A kind of inertia measurement apparatus based on weiss pendulum and its measurement method

Abstract

This invention provides a rotational inertia measuring instrument and method based on a Widmanstätten pendulum, belonging to the field of educational tools. The invention includes a stage and a spring. The stage includes an upper metal plate with four longitudinal threaded posts symmetrically fixed at its four corners. A lower metal plate, movable vertically, is mounted on the longitudinal threaded posts and secured by a fixing nut. The upper surface of the lower metal plate has a pre-set groove for fixing the cylinder to be measured. A connecting cylinder is fixed at the center of the upper surface of the upper metal plate, with a horizontal threaded post passing through it. Adjusting nuts are symmetrically located at both ends of the horizontal threaded post. The center of the upper surface of the connecting cylinder is connected to the lower end of the spring. By pulling the stage away from its equilibrium position and releasing it from rest, the stage undergoes both vertical and horizontal rotational motion. Utilizing the coupling motion law of the Widmanstätten pendulum, the rotational inertia of the stage and the total rotational inertia of the stage and the sample can be measured, thereby determining the rotational inertia of the cylinder to be measured. This effectively demonstrates the influence of rotational inertia on the motion of an object.

Description

Technical Field

[0001] This invention relates to the field of teaching aids technology, and in particular to a moment of inertia measuring instrument based on a Widmanstätten pendulum and its measuring method. Background Technology

[0002] The moment of inertia of a rigid body is a measure of its inertia during rotation. It depends on the body's mass, mass distribution, and the position and orientation of its axis of rotation. For geometrically regular rigid bodies, the moment of inertia about its axis of mass can be calculated using integral formulas, and the moment of inertia about any specific axis can be calculated using the parallel axis theorem. For rigid bodies with complex shapes, experimental methods are generally used for determination. Currently, university physics experiments mainly use the torsion pendulum method and the trifilar pendulum method to measure the moment of inertia of rigid bodies. However, the principles employed in these two methods have limited relevance to the knowledge taught in university physics courses, failing to effectively combine theoretical knowledge with experimental practice and hindering students' better understanding and mastery of the moment of inertia of rigid bodies and related knowledge. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a moment of inertia measuring instrument based on a Widmanstätten pendulum and its measurement method. This invention effectively demonstrates the influence of moment of inertia on the motion of an object, aiding students' understanding.

[0004] The technical means employed in this invention are as follows:

[0005] A rotational inertia measuring instrument based on a Wechsler pendulum includes a stage and a spring. The stage includes an upper metal plate, with four longitudinal threaded posts symmetrically fixed at its four corners. A lower metal plate, which can move up and down, is also mounted on the longitudinal threaded posts. The lower metal plate is fixed by a fixing nut. The upper surface of the lower metal plate has a preset groove for fixing a cylinder to be measured. A connecting cylinder is fixed at the center of the upper surface of the upper metal plate. A horizontal stud runs horizontally through the connecting cylinder. Adjusting nuts are symmetrically located at both ends of the horizontal stud. The center of the upper surface of the connecting cylinder is connected to the lower end of the spring.

[0006] The upper and lower metal plates are square, with a hole at each of the four corners for a longitudinally threaded post to pass through.

[0007] Furthermore, the moment of inertia of the stage is calculated in advance during the manufacture of the stage and is used as a known quantity I0.

[0008] Furthermore, the metal plate on the platform and the lower metal plate are square plates made of aluminum alloy.

[0009] Furthermore, the distance between the metal plate on the platform and the lower metal plate can be adjusted.

[0010] Furthermore, the longitudinal threaded posts are four in number and symmetrically inserted into four holes in the upper and lower metal plates, with the four holes being symmetrical about the central axis of the stage.

[0011] The coupled motion of this device is formed by the torsion of the spring itself, specifically by the periodic conversion between vertical motion and horizontal helical motion of the stage.

