Device and method for measuring gravitational acceleration by air flow compensation

Abstract

The application relates to a device and a method for measuring gravitational acceleration by air flow compensation method, which comprises a clamping device, a ball and a pipeline. The ball comprises a small ball and a large ball; the clamping device suspends the small ball through a thin rope, a thin rope is connected to the large ball below the small ball, the thin rope passes through the center through hole of the small ball above, is suspended to the thin rope, and the two balls are suspended in sequence without force action. The pipeline 5 is sequentially provided with a first light hole, a second light hole and a third light hole from top to bottom, and is respectively provided with a first photoelectric gate, a second photoelectric gate and a third photoelectric gate at the same horizontal height of the first light hole, the second light hole and the third light hole. The application has the beneficial effect that the application does not need complex equipment and operation, and can complete the reduction of the influence of air resistance through the falling of multiple balls, compared with other mechanical structures, the small ball does not contact other media except air, and therefore new errors need not be considered.

Description

Technical Field

[0001] This invention relates to the field of physical quantity measurement, and more specifically, to an apparatus and method for measuring gravitational acceleration using the airflow compensation method. Background Technology

[0002] In physics teaching, measuring gravitational acceleration is often a basic experimental project for middle and high school students. Measuring gravitational acceleration by free fall is a traditional method, but due to air resistance, the measurement results can have significant errors.

[0003] In experiments, the acceleration due to gravity is typically calculated by measuring the fall time and corresponding height of a metal ball. The falling metal ball is acted upon by several forces: 1. The resistance generated by air molecules colliding with the ball directly in front of it; 2. The viscous drag of the air during its fall; 3. The buoyancy of the air. Therefore, numerous experimental schemes have been proposed to eliminate air resistance. However, due to the high cost or complexity of sophisticated equipment, methods such as eliminating air resistance by creating a vacuum in a sealed space, or deriving approximate formulas to correct the results based on the relationship between air resistance and the ball's falling speed, are cumbersome and computationally complex. Therefore, new methods are needed to simplify these approaches. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and to provide an apparatus and method for measuring gravitational acceleration using the airflow compensation method.

[0005] In the first aspect, an apparatus for measuring gravitational acceleration using the airflow compensation method is provided, comprising: a clamping device, a sphere, and a pipe;

[0006] The spheres include a small sphere and a large sphere. The clamping device suspends the small sphere by a thin rope. A thin rope is connected to the large sphere below the small sphere, and the rope passes through the central hole of the small sphere and is suspended from the rope, so that there is no force between the two spheres, which are suspended sequentially from top to bottom. The diameter of the large sphere is smaller than and close to the inner diameter of the pipe. The center of the pipe is aligned with the small sphere and the large sphere. The pipe is provided with a first light-transmitting hole, a second light-transmitting hole and a third light-transmitting hole from top to bottom, and a first photoelectric gate, a second photoelectric gate and a third photoelectric gate are respectively provided at the same horizontal height of the first light-transmitting hole, the second light-transmitting hole and the third light-transmitting hole.

[0007] Preferably, the first photoelectric gate, the second photoelectric gate, and the third photoelectric gate are all connected to a time measuring instrument.

[0008] Preferably, a plumb line is provided next to the pipe.

[0009] Secondly, a method for measuring gravitational acceleration using airflow compensation is provided, including:

[0010] S1. Adjust the position of the pipe using a plumb line so that the pipe is parallel to the plumb line;

[0011] S2. A small ball is suspended by a clamping device. A thin rope is connected to the large ball below the small ball. The thin rope passes through the central hole of the small ball and is suspended by the thin rope, so that there is no force between the two balls. The balls are suspended one above the other and the positions and heights of the small ball and the large ball are adjusted so that they are located above the center of the pipe.

[0012] S3. After the small ball and the large ball are suspended and come to rest, the clamping device is released, and the small ball and the large ball fall freely at the same time, passing through the first light-transmitting hole, the second light-transmitting hole and the third light-transmitting hole in the pipe.

