediameter of the added cylinder	m
$L$	vzdálenost osy od přidaného válcedistance pivot – added cylinder	m
$T$	perioda bez přidaného tělesaperiod without the added body	s
$\overline{T}$	perioda s přidaným tělesemperiod with the added body	s
$g$	tíhové zrychlenígravitational acceleration	m·s^-2

Error calculationformulas without values

Metoda 1 · z periody samotné deskyMethod 1 · From the period of the plate alone

J = \frac{g}{4 π^{2}} a M T^{2}

ϑ (J) = \sqrt{{(\frac{\partial J}{\partial a} ϑ (a))}^{2} + {(\frac{\partial J}{\partial M} ϑ (M))}^{2}}

Partial derivatives

\frac{\partial J}{\partial a} = \frac{g}{4 π^{2}} M T^{2}

\frac{\partial J}{\partial M} = \frac{g}{4 π^{2}} a T^{2}

Metoda 3 · přímý výpočet (obdélníková deska)Method 3 · Direct calculation (rectangular plate)

J = \frac{1}{12} M (b^{2} + c^{2}) + M a^{2}

ϑ (J) = \sqrt{{(\frac{\partial J}{\partial a} ϑ (a))}^{2} + {(\frac{\partial J}{\partial b} ϑ (b))}^{2} + {(\frac{\partial J}{\partial c} ϑ (c))}^{2} + {(\frac{\partial J}{\partial M} ϑ (M))}^{2}}

Partial derivatives

\frac{\partial J}{\partial a} = 2 M a

\frac{\partial J}{\partial b} = \frac{1}{6} M b

\frac{\partial J}{\partial c} = \frac{1}{6} M c

\frac{\partial J}{\partial M} = \frac{1}{12} (b^{2} + c^{2}) + a^{2}

Metoda 2 · s přidaným tělesem (hlavní výsledek)Method 2 · With the added body (main result)

J = \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} m [\frac{g L {\overline{T}}^{2}}{4 π^{2}} - \frac{d^{2}}{8} - L^{2}]

ϑ (J) = \sqrt{{(\frac{\partial J}{\partial m} ϑ (m))}^{2} + {(\frac{\partial J}{\partial L} ϑ (L))}^{2} + {(\frac{\partial J}{\partial d} ϑ (d))}^{2}}

Partial derivatives

\frac{\partial J}{\partial m} = \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} [\frac{g L {\overline{T}}^{2}}{4 π^{2}} - \frac{d^{2}}{8} - L^{2}]

\frac{\partial J}{\partial L} = \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} m [\frac{g {\overline{T}}^{2}}{4 π^{2}} - 2 L]

\frac{\partial J}{\partial d} = \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} m (- \frac{2 d}{8}) = - \frac{T^{2}}{T^{2} - {\overline{T}}^{2}} \frac{m d}{4}