LEP Moments of inertia of different bodies / Steiner’s theorem 1.3.28 -01 Related topics Tasks Rigid body, moment of inertia, centre of gravity, axis of rota- 1. Determination of the angular restoring constant of the tion, torsional vibration, spring constant, angular restoring spiral spring. force. 2. Determination of the moment of inertia of a circular disc as a function of the perpendicular distance of the axis of rota- Principle tion from the centre of gravity. The period of vibration of a circular disc which performs tor- sional vibrations about various parallel axes, is measured. The Set-up and procedure moment of inertia of the disc is determined as a function of the The experimental set-up is arranged as shown in Fig. 1. In perpendicular distance of the axis of rotation from the centre order to measure the angular restoring factor, the disc is fixed of gravity. on the torsion axis at its centre of gravity. With the spring bal- ance, which acts in a hole in the disc, the force needed to deflect the disc through a given angle is measured. When Equipment doing this, the lever arm forms a right angle with the spring Rotation axle 02415.01 1 balance. It is convenient to select an angle of 180°, since the Disk, w. diametrical holes 02415.07 1 row of holes can thus be used as a “protractor”. Spring balance, transparent, 2 N 03065.03 1 Light barrier with Counter 11207.30 1 For measuring the vibration period of the disc, a mask (width ≤ Power supply 5 V DC/2.4 A 11076.99 1 3 mm) is stuck on, on the line of the row of holes. The light Tripod base -PASS- 02002.55 1 barrier is pushed over this mask with the disc in its position of Barrel base -PASS- 02006.55 1 rest. Rule, plastic, l = 200 mm 09937.01 1 Switch the light barrier to -mode. Fig. 1: Experimental set-up for measuring the moment of inertia (Steiner’s theorem). PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 21328-01 1 LEP 1.3.28 Moments of inertia of different bodies / Steiner’s theorem -01 The disc is deflected through about 180° and the half-cycle Fig. 2: Moment (torque) of a spiral spring as a function of the time of the vibration is measured with the counter, anticlock- angle of rotation. wise and clockwise measurements being averaged. For safe- ty and stability reasons, it is recommended that the spring should not be twisted beyond ±720°. Theory and evaluation S The relationship between the angular momentum L of a rigid body in a stationary coordinate systemS with its origin at the centre of gravity, and the moment T acting on it, is S d S T ϭ L . (1) dt The angular momentum is expressed by the angular velocity S ˆ v and the inertia tensor I from S ˆ S L ϭ I ᭪ v , that is, the reduction of the tensor with the vector. S In the present case,v hasS the direction of a principal iner- tia axis (z-axis), so that L has only one component: LZ = IZ · v, where IZ is the z-component of the principal inertia tensor of the plate. For this case, equation (1) reads: v 2f ϭ d ϭ d TZ IZ IZ . dt dt where f is the angle of rotation. In the Hooke’s law range, the moment of a spiral spring is: If r (x, y, z) is the density distribution of the body, the moment TZ = -D · f (2) of inertia IZ is obtained as where D is the angular restoring constant. ∫∫∫ 2 2 IZ = (x + y ) r (x, y, z) dx dy dz. From the regression line to the measured values of Fig. 2 with the linear statement The origin of coordinates is located at the centre of gravity. The same component of the inertia tensor, in relation to a Y = A · BX coordinate origin displaced by a, is the slope 2 IZ’=IZ m a , B = 0.0255 Nm/rad. (see (2)) where m is the mass of the body. The angular restoring factor, from (2) is Therefore 4p2 = 0.0255 Nm/rad. 2 ϭ ϩ 2 D T · IZ m a (3) D 1 2 The equation of motion reads: From the regression line to the measured values of Fig. 3 and the linear statement 2f d ϩ D f = 0. dt IZ Y = A · BX The period and frequency of this vibration are: one obtains 2 2 IZ A = 6.86 s ± 0.15 s (see (3)) T ϭ 2p BD and from this the moment. of inertia of the disc with the axis of rotation through the centre of gravity 1 ϭ D f –3 2 2p B IZ IZ = 4.52 · 10 kgm . 2 21328-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen LEP Moments of inertia of different bodies / Steiner’s theorem 1.3.28 -01 Fig. 3: Vibration period of a disc as a function of the perpen- dicular distance of the axis of rotation from the centre of gravity. PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 21328-01 3 LEP 1.3.28 Moments of inertia of different bodies / Steiner’s theorem -01 4 21328-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen.