A cylindrical-conical volume ratio experimental device using water displacement

Abstract

The utility model belongs to teaching experiment teaching aid technical field, and disclose a kind of cylinder cone volume ratio experimental device using water replacement, including horizontal bottom plate, main support pole, balance hanging beam, water basket and multiple teaching demonstration sink block, device is based on water replacement principle, for teaching experiment demonstration proof etc. The volume relationship of isobase isohigh cylinder and cone. Horizontal bottom plate ensures that device is stable, and main support pole and balance hanging beam constitute lever system, and leveling assembly can calibrate initial equilibrium state. When experiment, isobase isohigh cylinder and cone sink block are placed into water basket, and sink block extrudes clean water and discharges, and replacement water volume corresponds sink block volume. Replacement water amount is measured by measuring cylinder or weighing apparatus, and cylinder volume can be directly proved to be three times of cone volume. Device converts abstract volume relationship into observable data, and it is easy to operate, student participation is high, device stability is good, error is small, service life is high, teaching effect for student is good, and it has greater practical significance.

Description

Technical Field

[0001] This utility model relates to the field of teaching experimental teaching aids technology, specifically a cylindrical-cone volume ratio experimental device that utilizes water displacement. Background Technology

[0002] In mathematics education, teaching the formulas for the volumes of cylinders and cones is an important part of the elementary school mathematics curriculum. There is a fixed proportional relationship between the volumes of cylinders and cones with the same base and height; that is, the volume of a cone is one-third the volume of a cylinder. This conclusion has been proven through various scientific and rigorous methods throughout the history of mathematics. From a theoretical derivation perspective, calculus, as an important tool in modern mathematics, can precisely derive the volume formulas for cylinders and cones by infinitely subdividing geometric solids using definite integrals and then summing the results. Archimedes' method of exhaustion, centered on the concept of limits, continuously approximates the volume of a cone by using inscribed and circumscribed prisms and pyramids, establishing a rigorous foundation for the volume formulas through proof by contradiction. Zu Geng's principle, starting from a geometric intuition perspective, utilizes the property that geometric solids with equal cross-sectional areas have equivalent volumes to transform cylinders and cones into geometric solids with known volumes for derivation. These methods reveal the essence of the volume formulas for cylinders and cones from different angles and are of great significance in the mathematical theoretical system. However, these traditional proof methods have significant limitations in elementary school teaching practice. Elementary school students have limited cognitive abilities and mathematical knowledge; their thinking is still primarily concrete, making it difficult for them to grasp abstract mathematical concepts and complex derivation processes such as calculus and the method of exhaustion. Directly applying these methods in the classroom not only fails to help students understand the principles behind the formulas for the volumes of cylinders and cones, but may also instill a sense of difficulty, negatively impacting their enthusiasm and effectiveness in learning mathematics. Therefore, in teaching the volume of cylinders and cones in elementary school, there is an urgent need for a simple, direct teaching method that aligns with the cognitive characteristics of elementary school students, coupled with appropriate teaching aids. Through intuitive demonstrations and hands-on activities, abstract mathematical knowledge can be transformed into concrete content that students can easily understand. This helps students better grasp the relationship between the volumes of cylinders and cones, understand the essence of the volume formula, and thus improve the quality of classroom teaching, laying a solid foundation for their subsequent mathematical learning. Utility Model Content

[0003] To address the shortcomings of existing technologies, this invention provides an experimental device for comparing the volume ratio of a cylinder and a cone using water displacement, thereby solving the problems mentioned in the background section, such as the lack of intuitiveness of existing teaching methods and the difficulty for primary school students to understand them.

[0004] To achieve the above-mentioned objectives, this utility model provides the following technical solution: an experimental device for the volume ratio of a cylinder and a cone using water displacement, comprising: A horizontal base plate, which is a planar plate structure, and two sets of water containers are provided on the horizontal base plate; The main support rod is a rigid column structure, which is mounted on the horizontal base plate and is perpendicular to the horizontal base plate. The balance beam is a strip-shaped rigid square bar, which is located above the main support bar and is hinged to the main support bar at its middle part. Two sets of water baskets are respectively set at both ends of the balance beam, and the two sets of water baskets contain an equal amount of clean water; Multiple sets of teaching demonstration blocks, including multiple pairs of cylindrical blocks and conical blocks of equal height and depth, wherein the density of the teaching demonstration blocks is greater than the density of water.

