3.2.3 Experimental principles
1)Time counting
The general computer millisecond meter starts counting at the time when the photoelectric gate gets the shading signal.Since the block rod reaches the photoelectric gate after every half turn of the rotating system,the general computer millisecond meter can record the shading times and time of every half turn of the rotating system.
2)Principle of measuring moment of inertia
(1)The relationship between the moment of inertia J and the angular acceleration β.
If both an external torque Tr and a frictional torque M act on a body with the moment of inertia J to give it an angular acceleration β1,according to the law of rotation,it can be expressed as:
When the external force T is removed,only the friction torque M acts on this body,which will decelerate with the angular acceleration β2,according to Newton’s second law for rotational motion,it can be expressed as:
According to Newton’s law,the net external force mg-T provides the acceleration rβ1 for the weight m,which can be expressed as:
Combined with Equation(3.2.1)-Equation(3.2.3),it can be expressed as:
If J1 represents the moment of inertia about the axis of rotation,J2 represents the moment of inertia of the rotating system with the testing body,and J is the moment of inertia of the testing body,according to the superposition principle of the moment of inertia,it can be expressed as:
(2)The measuring principle of the angular acceleration β.
According to the law of uniformly accelerated rotational motion,it can be expressed as:
During the experiment,the photoelectric gate is used to obtain the time of every half circumference and generate the counting pulse when the shading slender rod goes through the photoelectric gate.
When the pulse number is Km and the corresponding time is tm,it can be expressed as:(https://www.daowen.com)
When the pulse number is Kn and the corresponding time is tn,it can be expressed as:
Combined with Equation(3.2.7)and Equation(3.2.8),it can be expressed as:
(3)Theoretical formula for the moment of inertia.
The moment of inertia for the solid cylinder with a mass of m1and a diameter of D:
The moment of inertia for the hollow cylinder with a mass of m1,inner diameter of D1 and outer diameter of D2:
The moment of inertia for the long uniform rod with a mass of m1,the length of L:
The moment of inertia for the uniform sphere with a mass of m1,and the radius of R:
(4)Parallel axis theorem.
If the moment of inertia of the body about an axis through the mass is J0,the moment of inertia J about any axis parallel to the axis through the center of mass can be expressed as:
where m2 is the mass of the body and d is the difference between the two axes.