[0012] The present invention also provides a measurement method for the above-mentioned moment of inertia measuring instrument based on the Widmanstätten pendulum, comprising the following steps:

[0013] S1. When the stage is unloaded, use vernier calipers to measure the distance between the nuts on both sides of the horizontal stud and the central axis of the stage, ensuring that the nuts on both sides are symmetrical about the central axis of the stage. Then, use vernier calipers to measure the distance between the outermost edges of the left and right nuts and record the data at this time. ;

[0014] S2. Connect the empty stage to the spring, with the upper end of the spring suspended from the iron frame. Pull the empty stage vertically downwards a suitable distance, release it from rest, and record the time at that point. When the horizontal velocity is observed to be zero, i.e., the horizontal motion stops, this moment is recorded as... Subsequently, horizontal acceleration can be observed. After reaching its peak speed, the velocity drops back to zero. This moment is recorded as... Continue to observe and record experimental data. , ;

[0015] S3. Adjust the distance of the nut to x2, repeat steps S1-S2, measure and record the second set of data under no-load conditions;

[0016] S4. Fix the cylinder to be tested on the preset groove of the lower metal plate, and repeat steps S1-S2.

[0017] S5. Using the method of successive differences, the vibration periods of the coupled motions of S2, S3, and S4 are calculated to be T1, T2, and T3, respectively; the calculation formulas are as follows:

[0018] (1)

[0019] S6. The frequency ω of the Weyspeare pendulum system is the absolute value of the difference between the natural frequencies of the simple harmonic motion in the vertical and horizontal directions, i.e.

[0020] (2)

[0021] Where k1 is the bending coefficient of the spring, k2 is the torsional coefficient of the spring, m is the total mass of the object suspended by the spring, the mass of the spring is neglected, and the moment of inertia of the system about its central axis is I.

[0022] If we conduct experiments on the stage with the cylinder to be tested both unloaded and with the cylinder placed on it, and substitute multiple sets of data into the equation, then combine this with relevant mathematical derivations to eliminate some terms, we can obtain the equation:

[0023] (3)

[0024] Where T1 and T2 are the vibration periods of the coupled motion when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; I1 and I2 are the total moments of inertia of the system when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; m is the mass of the stage when unloaded; T3 is the vibration period of the coupled motion when the cylinder to be measured is placed; and I3 is the total moment of inertia of the system when the cylinder to be measured is placed. , Let I be the mass of the stage on which the cylinder to be measured is placed; then the moment of inertia I of the cylinder to be measured is... 物 for:

[0025] (4)

[0026] Where I0 and I 螺 These are the moments of inertia of the stage and the adjusting nut, respectively. I0 is the moment of inertia of the stage itself about its central axis (pre-calculated during stage fabrication and known as a known quantity). 螺 Can be measured in experiments The calculation yields the following formula:

[0027] (5)

[0028] Where m is the mass of a single nut, x3 is the distance between the outermost edges of the left and right nuts measured in the experiment, and d is the thickness of a single nut.

[0029] To improve accuracy, multiple sets of rotational inertia of the cylinders under test can be measured in step S4 to obtain multiple T3 values. That is, multiple sets of rotational inertia of the cylinders under test can be measured by substituting them into the formula and then calculating the average value.

[0030] The device of this invention mainly comprises a stage and a spring. By measuring the period of the coupled motion of the stage, the moment of inertia of the rigid body is calculated using the laws of Vieira's pendulum coupled motion; the stage is symmetrical about its central axis. During measurement, the moment of inertia of the stage can be changed by adjusting the position of the adjusting nut on the horizontal stud, thereby obtaining multiple sets of data and improving the accuracy of the measurement results. This invention can effectively demonstrate the influence of moment of inertia on the motion of an object. This invention can be widely applied in the field of teaching aids technology. Attached Figure Description

[0031] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0032] Figure 1 This is a front view of the rotational inertia measuring instrument based on the Wechsler pendulum of the present invention.

[0033] Figure 2 This is a schematic diagram of the bottom of the rotational inertia measuring instrument based on the Wechsler pendulum of the present invention.

[0034] Figure 3 This is a schematic diagram of the top of the rotational inertia measuring instrument based on the Wechsler pendulum of the present invention.

[0035] In the diagram: 1. Spring; 2. Connecting cylinder; 3. Adjusting nut; 4. Horizontal stud; 5. Upper metal plate; 6. Longitudinal threaded stud; 7. Lower metal plate; 8. Pre-set groove; 9. Fixing nut. Detailed Implementation

[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0037] like Figures 1-3 As shown, this embodiment discloses a rotational inertia measuring instrument based on a Wechsler pendulum, including a stage and a spring. The stage includes an upper metal plate, and four longitudinal threaded columns are symmetrically fixed to the four corners of the upper metal plate. A lower metal plate that can move up and down is also on the longitudinal threaded columns. The lower metal plate is fixed by a fixing nut. The upper surface of the lower metal plate is provided with a preset groove for fixing the cylinder to be measured. A connecting cylinder is fixed at the center of the upper surface of the upper metal plate. A horizontal stud is horizontally inserted through the connecting cylinder. Adjusting nuts are symmetrically attached to both ends of the horizontal stud. The center of the upper surface of the connecting cylinder is connected to the lower end of the spring.