[0005] Preferably, the balance beam has a hinge hole in the middle, and the main support rod has a hinge rod on the upper part. The hinge rod passes through the hinge hole and is rotatably connected to the balance beam. The balance beam is hinged to the main support rod through the hinge rod.

[0006] Preferably, the bottom of the balance beam is fixedly connected to two sets of hooks, which are respectively set on the left and right arms of the balance beam. A leveling component is provided on the balance beam between the hooks and the hinge rods, and a balance pointer is provided on the balance beam at the hinge rod.

[0007] Preferably, the leveling component includes a counterweight slider and slider markings. The counterweight slider is slidably engaged with the balance beam, and slider markings are provided on the balance beam at the position corresponding to the counterweight slider.

[0008] Preferably, a pointer mark is provided on the main support rod at the position corresponding to the balance pointer.

[0009] Preferably, the water basket includes a water tank, a drain pipe, a single-leaf valve, and a hook. The water tank is a cylindrical hollow barrel structure with an open top and side. A horizontal drain pipe is provided at the waist of the water tank, and a single-leaf valve is provided on the drain pipe. A hook is provided at the top of the water tank. Two sets of the water baskets are suspended on both sides of the balance beam by the hook and hook. A deformation-resistant reinforcement component is provided at the connection between the hook and the water tank.

[0010] Preferably, the anti-deformation reinforcement component includes an outer ring reinforcement rib and an inner ring reinforcement rib. The outer ring reinforcement rib and the inner ring reinforcement rib are made of metal with good mechanical properties. The outer ring reinforcement rib and the inner ring reinforcement rib are respectively fixedly installed on the inner and outer sides of the water tank body. The end of the hook passes through the outer ring reinforcement rib, the water tank body and the inner ring reinforcement rib in sequence.

[0011] Compared with the prior art, this utility model provides an experimental device for the volume ratio of a cylinder and a cone using water displacement, which has the following beneficial effects: This experimental apparatus for comparing the volume of a cylinder and a cone using water displacement is equipped with a horizontal base plate, main support rod, balance beam, water basket, and demonstration sink. Combined with purified water, collection containers, and weighing equipment, it demonstrates how water can displace the volume of a cylinder or cone. The process is clear and easy to understand, facilitating demonstrations for students. Most operations are simple and can be performed by students themselves, resulting in strong student participation and excellent teaching effectiveness. The apparatus is stable, has low error, and a long service life, making it highly practical and effective for student learning. Attached Figure Description

[0012] Figure 1 This is a schematic diagram of the overall structure of this utility model; Figure 2 This is a schematic diagram of the overall structure of this utility model; Figure 3 This is a schematic diagram of the balance beam structure of this utility model; Figure 4 This is a schematic diagram of the structure of the water-filled suspended platform of this utility model.

[0013] In the diagram: 1. Horizontal base plate; 2. Main support rod; 3. Balance hanging beam; 4. Water basket; 5. Teaching demonstration sinker; 6. Hinge rod; 7. Hook; 8. Leveling component; 9. Counterweight slider; 10. Balance pointer; 11. Slider mark; 12. Pointer mark; 13. Water tank; 14. Drain pipe; 15. Single-leaf valve; 16. Lifting device; 17. Deformation-resistant reinforcement component; 18. Outer ring reinforcement; 19. Inner ring reinforcement. Detailed Implementation

[0014] The technical solutions of the present utility model will be clearly and completely described below with reference to the accompanying drawings of the embodiments. Obviously, the described embodiments are only some embodiments of the present utility model, and not all embodiments. Based on the embodiments of the present utility model, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the protection scope of the present utility model.

[0015] Please see Figure 1-4 This utility model provides a technical solution: An experimental apparatus for determining the volume ratio of a cylinder to a cone using water displacement includes: The horizontal base plate 1 is a flat plate structure. Two sets of water containers are set on the horizontal base plate 1. The water containers can be graduated cylinders or beakers. Using graduated cylinders can quickly obtain results, but the disadvantage is that the capacity is small and the accuracy is insufficient. Using beakers requires the use of a weighing device to weigh the weight of the displaced water to obtain its quantity. The accuracy is high (the accuracy is determined by the weighing device, and the existing weighing devices are very accurate).

