CN103985290A - Method for measuring gravity acceleration by sleeving cylinder to increase compound pendulum length - Google Patents

Abstract

The invention relates to the field of physical experiment equipment and experimental methods by inserting the cylinder into the prolonging pendulum length of the compound pendulum to measure the gravitational acceleration. It is designed to overcome the tedious and time-consuming process of data measurement, data processing and reduce the deviation introduced by calculation. The technical solution is: fix the hanging rod on the thin rod, the hanging rod is cylindrical, the central axis of the thin rod is perpendicular to and intersects with the central axis of the hanging rod, the two ends of the hanging rod are placed in a pair of grooves, and after being placed in the groove , the central axis of the hanging rod is horizontal, and the thin rod can swing back and forth around the central axis of the hanging rod; the distance h between the center of mass C of the thin rod and the central axis of the hanging rod is greater than 0.30 times the length L of the thin rod and less than 0.35 times the length L of the thin rod Double; the sleeve rod is fixed on the upper end of the thin rod, the sleeve rod and the thin rod have a common central axis, the cylinder, the sleeve rod, and the thin rod are made of the same material, and the peripheral shapes and sizes of the cylinder and the thin rod cross-section are the same, The gap between the cylinder and the sleeve rod is so small that it can be ignored. The beneficial effect is that the measuring device and the measuring process are simple and clear, and the data processing is simple and fast.

Description

Measuring the Acceleration of Gravity by Inserting the Cylinder to Extend the Pendulum Length of the Compound Pendulum

technical field

The invention relates to the field of experimental equipment and experimental methods, in particular to the experiment of measuring the acceleration of gravity with a compound pendulum in a physical experiment.

Background technique

In the experiment of measuring the acceleration of gravity with the compound pendulum in the university physics experiment, a long pendulum rod with equidistant circular holes is generally used. There are scales on the surface of the pendulum rod. The hole is hung on the hanging rod (the hanging rod is fixed on the bracket) to swing, measure the swing cycle, record the position coordinates of each hole when hanging (the distance from the round hole to the center of the swing rod, or the origin of other coordinates) and the The swing period when the circular hole is suspended, and then draw a graph on the coordinate paper, and use the graphical method to calculate the acceleration of gravity.

"University Physics Experiment", edited by Wang Zhiheng, He Yuan, and Zhu Jun, first edition in December 2008, Higher Education Press, page 41, the moment of inertia of a thin rod with uniform mass distribution is I=(1/12)mL ² , where L is the length of the thin rod, m is the mass of the thin rod; the periodic formula (9) of the compound pendulum on page 106 of "University Physics Experiments", T=2*p*[(a ² +h ² )/(gh)] ^0.5 , where, define I=ma ² , for a thin rod with uniform mass distribution, I=(1/12)mL ² =ma ² , h is the distance from the suspension point to the center of mass, so T=2*p*{ 【(1/12)L ² +h ² 】/(gh)} ^0.5 , take the derivative with respect to h on both sides, and make the derivative equal to 0, then h _Tmin = L/(12 ^0.5 )= 0.288675L »0.289L, the suspension point The swing period located here is an extreme value (minimum), and the formula suggests: first, there is a minimum swing period; second, on the side of the center of mass of the compound pendulum, at the minimum swing period of 0.289L from the center of mass of the pendulum There may be two positions h1 and h2 on both sides of the point with the same period; third, because the slender pendulum with uniform mass distribution is symmetrical about the center, it is possible to find four suspension points with the same swing period on the compound pendulum .

The data processing of the complex pendulum generally adopts the drawing method, as shown in Figure 1, trace the measurement points on the coordinate paper, and then draw a smooth curve to connect each measurement point or make the measurement points evenly distributed on both sides of the curve to form a hT relationship diagram , since the relationship diagram is a curve, it is difficult to draw an accurate curve, then draw a straight line parallel to the abscissa, the line intersects the curve at four points, and pass the four points corresponding to the coordinates of the abscissa h ₁ , Calculate the equivalent pendulum length L' from h ₂ , h ₃ , and h ₄ , and then calculate the gravitational acceleration g from the formula gT ² /(4p ² )= L'.