[0013] S4. Measure and record the times when the ball falls into the first photogate, the second photogate, and the third photogate;

[0014] S5. Calculate the gravitational acceleration based on the distance the ball falls and the time of fall.

[0015] Preferably, in S4, the times when the ball falls onto the first photogate, the second photogate, and the third photogate are t1, t2, and t3, respectively, and the distance between the first photogate and the second photogate is h. 12 The distance between the first photogate and the third photogate is h. 13 In S5, the formula for calculating gravitational acceleration is:

[0016] g = 2 × (h) 13 ×△t 12 -h 12 ×△t 13 ) / (△t 12 ×△t 13 2 -△t 13 ×△t 12 2 )

[0017] Wherein, △t 12 = t2-t1, Δt 13 = t3-t1.

[0018] The beneficial effects of this invention are as follows: This invention proposes a novel method for measuring gravitational acceleration that does not require complex equipment or operations. It reduces the influence of air resistance simply by dropping multiple spheres. Compared to other mechanical structures, the spheres do not come into contact with any medium other than air, thus eliminating the need to consider new errors. Therefore, this invention reduces equipment costs and achieves higher accuracy. Attached Figure Description

[0019] Figure 1A schematic diagram illustrating the method for measuring gravitational acceleration using the airflow compensation method;

[0020] Figure 2 This is a force diagram showing the forces acting on the smaller and larger balls.

[0021] Figure 3 This is a schematic diagram of airflow compensation for a single large sphere;

[0022] Figure 4 A schematic diagram of airflow compensation for multiple large spheres;

[0023] Figure 5 A schematic diagram of a device for measuring gravitational acceleration using the airflow compensation method;

[0024] Explanation of reference numerals in the attached drawings: 1. Clamping device; 2. Thin rope; 3. Small ball; 4. Large ball; 5. Pipe; 6. First photoelectric gate; 7. Second photoelectric gate; 8. Third photoelectric gate; 9. First light-transmitting hole; 10. Second light-transmitting hole; 11. Third light-transmitting hole. Detailed Implementation

[0025] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0026] Example 1:

[0027] To reduce the interference of air resistance during the free fall of the measured ball, it is necessary to create a scenario where the downward flow of air is synchronized with the falling speed of the ball. In such a scenario, the falling of the measured metal ball and the airflow occur simultaneously, making the relative velocity between the measured metal ball and the air very small or even negligible, thus improving measurement accuracy.

[0028] To create a scenario where the downward flow of air is synchronized with the falling speed of the ball being measured, an automatic air pump can be used to inject air into the upper part of the pipe to accelerate the downward flow of air. However, controlling the air pump brings certain difficulties and complexities. Therefore, this invention utilizes the airflow compensation method to measure gravitational acceleration.

[0029] Specifically, the present invention provides an apparatus for measuring gravitational acceleration using the airflow compensation method, such as... Figure 5 As shown, it includes: clamping device 1, ball and pipe 5;

[0030] The spheres include a small ball 3 and a large ball 4. The clamping device 1 suspends the small ball 3 via a thin rope 2. A thin rope is connected to the large ball 4 below the small ball 3. The thin rope passes through the central through hole of the small ball 3 and is suspended by the thin rope 2, so that there is no force between the two spheres. They are suspended sequentially from top to bottom. The diameter of the large ball 4 is smaller than and close to the inner diameter of the pipe 5. The center of the pipe 5 is aligned with the small ball 3 and the large ball 4. The pipe 5 is provided with a first light-transmitting hole 9, a second light-transmitting hole 10 and a third light-transmitting hole 11 from top to bottom. At the same horizontal height of the first light-transmitting hole 9, the second light-transmitting hole 10 and the third light-transmitting hole 11, a first photoelectric gate 6, a second photoelectric gate 7 and a third photoelectric gate 8 are respectively provided.