[0038] In this embodiment, the upper metal plate 5 and the lower metal plate 7 are made of metal, specifically aluminum alloy.

[0039] Depending on the shape of the rigid body to be tested, in this embodiment, the preset groove 8 is circular. In order to secure the rigid body to be tested, in this embodiment, the cylinder to be tested is placed in the preset groove, and the lower metal plate is moved upward along the longitudinal threaded post so that the cylinder to be tested is clamped by the upper and lower metal plates. Then, the fixing nut on the longitudinal threaded post is tightened to fix the cylinder to be tested.

[0040] This invention also provides a method for measuring the moment of inertia of the aforementioned Widmanstätten-based moment of inertia measuring instrument. The upper end of a spring is fixed to an iron frame. After the stage is stretched vertically downwards and released from rest, the spring's cross-section contracts upon stretching, generating an outward expanding force in the horizontal direction, causing the stage to undergo horizontal helical motion. This motion couples with its vertical motion. By measuring the period of this coupled motion, the moment of inertia of the rigid body is calculated using the laws governing the coupled motion of the Widmanstätten. The stage and spring are symmetrical about their central axis. The moment of inertia of the stage can be changed by adjusting the position of the adjusting nut on the horizontal stud, thereby allowing for multiple sets of data to be measured and improving the accuracy of the measurement results.

[0041] S1. With the stage unloaded, use vernier calipers to measure the distance between the nuts on both sides of the horizontal stud and the central axis, ensuring that the nuts on both sides are symmetrical about the central axis of the stage. Then, use vernier calipers to measure the distance between the outermost edges of the left and right adjusting nuts and record the data. ;

[0042] S2. Connect the empty stage to the spring, with the upper end of the spring suspended from the iron frame. Pull the stage vertically downwards a suitable distance, release it from rest, and record the time at that point. When the horizontal velocity is observed to be zero, i.e., the horizontal motion stops, this moment is recorded as... Subsequently, horizontal acceleration can be observed. After reaching its peak speed, the velocity drops back to zero. This moment is recorded as... Continue to observe and record experimental data. , ;

[0043] S3. Adjust the distance between the two nuts to x2, repeat steps S1-S2, and measure and record the second set of multiple data points under no-load conditions.

[0044] S4. Fix the rigid body to be tested on the groove of the chassis, and repeat steps S1-S2.

[0045] S5. The period of each group of data is calculated as T1, T2, T3 using the method of successive differences.

[0046] (1)

[0047] S6. The frequency of the Vibration of the Widmanstätten pendulum system is the absolute value of the difference between the natural frequencies of simple harmonic motion in the vertical and horizontal directions, i.e.

[0048] (2)

[0049] Where k1 is the bending coefficient of the spring, k2 is the torsional coefficient of the spring, m is the total mass of the object suspended by the spring, the mass of the spring is neglected, and the moment of inertia of the system about its central axis is I.

[0050] If we conduct experiments on the stage with the cylinder to be tested both unloaded and with the cylinder placed on it, and substitute multiple sets of data into the equation, then combine this with relevant mathematical derivations to eliminate some terms, we can obtain the equation:

[0051] (3)

[0052] Where T1 and T2 are the vibration periods of the coupled motion when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; I1 and I2 are the total moments of inertia of the system when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; m is the mass of the stage when unloaded; T3 is the vibration period of the coupled motion when the cylinder to be measured is placed; I3 is the total moment of inertia of the system when the cylinder to be measured is placed; m , Let I be the mass of the stage on which the cylinder to be measured is placed. Then the moment of inertia I of the cylinder to be measured is... 物 for:

[0053] (4)

[0054] Where I0 and I 螺 These are the moments of inertia of the device and the nut, respectively. I0 is the moment of inertia of the stage itself about its central axis (pre-calculated during the manufacture of the stage and is known). 螺 Can be measured in experiments The calculation yields the following formula:

[0055] (5)

[0056] Where m is the mass of a single nut, x3 is the distance between the outermost edges of the left and right nuts measured in the experiment, and d is the thickness of a single nut.