Since the measurement points are discrete data, the abscissa interval of the discrete data is generally 2cm, and the interval is relatively long. Generally, it is impossible to find that the cycle of the four measurement points is exactly the same. The cycle curve is a traced curve, which deviates from the real curve, so make a The four points where the horizontal line intersects the deviated h-T curve, the coordinate values of the abscissa read from the four points also have deviations, so in addition to the error of the experiment itself, the graphical calculation method also brings a large deviation, Moreover, the data measurement process and processing process are cumbersome and time-consuming.

Contents of the invention

In the experiment of measuring the acceleration of gravity with the compound pendulum, in order to overcome the cumbersome and time-consuming process of data measurement and data processing and reduce the deviation introduced by calculation, the present invention designs a device for measuring the acceleration of gravity by extending the pendulum length of the compound pendulum by inserting it into the cylinder, and Describes how to use it.

The technical solution adopted by the present invention to achieve the purpose of the invention is: a device for measuring the acceleration of gravity by extending the pendulum length of the compound pendulum by being inserted into the cylinder, which is characterized in that the pendulum rod of the compound pendulum is composed of a thin rod and a cylinder inserted into the sleeve rod In the initial state, the sleeve rod of the thin rod is not inserted into the cylinder, and the swing rod at this time is the thin rod; the hanging rod is fixed on the thin rod, and the hanging rod is cylindrical, and the central axis of the thin rod is perpendicular to the central axis Intersect, the two ends of the hanging rod are placed in a pair of grooves, after being put into the grooves, the central axis of the hanging rod is horizontal, and the thin rod can swing back and forth around the central axis of the hanging rod; The distance h of the axis is greater than 0.30 times the length L of the thin rod and less than 0.35 times the length L of the thin rod, that is, h is 0.30L-0.35L; the sleeve rod is fixed on the upper end of the thin rod, and the sleeve rod and the thin rod have a common central axis , the cylinder, sleeve rod, and thin rod are made of the same material, the peripheral shape and size of the cross section of the cylinder and the thin rod are the same, and the gap between the cylinder and the sleeve rod is so small that it can be ignored.

The cross-sectional area of the sleeve rod is 0.01-0.2 times that of the thin rod. The sleeve rod has an axisymmetric shape, the cylinder has a through hole, and the shape of the through hole is consistent with the shape of the sleeve rod. The height specification of the cylinder is that there are 10 cylinders with a height of 1cm, 4-8 cylinders with a height of 5mm, 5-20 cylinders with a height of 2mm, and 5-20 cylinders with a height of 1mm.

The measurement method adopts the device for measuring the acceleration of gravity by inserting into the cylinder to prolong the length of the compound pendulum. For a cylinder with a height of 1cm, measure the swing cycle T of the pendulum every time a cylinder is added. As the center of mass of the pendulum gradually moves up, the cycle first gradually decreases and then gradually increases. When the period just changes from small to When it becomes larger, remove the 1cm cylinder placed twice before, and place a cylinder with a height of 5mm or 2mm instead, and measure again when the period just changes from decreasing to increasing, remove the previous two cylinders again. A cylinder with a height of 5mm or two cylinders with a height of 2mm is placed instead of a cylinder with a height of 1mm until the cycle is just changed from smaller to larger when measured again. At this time, the length accurate to mm is measured. The smallest swing period Tmin, at this time the height of all the cylinders is recorded as h _ny , the length of the swing rod=L+h _ny , and the moment of inertia of the swing rod around the center of mass is I=(1/12)m(L+ h _ny ) ² , h=0.289(L+ h _ny ), Tmin=2*p*{[(1/12)* (L+ h _ny ) ² +h ² ]/g/h} ^0.5 =4.773*{ (L+ h _ny )/ g} ^0.5 , the gravitational acceleration g=22.78 (L+ h _ny )/ T ² min can be calculated from this formula.

The beneficial effects of the present invention are: due to the minimum period of compound pendulum swing, a slender pendulum with uniform mass distribution, the distance between the suspension point corresponding to the minimum period and the center of mass is 0.289 times of the total length, reducing the diameter of the sleeve rod and setting a suitable Length, so that the influence of the sleeve rod is negligible, insert a cylinder into the sleeve rod, and the linear density formed by the combination of the inserted cylinder and the sleeve rod is equivalent to the linear density of the compound pendulum swing rod. Thereby, the length of the pendulum pendulum can be extended evenly, and the minimum period can be found directly by continuously increasing or decreasing or changing the cylinder, and then its length can be measured or calculated (the length of the thin rod and the height of the cylinder can be known amount), so as to directly calculate the acceleration of gravity, the measuring device and the measuring process of the present invention are simple and clear, and the data processing is simple and fast.

