Moment of inertia and torsional vibrations (Item No.: P )

Transcription

1 Moent of inertia and torsional vibrations (Ite No.: P ) Curricular Relevance Area of Expertise: Physics Education Level: University Topic: Mechanics Subtopic: Static Equilibriu and Elasticity Experient: Moent of inertia and torsional vibrations Difficulty Preparation Tie Execution Tie Recoended Group Size Difficult 10 Minutes 20 Minutes 1 Student Additional Requireents: Experient Variations: Keywords: Rigid body, oent of inertia, axis of rotation, torsional vibration, spring constant, angular restoring oent, oent of inertia of a sphere, oent of inertia of a disc, oent of inertia of a cylinder, oent of inertia of a long bar, oent of inertia of 2 point asses Introduction Overview Various bodies perfor torsional vibrations about axes through their centres of gravity. The vibration period is easured and the oent of inertia deterined fro this. Fig. 1: Experiental set-up for deterining oents of inertia of various bodies.

2 Equipent Tasks The following will be deterined: Position No. Material Order No. Quantity 1 Rotation axle Sphere Disk Hollow cylinder Solid cylinder Rod with ovable asses Spring balance 2,5 N Light barrier with counter Power supply 5 V DC/2.4 A with 4 plugs Tripod base PHYWE Barrel base PHYWE The angular restoring oent of the spiral spring. 2. The oent of inertia of a disc, two cylinder, a sphere and a bar, of two point asses, as a function of the perpendicular distance to the axis of rotation. The centre of gravity lies in the axis of rotation.

3 Set-up and procedure The experiental set-up is arranged as shown in Fig. 1. To deterine the angular restoring oent, the bar is claped in the torsion spindle and the two asses fixed syetrically at a defined distance fro the axis of rotation. With the spring balance, the bar is rotated by 180 about the axis in each case and the force is easured. During this easureent, the spring balance is held at right angles to the lever ar. To easure the vibration period of the various bodies, a ask (width 3 ) is stuck on. The light barrier is pushed over the ask while the body is at rest. Switch the light barrier to -ode. Now the body is deflected through about 180. In each case, the tie of a half-cycle is easured, several easureents being averaged. For reasons of safety and stability, it is recoended that the spring should not be twistedt beyond ± 720.

4 Theory and evaluation The relationship between the angular oentu of gravity, and the oent acting on it, is of a rigid body in a stationary coordinate syste with its origin at the centre (1) The angular oentu is expressed by the angular velocity and the inertia tensor : that is, the reduction of the tensor with the vector. In the present case, that has only one coponent: has the direction of a principal inertia axis (z-axis), so where is the z-coponent of the principal inertia tensor of the body. For this case, equation (1) reads: where is the angle of rotation. The oent of a spiral spring, in the Hooke s law range, is: (2) where is the angular restoring constant. Fro the regression line to the easured values of Fig. 2 with the linear stateent the slope N/rad (see (2)) is obtained. The angular restoring factor is N/rad. The equation of otion reads: The period and frequency of this vibration are respectively (3) If (x, y, z) is the density distribution of the body, the oent of inertia is obtained as

5 Fig. 2: Moent of a spiral spring as a function of angle of rotation. For a sphere of radius and of ass kg The easured value is kg For a disc of radius and of ass kg The easured value is kg For a solid cylinder of radius and of ass kg The easured value is kg For a hollow cylinder with the two radii and of ass kg The easured value is kg

6 For a thin rod of length and of ass kg The easured value is kg Fig. 3: Moent of inertia of two equal asses, of kg each, as a function of the distance between the. For a point ass of ass, at a distance fro the axis of rotation, one obtains: (4) Fro the regression line to the easured values of Fig. 3, with the stateent (see(4)) the slope kg and the axis intercept kg/ are obtained.