[0031] In the aforementioned device, a plumb line is installed beside pipe 5, and photoelectric gates (including transmitters and receivers) are installed on the outer sides of three pairs of opposing light-transmitting holes on the pipe wall. The side of pipe 5 is sealed, and the photoelectric gates fit tightly with the light-transmitting holes to ensure that the side of pipe 5 is airtight. The first photoelectric gate 6, the second photoelectric gate 7, and the third photoelectric gate 8 are all connected to a time measuring instrument. The small ball 3 is a sphere with a small hole in the middle. A thin rope 2 is connected above the large ball 4, passes through the small hole in the small ball 3, and is suspended from the clamping device 1. Another thin line is connected at one end to the top of the small ball 3 and at the other end to the thin line suspending the large ball 4. The large ball 4 is made of high-density material and can be a metal ball (such as a lead ball with a radius of r). The inner wall of pipe 5 is smooth and is a hollow pipe (with an inner diameter of R, where R is slightly larger than r). Figure 3 As shown, if the hollow pipe is placed vertically, and the metal ball falls inside the pipe without friction against the pipe wall, the acceleration of the airflow moving downwards inside the pipe is close to the acceleration due to gravity, g. The ball being measured falls synchronously with the metal ball, and the relative velocity between the ball being measured and the air is very small or even negligible. Furthermore, the aforementioned metal ball can also be replaced with a metal cylinder (such as a lead cylinder with radius r).

[0032] Based on the above-described apparatus, the present invention provides a method for measuring gravitational acceleration using the airflow compensation method, comprising:

[0033] S1. Adjust the position of pipe 5 using a plumb line so that pipe 5 is parallel to the plumb line.

[0034] In S1, there is a plumb line on the side of the pipe for adjusting the plumbness of the pipe. The staff observes whether the pipe is vertical and adjusts the pipe to be parallel to the plumb line.

[0035] S2. The thin rope 2 is clamped by the clamping device 1 to suspend the small ball 3. A thin rope is connected to the large ball 4 below the small ball 3. The thin rope passes through the central through hole of the small ball 3 and is suspended by the thin rope 2, so that there is no force between the two balls. They are suspended one above the other, and the position and height of the small ball 3 and the large ball 4 are adjusted so that they are located above the center of the pipe.

[0036] S3. After the small ball 3 and the large ball 4 are suspended and come to rest, the clamping device 1 is released, and the small ball 3 and the large ball 4 fall freely at the same time, passing through the first light-transmitting hole 9, the second light-transmitting hole 10 and the third light-transmitting hole 11 in the pipe 5.

[0037] In S2 and S3, the device suspending the large ball 4 and the small ball 3 is inserted into the pipe 5 from the top of the pipe 5. The bottom height of the large ball 4 is greater than the height of the first photoelectric gate 6. When it is inserted, the center of the large ball 4 is adjusted to be aligned with the center of the pipe 5 by the adjustment device to prevent the large ball 4 from falling and rubbing against the pipe wall.

[0038] When large ball 4 and small ball 3 fall freely at the same time, large ball 4 accelerates downwards below small ball 3, causing the air to accelerate downwards, thus reducing the air resistance interference on small ball 3. Because large ball 4 experiences greater air friction resistance, its downward acceleration is less than that of small ball 3.

[0039] S4. Measure and record the time when ball 3 falls into the first photogate 6, the second photogate 7 and the third photogate 8.

[0040] S5. Calculate the gravitational acceleration based on the distance the ball falls and the time of fall.

[0041] In S4, such as Figure 1 As shown, the times when ball 3 falls onto the first photogate 6, the second photogate 7, and the third photogate 8 are t1, t2, and t3, respectively, and the distance between the first photogate 6 and the second photogate 7 is h. 12 The distance between the first photogate 6 and the third photogate 8 is h. 13 In S5, the formula for calculating gravitational acceleration is:

[0042] g = 2 × (h) 13 ×△t 12 -h 12 ×△t 13 ) / (△t 12 ×△t 13 2 -△t 13 ×△t 12 2 )

[0043] Wherein, △t 12 =t2-t1, Δt 13 = t3-t1.