Description of drawings

Fig. 1 is a schematic diagram of the existing data processing of the acceleration of gravity measured by the compound pendulum; Fig. 2 is a schematic diagram of the swing bar of the compound pendulum; Fig. 3 is a schematic diagram of the cylinder.

Among them, 1. thin rod, 2. hanging rod, 3. sleeve rod, 4. cylinder, 5. through hole; C is the center of mass of the thin rod, L is the total length of the thin rod, and h is the centroid of the swing rod to the hanging rod The distance from the axis, h _y is the height of the cylinder.

Detailed ways

The pendulum rod of compound pendulum consists of a thin rod 1 and a cylinder 4 inserted into the sleeve rod 3;

The hanging rod 2 is fixed on the thin rod 1, the central axis (center line) of the thin rod 1 is perpendicular to and intersects with the central axis (center line) of the hanging rod 2, and the two ends of the hanging rod 2 are placed in a pair of grooves, and put in After the groove, the central axis of the hanging rod 2 is horizontal, and the hanging rod is cylindrical, and the thin rod 1 can swing back and forth around the central axis of the hanging rod 2.

The distance between the central axis of the hanging rod 2 and the centroid C of the thin rod 1 is h, h>0.289L, preferably h is 0.30L-0.35L, the sleeve rod 3 is fixed on the upper end of the thin rod 1, and the sleeve rod 3 and the thin rod 1 have The common central axis (central line), the cylinder 4, the sleeve rod 3, and the thin rod 1 are made of the same material, and the cross-sectional area of the sleeve rod 3 is 0.01-0.2 times the cross-sectional area of the thin rod 1. The external dimensions of the cylinder 4 are exactly the same as those of the thin rod 1 (the cross-sectional shape is the same and the size is equal), and the gap between the cylinder 4 and the sleeve rod 3 is so small that it can be ignored (basically in close contact and can be inserted freely and take out, related to manufacturing accuracy).

Cover rod 3 is preferably axisymmetric shape, such as cylinder, cube, cuboid, triangular prism, the central axis of cover rod 3 coincides with the central axis of thin rod 1, and cylinder 4 has a through hole 5, and the shape of through hole 5 and The shape of the sleeve rod 3 is consistent, and the height of the cylinder 4 is 1cm (10 pieces), 5mm (4 pieces), 2mm (10 pieces), 1mm (10 pieces).

Through the swing of the thin rod 1, use a timing instrument such as a photoelectric gate to measure the swing cycle, gradually place a cylinder 4 with a height of 1 cm on the sleeve rod 3 at the upper end of the thin rod 1, and measure the swing cycle of the swing rod every time a cylinder 4 is added. T, as the center of mass C of the pendulum gradually moves up (because it is inserted into the cylinder), h becomes smaller, and the period first gradually decreases, and then the period gradually increases after reaching the minimum swing period. When the period is just right (appears for the first time) When changing from smaller to larger, remove the 1cm cylinder placed twice before, and place a cylinder with a height of 5mm or 2mm, and measure again just when the cycle changes from smaller to larger, and move In addition to the previous two cylinders with a height of 5mm or two cylinders with a height of 2mm, place a cylinder with a height of 1mm until the time when the cycle changes from smaller to larger is measured again, at this time, it is accurate to mm The length of the swing rod is measured to the minimum swing period Tmin. At this time, the height of all the cylinders is recorded as h _ny , the length of the swing rod=L+h _ny , and the moment of inertia of the swing rod around the center of mass is I=(1/12)m(L+ h _ny ) ² , h=0.289(L+ h _ny ), at this time, the formula (9) in the "University Physics Experiment" textbook becomes Tmin=2*p*{[(1/12)* (L+ h _ny ) ² + h ² ]/g/h} ^0.5 =4.773*{ (L+ h _ny )/g} ^0.5 , the gravitational acceleration g=22.78 (L+ h _ny )/ T ² min can be calculated from this formula.

