CN107490806A - It is a kind of to utilize the method for increasing pendulum radius measurement acceleration of gravity - Google Patents

Abstract

The present invention provides a method for measuring the gravitational acceleration by increasing the pendulum radius, the method studies the cylindrical weights whose radii are _r2 and _r1 by analyzing the swing period of the compound pendulum system composed of the pendulum rod and the cylindrical weight The change of the swing period of the compound pendulum system when the radius is gradually increased by the step length r ₂ and r ₁ respectively, and the influence of different step lengths on the swing period is compared, and then the calculation formula of the gravitational acceleration g is obtained. Compared with the widely used experimental method, the present invention no longer relies on changing the position of the center of mass of the compound pendulum system to the suspension point to obtain the equivalent pendulum length from the experimental principle, but by analyzing the influence of the change step of the pendulum radius on the swing period, it is realized to fit the straight line The slope of the method is used to obtain the value of the acceleration of gravity, which reduces the complicated and tedious data processing work, improves the data processing speed, and reduces the data processing error. The principle of this method is simple and clear, easy to understand, and suitable for students to operate.

Description

A Method of Measuring Gravitational Acceleration by Increasing Pendulum Radius

technical field

The invention relates to an experimental device, in particular to a method for measuring the acceleration of gravity by increasing the radius of a pendulum.

Background technique

In the experiment of measuring the acceleration of gravity with the compound pendulum in the university physics experiment, a slender pendulum rod with equidistant round holes is generally used. The surface of the pendulum rod is marked with vertical scales. In the experiment, one of the round holes of the pendulum rod is hung on the bracket Swing, record the distance from the circular hole to the center of the swing rod and the swing cycle when the circular hole is suspended, then draw a graph on the coordinate paper, use the graphic method to process data and calculate the acceleration of gravity. The principle is to change the position of the suspension point to obtain the swing period of the pendulum at different distances from the suspension point to the center of mass, and obtain the equivalent pendulum length.

The method of processing the experimental data of the compound pendulum to measure the acceleration of gravity is to take the distance h from the suspension point to the center of mass of the pendulum as the abscissa, and take the swing period T measured at the suspension point as the ordinate, trace the measurement point on the coordinate paper, and then draw A smooth curve connects the measurement points or evenly distributes the measurement points on both sides of the curve to form a hT relationship diagram, and then draws a straight line parallel to the abscissa axis. The straight line intersects the curve at four points to obtain four swing cycles Same suspension points. Calculate the equivalent pendulum length L` through the coordinate values h <sub>1</sub> , h <sub>2</sub> , h <sub>3</sub> , and h <sub>4</sub> corresponding to the abscissa of the four points, and then calculate the gravitational acceleration g by the formula gT <sup>2</sup> /(4p <sup>2</sup> )=L`.

In the actual operation process, the above-mentioned widely used compound pendulum measurement gravitational acceleration experiment has the following difficulties. The first is that the periodic curve is a traced curve. Unlike the fitted straight line, it is very difficult for beginners to describe an accurate curve; There is a deviation between the traced periodic curve and the real curve, so the coordinate values of the four points where the straight line parallel to the abscissa axis intersects the periodic curve must also have deviations; third, because the measurement points are discrete data, the abscissa interval is generally 2cm , such an interval is difficult to find exactly the same measurement points for four periods in reality. To sum up, there are large deviations in the existing acceleration measurement experiments, and the data measurement and processing process is also cumbersome and time-consuming.

Contents of the invention

The invention provides a method for measuring the acceleration of gravity, which can quickly and accurately obtain the acceleration of gravity. In order to achieve the above object, a method for measuring the acceleration of gravity by increasing the radius of the pendulum provided by the invention is composed of the following steps:

1. Select a cylindrical weight I with a radius of r ₁ and a height of h as a pendulum. The center of its top surface is vertically connected to the bottom of a rigid and light pendulum with a length of H, and a compound pendulum composed of the pendulum. Measuring the swing period of a compound pendulum when it swings at a small angle around the apex of the pendulum ;

2. The outer wall of the cylindrical weight I in step 1 is sequentially inserted into the hollow cylindrical weight I of the same material with an inner diameter of 2(x-1)r ₁ , an outer diameter of 2xr ₁ , and a height of h to obtain For a cylindrical pendulum with a radius of xr ₁ , each set of a hollow cylindrical weight I measures the swing period of the compound pendulum, and obtains the swing period of the compound pendulum system when the pendulum radius increases by r ₁ ~ , it is known that x is an integer between 2 and n, and n is an integer not greater than 15 and not less than 10;

3. Select a cylindrical weight II with a radius of r ₂ and a height of h as a pendulum, and connect the center of its top surface to the bottom of a rigid and lightweight pendulum with a length of H, and form a compound pendulum with the pendulum, and measure complex The swing period when the pendulum swings at a small angle around the apex of the pendulum ;

4. Insert a hollow cylindrical weight II of the same material with an inner diameter of 2(x-1)r ₂ , an outer diameter of 2xr ₂ , and a height of h into the outer wall of the cylindrical weight II in step 3 to Obtain a cylindrical pendulum with a radius of xr ₂ , put a hollow cylindrical weight II in each set to measure the swing period of the compound pendulum, and obtain the swing period of the compound pendulum system when the pendulum radius increases with a step size of r ₂ ~ , it is known that r ₂ is greater than r ₁ ;

5. Let y= , with x as the horizontal axis and y as the vertical axis to establish a coordinate system, the point (x, ) is plotted in the coordinate system, and the fitting obtains the oblique line ,according to , get .