[0044] When a large ball 4 is suspended, the airflow compensation has a good effect. According to theoretical calculations, the error Δg caused by the air resistance and viscous force on the measured small ball 3 is negligible. If enough large balls 4 are added, the measurement accuracy can be further improved.

[0045] The principle of this invention is as follows: Based on the above-mentioned device, in order to achieve a better airflow compensation effect, the experiment is conducted in a long straight pipe with openings at both ends. A high-density large sphere and a small sphere to be measured fall freely at the same time. The large sphere accelerates its fall below the small sphere, causing the internal air to flow downwards at an acceleration close to g. The small sphere falls freely behind the large sphere (at a certain distance to reduce the influence of complex airflow on the small sphere), and its relative speed with the air is very small, that is, the air resistance experienced by the small sphere is almost eliminated.

[0046] Example 2:

[0047] In a device for measuring gravitational acceleration using the airflow compensation method, let the height of the long straight pipe be h, the internal cross-sectional area be s, and the mass of air inside the pipe be ρ. 空 sh, ρ 空 Let be the air density. When a large ball is placed inside, because the outer diameter (r) of the large ball is smaller than and similar to the inner diameter (R) of the pipe, [the density is determined by...]. Figure 3 It can be seen that when h is much greater than r, the air mass inside the pipe is approximately:

[0048] m 空 =ρ 空 sh

[0049] The falling ball pushes the airflow in the lower section, compressing the air there and stretching the air in the upper section. At low speeds, assuming the stretching and compression of the air in the pipe are negligible, the total air mass in the pipe remains constant, and the friction between the air and the pipe wall is negligible, the force required to accelerate the air is F = m 空 a. Because the air in the pipe is rapidly pushed out of the lower port, it experiences frontal resistance from the external air, the magnitude of which is equivalent to the frontal resistance f of the air on the metal sphere. v f v =kv 2 v is the velocity of the large ball, k is the drag coefficient, viscous drag from the air is negligible, and the buoyant force on the large ball in the air is f. 浮 =m 大球 gρ 空 / ρ 大球 ,according to Figure 2 The forces acting on the medium and large spheres, and the equation of motion for the large sphere are:

[0050] m 大球 a = m 大球 gm 空 af v -f 浮

[0051] 'a' represents the acceleration of the air and the falling metal ball in the pipe, i.e.

[0052] a=(m 大球 gf v -F 浮 ) / (m大球 +m 空 )

[0053] =[m 大球 g-(1 / 2)×Cρ 空 S 迎风 ×(2gh)-m 大球 gρ 空 / ρ 大球 ] / (m 大球 +m 空 )

[0054] =g[1-Cρ 空 S 迎风 h / m 大球 -ρ 空 / ρ 大球 ] / (1+m 空 / m 大球 )

[0055] =g[1-Cρ 空 πR 2 h / m 大球 -ρ 空 / ρ 大球 ] / (1+m 空 / m 大球 )

[0056] =g[1-Cρ 空 πR 2 h / (4 / 3)πr 3 ρ 大球 -ρ 空 / ρ 大球 ] / (1+m 空 / m 大球 )

[0057] =g[1-(3CR)] 2 h / 4r 3 )×(ρ 空 / ρ 大球 )-ρ 空 / ρ 大球 ] / (1+m 空 / m 大球 )

[0058] C is the air drag coefficient, S 迎风 This is the area of ​​the sphere's windward side.

[0059] If the metal sphere is a steel sphere, then the density of air is approximately 1 / 6000 of the density of a steel sphere.