According to the design of h is 0.30L-0.35L, the change of h is 0.05L, then the change of the center of mass is half of the change of length, so the change of length is 0.1L, and the change of length is mainly caused by the height of 1cm It is caused by the cylinder, so to be able to meet this requirement, that is, 0.1L<10cm, the length of the thin rod is less than 10cm/0.1=100cm. Under the premise of less than 100cm, the user can choose other sizes, such as 50cm, 60cm, etc. .

Claims (4)

1. A device for measuring gravitational acceleration by inserting a cylinder to extend the pendulum length of a compound pendulum, the feature is: the pendulum rod of the compound pendulum is composed of a thin rod (1) and a cylinder (4) inserted into the sleeve rod (3) ;In the initial state, the sleeve rod (3) of the thin rod (1) is not inserted into the cylinder (4), and the swing rod at this time is the thin rod (1); the hanging rod (2) is fixed on the thin rod (1), and the hanging The rod is cylindrical. The central axis of the thin rod (1) is perpendicular to and intersects with the central axis of the hanging rod (2). The two ends of the hanging rod (2) are placed in a pair of grooves. After being placed in the groove, the hanging rod The central axis of (2) is horizontal, and the thin rod (1) can swing back and forth around the central axis of the hanging rod (2); the distance h between the center of mass C of the thin rod (1) and the central axis of the hanging rod (2) is greater than the length of the thin rod 0.30 times of L, less than 0.35 times of the length L of the thin rod, that is, h is 0.30L-0.35L; the sleeve rod (3) is fixed on the upper end of the thin rod (1), and the sleeve rod (3) and the thin rod (1) have The common central axis, the cylinder (4), the sleeve rod (3), and the thin rod (1) are made of the same material, the peripheral shape and size of the cross section of the cylinder (4) and the thin rod (1) are the same, and the cylinder The gap between (4) and sleeve rod (3) is so small that it can be ignored. 2. A device for measuring gravitational acceleration by inserting into a cylinder to extend the pendulum length of a compound pendulum according to claim 1, characterized in that: the cross-sectional area of the sleeve rod (3) is the cross-sectional area of the thin rod (1) 0.01-0.2 times. 3. A device for measuring gravitational acceleration by inserting into a cylinder to extend the pendulum length of a compound pendulum according to claim 1, characterized in that: the sleeve rod (3) has an axisymmetric shape, and the cylinder (4) has a through hole (5), the shape of the through hole (5) is consistent with the shape of the sleeve rod (3). A device for measuring gravitational acceleration by inserting into a cylinder to extend the length of a compound pendulum according to claim 1, characterized in that: the height specification of the cylinder (4) is as follows: there are 10 cylinders (4) with a height of 1 cm and a height of 5 mm There are 4-8 pieces, 5-20 pieces with a height of 2mm, and 5-20 pieces with a height of 1mm. 4. The method for measuring the acceleration of gravity by inserting the device for measuring the acceleration of gravity by extending the length of the compound pendulum as claimed in claim 4 is characterized in that: the swing of the thin rod (1) is measured by a timing instrument such as a photoelectric gate Swing cycle, gradually place a cylinder (4) with a height of 1cm on the sleeve rod (3) at the upper end of the thin rod (1), and measure the swing cycle T of the swing rod every time a cylinder (4) is added. The center of mass of the center of gravity gradually moves up, and the cycle first gradually decreases, and then gradually increases. When the cycle just changes from small to large, remove the 1cm cylinder placed twice before, and place it at a height of 5mm or 2mm. When the cylinder is measured again and the period just changes from smaller to larger, remove the previous two cylinders with a height of 5mm or two cylinders with a height of 2mm and replace them with cylinders with a height of 1mm until Measure again when the cycle just changes from smaller to larger, at this time the minimum swing cycle Tmin is measured at a length accurate to mm, at this time the height of all cylinders is recorded as h _ny , the length of the pendulum = L+ h _ny , the moment of inertia of the pendulum around the center of mass is I=(1/12)m(L+ h _ny ) ² , h=0.289(L+ h _ny ), Tmin=2*p*{[(1/12)* ( L+ h _ny ) ² +h ² ]/g/h} ^0.5 =4.773*{ (L+ h _ny )/g} ^0.5 , the gravitational acceleration g=22.78 (L+ h _ny )/ T ² min can be calculated from this formula.