Experimental principle of the present invention is as follows:

It is known that when a compound pendulum with a cylindrical weight I as a pendulum and a pendulum length of H-h/2 swings at a small angle, the vibration equation of the compound pendulum system is:

In the formula, is the moment of inertia of the cylindrical weight I around the horizontal axis of its center of mass, which can be obtained according to the calculation formula of the moment of inertia of the cylinder ,Will Substitute into the above formula to get:

According to the above formula, the natural period of free vibration of the compound pendulum can be obtained as:

After inserting a hollow cylindrical weight I with an inner diameter of 2(x-1)r ₁ , an outer diameter of 2xr ₁ , and a height of h on the outer wall of the cylindrical weight I, the cylinder is gradually enlarged with a step size of r ₁ The radius of the shape weight I, when the pendulum radius increases to xr ₁ , the vibration equation of the compound pendulum system is:

substitute ,have to:

The natural period of free vibration of the system when the pendulum radius is xr ₁ can be obtained as:

Similarly, when the cylindrical weight II with radius r ₂ and height h is used as the pendulum, and the radius of the cylindrical weight II is gradually increased with the step size r ₂ , when the pendulum radius increases to xr ₂ After that, the natural period of free vibration of the compound pendulum system is:

Assume ,but

Take the number of weights x as the horizontal axis, and take To establish a coordinate system for the vertical axis, the point (x, ) plotted in the coordinate system, fitting the oblique line , the slope of the oblique line is k, which can be obtained according to , get .

The present invention composes the compound pendulum system by the pendulum rod and the cylindrical weight, and studies the change of the swing cycle of the compound pendulum system when the radius of the cylindrical weights whose radii are r ₂ and r ₁ is gradually increased by steps r ₂ and r ₁ respectively, By comparing the influence of different step lengths on the swing period, the calculation formula of the gravitational acceleration g is obtained, and the equivalent pendulum length is obtained by changing the position of the center of mass of the compound pendulum system to the suspension point from the current method.

Compared with the experimental device currently used, the present invention obtains the value of the acceleration of gravity by fitting the slope of the straight line, which reduces the complex and cumbersome drawing work caused by tracing the curve, and reduces the cost of data processing while increasing the data processing speed. Human error improves the accuracy and efficiency of the experiment; at the same time, it avoids the problem that the traced curve cannot accurately find the data points with the same period due to the discrete data; the calculation formula of the acceleration of gravity is obtained by changing the radius of the pendulum, the principle of the experimental method is simple and clear, and it is easy to understand. Suitable for students to operate.

Description of drawings

Fig. 1 is the schematic diagram of cylindrical weight I and cylindrical weight II of the present invention;

Fig. 2 is the schematic diagram of hollow cylindrical weight I of the present invention;

Fig. 3 is the schematic diagram of hollow cylindrical weight II of the present invention;

Fig. 4 is the schematic diagram that utilizes cylindrical weight 1 and hollow cylindrical weight 1 to carry out swing experiment;

Fig. 5 is a schematic diagram of experimental data processing in the present invention.

In the figure: 1. Cylindrical weight I; 2. Cylindrical weight II; 3. Hollow cylindrical weight I with an inner diameter of 2r ₁ ; 4. Hollow cylindrical weight with an inner diameter of 2(x-1)r ₁ Code I; 5. Hollow cylindrical weight I with an inner diameter of 2(n-1)r ₁ ; 6. Hollow cylindrical weight II with an inner diameter of 2r ₂ ; 7. A hollow cylindrical weight with an inner diameter of 2(x-1)r ₂ Hollow cylindrical weight II; 8. Hollow cylindrical weight II with an inner diameter of 2(n-1)r ₂ ; 9. Swing rod.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings and a specific embodiment.

As shown in Figure 1, the cylindrical weight I 1 is a solid cylindrical weight with a height of h and a radius of r1; the cylindrical weight II 2 is a solid cylindrical weight with a height of h and a radius of r2 .