[0057] To improve accuracy, the distance between the two nuts can be adjusted multiple times in step S4 to obtain multiple T3 values, which can then be substituted into the formula to calculate the rotational inertia of the multiple sets of cylinders to be measured.

[0058] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A moment of inertia measuring instrument based on a Wechsler pendulum, characterized in that, The device includes a stage and a spring. The stage includes an upper metal plate, with four longitudinal threaded posts symmetrically fixed at the four corners of the upper metal plate. A lower metal plate that can move up and down is also attached to the longitudinal threaded posts. The lower metal plate is fixed by a fixing nut. The upper surface of the lower metal plate has a preset groove for fixing the cylinder to be measured. A connecting cylinder is fixed at the center of the upper surface of the upper metal plate. A horizontal stud is horizontally inserted through the connecting cylinder. Adjusting nuts are attached to both ends of the horizontal stud. The center of the upper surface of the connecting cylinder is connected to the lower end of the spring.

2. The moment of inertia measuring instrument based on a Widmanstätten pendulum according to claim 1, characterized in that, The moment of inertia of the stage is calculated in advance during the manufacture of the stage and is known as I0.

3. The moment of inertia measuring instrument based on a Widmanstätten pendulum according to claim 1, characterized in that, The metal plate on the platform and the lower metal plate are square plates made of aluminum alloy.

4. The moment of inertia measuring instrument based on a Widmanstätten pendulum according to claim 1, characterized in that, The four holes in the upper and lower metal plates are symmetrical about the central axis of the platform.

5. The measurement method of the rotational inertia measuring instrument based on the Wechsler pendulum as described in claim 1, characterized in that, Includes the following steps: S1. With the stage unloaded, use vernier calipers to measure the distance between the nuts on both sides of the horizontal stud and the central axis of the stage, ensuring the nuts are symmetrical about the central axis. Then, use vernier calipers to measure the distance between the outermost edges of the left and right nuts and record the data. ; S2. Connect the empty stage to the spring, with the upper end of the spring suspended from the iron frame. Pull the empty stage vertically downwards a suitable distance, release it from rest, and record the time at that point. When the horizontal velocity is observed to be zero, i.e., the horizontal motion stops, this moment is recorded as... Subsequently, horizontal acceleration can be observed. After reaching its peak speed, the velocity drops back to zero. This moment is recorded as... Continue to observe and record experimental data. , ; S3. Adjust the distance of the nut to x2, repeat steps S1-S2, measure and record the second set of data under no-load conditions; S4. Fix the cylinder to be tested on the preset groove of the lower metal plate, and repeat steps S1-S2. S5. Using the method of successive differences, the vibration periods of the coupled motions of S2, S3, and S4 are calculated to be T1, T2, and T3, respectively; the calculation formulas are as follows: （1） S6. The frequency ω of the Weyspeare pendulum system is the absolute value of the difference between the natural frequencies of the simple harmonic motion in the vertical and horizontal directions, i.e. （2） Where k1 is the bending coefficient of the spring, k2 is the torsional coefficient of the spring, m is the total mass of the object suspended by the spring, the mass of the spring is neglected, and the moment of inertia of the system about its central axis is I. If we conduct experiments on the stage with the cylinder to be tested both unloaded and with the cylinder placed on it, and substitute multiple sets of data into the equation, then combine this with relevant mathematical derivations to eliminate some terms, we can obtain the equation: （3） Where T1 and T2 are the vibration periods of the coupled motion when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; I1 and I2 are the total moments of inertia of the system when the nut spacing is x1 and x2 respectively, respectively, when the stage is unloaded; m is the mass of the stage when unloaded; T3 is the vibration period of the coupled motion when the cylinder to be measured is placed; and I3 is the total moment of inertia of the system when the cylinder to be measured is placed. , Let I be the mass of the stage on which the cylinder to be measured is placed; then the moment of inertia I of the cylinder to be measured is... 物 for: （4） Where I0 and I 螺 These are the moments of inertia of the stage and the adjusting nut, respectively; I0 is the moment of inertia of the stage itself about its central axis. 螺 Measured in the experiment The calculation yields the following formula: （5） Where m is the mass of a single nut, x3 is the distance between the outermost edges of the left and right nuts measured in the experiment, and d is the thickness of a single nut.

6. The method according to claim 5, characterized in that... In step S4, the distance between the two nuts is adjusted multiple times to obtain multiple T3 values. The values ​​are then substituted into the formula to calculate the rotational inertia of the multiple sets of cylinders to be measured, and then the average value is calculated.

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