[0060] m 空 / m 大球 =πR 2 Lρ 空 / (4 / 3)πr 3 ρ 大球 =(3R) 2 L / 4r 3 )×(ρ 空 / ρ 大球 )

[0061] Assuming the pipe length L = 1.600 meters, h = 1.500 meters, the steel ball radius r = 0.010 meters, R ≈ 0.010 meters, and C = 0.4, then...

[0062] m 空 / m 大球 ≈(3L / 4r)×(ρ 空 / ρ 大球 = [(3×1.600) / (4×0.010)]×(1 / 6000) = 0.02

[0063] 3CR 2 h / 4r 3 =3Ch / 4r≈3×0.4×1.500 / (4×0.010)=45

[0064] From the formula for calculating acceleration a, we get:

[0065] a=g[1-45×(ρ 空 / ρ 大球 )-ρ 空 / ρ 大球 ] / (1+m 空 / m 大球 )

[0066] = g[1-46×(1 / 6000)] / (1+0.02)

[0067] = (5954 / 6000)g / (1+0.02)

[0068] =0.9729g

[0069] That is, the large ball causes the air inside the pipe to accelerate downwards at a rate of approximately 0.9729g.

[0070] Combination Figure 4 It can be seen that if two steel balls are used, then m 空 / 2m 大球 ≈0.01, that is, a≈0.9825g;

[0071] If n steel balls are used, then m 空 / m n个大球 ≈0.02 / n, that is, a≈(5954 / 6000)g / (1+0.02 / n)).

[0072] Let the sphere being measured have a radius r.小 A ball falls freely in a pipe. Under the influence of gravity, its relative velocity with the air inside the pipe is v'. The ball experiences frontal air resistance f. v' and subject to the viscous force f of air 粘滞 ≈6πηr 小 [v'R] 2 / (R 2 -r 小 2 )] and buoyancy f 浮 The function, according to Figure 2 Forces acting on the small ball, and the equation of motion for the small ball:

[0073] m 小球 a 小球 =m 小球 gf v '-f 粘滞 -f 浮

[0074] a 小球 =gf 浮 / m 小球 -f v ' / m 小球 -f 粘滞 / m 小球

[0075] =gf 浮 / m 小球 -k(v') 2 / m 小球 -6πηr 小 [v'R] 2 / (R 2 -r 小 2 )] / m 小球

[0076] =gf 浮 / m 小球 -1 / 2×Cρ 空 S 迎风 (v') 2 / m 小球 -6πηr 小 [v'R] 2 / (R 2 -r 小 2 )] / m 小球

[0077] Taking C as 0.4, the velocity difference v' of the ball falling 1.500 meters does not exceed (1-0.9729)g×(2×1.500 / g). 1 / 2 η is the air viscosity coefficient, taken as η = 1.87 × 10⁻⁶. -5Pasec, r 小 = 0.005 m, then

[0078] f 浮 / m 小球 ≈ (1 / 6000)g = 1.67×10 -4 g

[0079] f v' / m 小球 = k(v') 2 / m 小球 = 1 / 2×Cρ 空 S 迎风 (v') 2 / m 小球

[0080] = 0.2[(ρ 空 / ρ 小球 (v') 2 ×πr 小球 2 / [(4 / 3)πr 小球 3

[0081] = (1 / 6000)×0.15×(1 / 0.005)×(v') 2

[0082] = (1 / 6000)×0.15×(1 / 0.005)×(a 小球 -0.9729g) 2 ×(2h / g)

[0083] Since a 小球 -0.9729g ≤ 0.0271g

[0084] So f<000​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​2 )] / 4πr 小球 3 ρ 小球 / 3

[0087] Let R = 0.01000(m), r = 0.00500(m), v' = (0.0271g) × (2 × 1.500 / g) 1 / 2

[0088] ρ 小球 =7.8×10 3 (kg / m)

[0089] f 粘滞 / m 小球 ≈8.63×10 -6 g

[0090] Final result:

[0091] a 小球 =gf 浮 / m 小球 -f v ' / m 小球 -f 粘滞 / m 小球

[0092] ≈g-1.67×10 -4 g-1.10×10 -5 g-8.63×10 -6 g

[0093] If the conventional method is used, removing the straight hollow tube (without utilizing the airflow compensation method), the acceleration of the metal sphere being measured is:

[0094] a 小球 =gf 浮 / m 小球 -f v ' / m 小球 -f 粘滞 / m 小球

[0095] ≈g-1.67×10 -4 g-1.50×10 -2 g-3.18×10 -4 g

[0096] Therefore, it can be concluded that the error in measuring gravitational acceleration caused by air buoyancy using the airflow compensation method is 1.67 × 10⁻⁶. -4 g; The error in gravitational acceleration caused by air resistance is 1.10 × 10⁻⁶. -5 g(error without airflow compensation method: 1.50 × 10) -2g); the error in gravitational acceleration caused by air viscosity is only 8.63 × 10⁻⁶. -6 g(The error without airflow compensation is: 3.18 × 10⁻⁶) -4 g).

Claims

1. Method for measuring the acceleration of gravity by the airflow compensation method, characterized in that, The device for measuring the acceleration of gravity by air flow compensation method is executed, and the device for measuring the acceleration of gravity by air flow compensation method comprises clamping devices (1), a ball and a pipeline (5); The ball comprises a small ball (3) and a large ball (4); the clamping device (1) suspends the small ball (3) through a thin rope (2), a thin rope is connected to the large ball (4) below the small ball (3), the thin rope passes through the center through hole of the small ball above, is suspended to the thin rope (2), so that there is no force between the two balls, and the two balls are suspended in turn; the diameter of the large ball (4) is smaller than and close to the inner diameter of the pipeline (5); the central position of the pipeline (5) is aligned with the small ball (3) and the large ball (4); the pipeline (5) is sequentially provided with a first light hole (9), a second light hole (10) and a third light hole (11) from top to bottom, and is respectively provided with a first photoelectric gate (6), a second photoelectric gate (7) and a third photoelectric gate (8) at the same horizontal height of the first light hole (9), the second light hole (10) and the third light hole (11); the first photoelectric gate (6), the second photoelectric gate (7) and the third photoelectric gate (8) are connected with a time measuring instrument; a plumb line is arranged beside the pipeline (5); The method comprises: S1, adjusting the position of the pipeline (5) through the plumb line, so that the pipeline (5) is parallel to the plumb line; S2, suspending the small ball (3) through the clamping device (1), connecting a thin rope to the large ball (4) below the small ball (3), passing the thin rope through the center through hole of the small ball above, suspending the thin rope to the thin rope (2), so that there is no force between the two balls, and the two balls are suspended in turn, and adjusting the position and height of the small ball (3) and the large ball (4) so that they are located above the center of the pipeline; S3, after the small ball (3) and the large ball (4) are suspended and stationary, releasing the clamping device (1), the small ball (3) and the large ball (4) fall freely at the same time, passing through the first light hole (9), the second light hole (10) and the third light hole (11) in the pipeline (5); S4, measuring and recording the time when the small ball (4) falls to the first photoelectric gate (6), the second photoelectric gate (7) and the third photoelectric gate (8); S5, calculating the acceleration of gravity according to the falling distance and falling time of the small ball.

2. The method of claim 1, wherein In S4, the time when the ball (3) falls to the first photoelectric gate (6), the second photoelectric gate (7) and the third photoelectric gate (8) is t1, t2, t3 respectively, the distance between the first photoelectric gate (6) and the second photoelectric gate (7) is h 12 , the distance between the first photoelectric gate (6) and the third photoelectric gate (8) is h 13 ; in S5, the calculation formula of the gravitational acceleration is: g = 2 x (h 13 x Δt 12 - h 12 x Δt 13 ) / (Δt 12 x Δt 13 2 - Δt 13 x Δt 12 2 ) wherein, Δt 12 = t2-t1, Δt 13 = t3-t1.

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