As shown in Figure 2, the hollow cylindrical weight I 3 with an inner diameter of 2r ₁ is made of the same material as the cylindrical weight I 1 , the height is h, the inner diameter is 2r ₁ , and the wall thickness is r ₁ . Can be closely connected with the outer wall of the cylindrical weight I 1; the hollow cylindrical weight I 4 with an inner diameter of 2(x-1)r ₁ , whose wall thickness is r ₁ ; the inner diameter of 2(n-1)r ₁ The hollow cylindrical weight I 5 has an inner diameter of 2(n-1)r ₁ and a wall thickness of r ₁ ; it is known that the inner wall of each hollow cylindrical weight I can be compared with a hollow cylinder whose inner diameter is 2r ₁ smaller The outer walls of the shaped weight I are closely connected, x is an integer between 2 and n, and n is an integer not greater than 15 and not less than 10;

As shown in Figure 3, hollow cylindrical weight II 6 with inner diameter 2r2, hollow cylindrical weight II7 with inner diameter ₂ (x-1) _r2 and hollow cylinder with inner diameter ₂ (n-1)r2 Shaped weight II 8, the weights above are all made of the same material as cylindrical weight II 2, and the height is h. It is known that the inner wall of each hollow cylindrical weight II can be compared with a hollow cylinder whose inner diameter is 2r ₁ smaller. The outer walls of the shape weight II are closely connected, and it is known that r ₂ is greater than r ₁ .

As shown in Figure 4, the experimental method of the present invention comprises the following steps:

1. Take the cylindrical weight I 1 as the pendulum, make the center of the top surface vertically connected to the bottom of the pendulum 9, measure the compound pendulum composed of the cylindrical weight I and the pendulum 9 at a small angle around the apex of the pendulum 9 Swing period when θ swings ;

2. Insert a hollow cylindrical weight I 3 with an inner diameter of 2r ₁ on the outer wall of the cylindrical weight I 1, and measure the swing period T ₂ of the compound pendulum system at this time; insert the weight in turn with a wall thickness of r ₁ and an inner diameter of 2r ₁ For the hollow cylindrical weight increased by the step length, when the hollow cylindrical weight I 4 with an inner diameter of 2(x-1)r ₁ is inserted, the swing period of the compound pendulum system is measured as , so as to obtain n swing cycle data ~ ;

3. Select the cylindrical weight II 2 as the pendulum, form a compound pendulum with the pendulum rod 9, repeat the experimental step (1), and measure the system swing period ;

4. Select the hollow cylindrical weight II shown in Figure 3, repeat the experiment according to the experimental step (2), and obtain n swing cycle data ~ ;

5. As shown in Figure 5, set y= , with x as the horizontal axis and y as the vertical axis to establish a coordinate system, the point (x, ) is plotted in the coordinate system, and the fitting obtains the oblique line ,according to , get .

The above embodiment is only an application of the experimental device of the present invention, not a limitation thereof.

The present invention achieves the purpose of gradually increasing the radius of the compound pendulum pendulum by adding hollow cylindrical weights with gradually increasing inner diameter and constant wall thickness, and then analyzes the swing cycle changes caused by the increasing speed of the pendulum radius with different step lengths, and then Get the formula for calculating the acceleration of gravity g. Compared with the currently used curve fitting experiment method, it has the characteristics of high data processing efficiency and high precision. The principle of the invention is simple and clear, easy to understand, and suitable for physics teaching.

Claims (1)

1. a method utilizing to increase the pendulum radius to measure the acceleration of gravity is characterized in that comprising following experimental steps: (1) Select a cylindrical weight I with a radius of r ₁ and a height of h as a pendulum, and make the center of its top surface vertically connected to the bottom of a rigid and lightweight pendulum with a length of H, and form a compound pendulum with the pendulum , to measure the swing period of the compound pendulum when it swings at a small angle around the apex of the pendulum ; (2) The outer wall of the cylindrical weight I in step 1 is sequentially inserted into the hollow cylindrical weight I of the same material with an inner diameter of 2(x-1)r ₁ , an outer diameter of 2xr ₁ , and a height of h, to Obtain a cylindrical pendulum with a radius of xr ₁ , put a hollow cylindrical weight I in each set to measure the swing period of the compound pendulum, and obtain the swing period of the compound pendulum system when the pendulum radius increases by r ₁ ~ , it is known that x is an integer between 2 and n, and n is an integer not greater than 15 and not less than 10; (3) Select a cylindrical weight II with a radius of r ₂ and a height of h as a pendulum, and connect the center of its top surface to the bottom of a rigid and lightweight pendulum with a length of H to form a compound pendulum with the pendulum. Measure The swing period of a compound pendulum when it swings at a small angle around the apex of the pendulum ; (4) Through the outer wall of the cylindrical weight II in step 3, insert the hollow cylindrical weight II of the same material with an inner diameter of 2(x-1)r ₂ , an outer diameter of 2xr ₂ , and a height of h, To obtain a cylindrical pendulum with a radius of xr ₂ , each set of a hollow cylindrical weight II measures the swing period of the compound pendulum, and obtains the swing period of the compound pendulum system when the pendulum radius increases by r ₂ ~ , it is known that r ₂ is greater than r ₁ ; (5) Let y= , with x as the horizontal axis and y as the vertical axis to establish a coordinate system, the point (x, ) is plotted in the coordinate system, and the fitting obtains the oblique line ,according to , get .