[0016] Main support rod 2 is a rigid column structure. Main support rod 2 is set on the horizontal base plate 1 and is perpendicular to the horizontal base plate 1. Balance beam 3 is a strip-shaped rigid square bar. Balance beam 3 is set above the main support 2, and the middle part of balance beam 3 is hinged to the main support 2. Two sets of water baskets 4 are respectively set at both ends of the balance beam 3, and the two sets of water baskets 4 contain an equal amount of clean water. Multiple sets of teaching demonstration sinking blocks 5, comprising multiple pairs of cylindrical and conical blocks of equal height and depth, are used. The density of the teaching demonstration sinking blocks 5 is greater than that of water. During the teaching experiment demonstration, the cylindrical and conical sinking blocks of equal height and depth can be placed in two sets of water baskets 4 respectively. Because the density of the teaching demonstration sinking blocks 5 is higher than that of water, they will sink to the bottom, and the purified water will be squeezed out. The squeezed-out purified water is collected, which is equivalent to replacing the teaching demonstration sinking blocks 5. Its volume is the same as that of the teaching demonstration sinking blocks 5. By comparing the amount of water replaced on both sides, the volume relationship between the cylinders and cones can be clearly and intuitively determined.

[0017] Furthermore, a hinge hole is provided in the middle of the balance beam 3, and a hinge rod 6 is provided on the upper part of the main support rod 2. The hinge rod 6 passes through the hinge hole and is rotatably connected to the balance beam 3. The balance beam 3 is hinged to the main support rod 2 through the hinge rod 6.

[0018] Furthermore, two sets of hooks 7 are fixedly connected to the bottom of the balance beam 3. The two sets of hooks 7 are respectively set on the left and right arms of the balance beam 3. A leveling component 8 is set on the balance beam 3 between the hooks 7 and the hinge rod 6. A balance pointer 10 is set on the balance beam 3 at the position corresponding to the hinge rod 6. The leveling component 8 is used for leveling. With repeated use of the equipment, it cannot guarantee straightness. That is, when the two water baskets 4 contain equal amounts of clean water, the entire balance beam 3 may not be balanced. The balance pointer 10 is used to indicate the balance status.

[0019] Furthermore, the leveling component 8 includes a counterweight slider 9 and slider markings 11. The counterweight slider 9 is slidably engaged with the balance beam 3, and slider markings 11 are provided on the balance beam 3 corresponding to the counterweight slider 9. Based on the lever principle, leveling can be achieved by adjusting the lever arm length of the counterweight slider 9, i.e., the position of the counterweight slider 9 on the balance beam 3. Slider markings 11 are used to indicate the leveling point, facilitating multiple leveling operations in a short period. In actual teaching experiments for students, it is not necessary to explain the leveling principle; it is recommended to briefly mention it to introduce the concept of error to students.

[0020] Furthermore, a pointer mark 12 is provided on the main support rod 2 at the position corresponding to the balance pointer 10. The balance pointer 10 rotates around the hinged rod 6 with the balance beam 3, and the pointer mark 12 facilitates the indication of the tilt direction and the determination of whether the level has been achieved.

[0021] Furthermore, the water basket 4 includes a water tank 13, a drain pipe 14, a single-leaf valve 15, and a hook 16. The water tank 13 is a cylindrical hollow barrel structure with an opening on the top side. A horizontal drain pipe 14 is provided at the waist of the water tank 13. A single-leaf valve 15 is provided on the drain pipe 14. A hook 16 is provided at the top of the water tank 13. The two sets of water baskets 4 are suspended on both sides of the balance beam 3 through the hook 16 and the hook 7. A deformation-resistant reinforcing component 17 is provided at the connection between the hook 16 and the water tank 13. In use, fill the water basket 4 with clean water, then open the single-leaf valve 15 to allow excess clean water to drain naturally. After draining, close the single-leaf valve 15. Repeat the same operation for both sets of water baskets 4. Then suspend them on both sides of the balance beam 3. At this time, place the teaching demonstration sink 5. The teaching demonstration sink 5 will squeeze the liquid, causing the water level to rise. Then open the single-leaf valve 15 to collect the clean water squeezed out by the teaching demonstration sink 5. The volume of the newly discharged clean water is equivalent to the volume of the teaching demonstration sink 5 (although there may be errors due to temperature fluctuations, evaporation, etc., the error is extremely small and does not affect the teaching experiment demonstration for elementary school students). The volume of the collected replacement water can be obtained by reading a graduated cylinder or by weighing (the density of water is fixed, and the weight is equal to the volume). This is convenient for teaching experiment demonstrations. The operation is simple and convenient. Students can operate the teaching demonstration sink 5, open the single-leaf valve 15 to receive the replacement water, and read or weigh the water. Students have a strong sense of participation and the teaching effect is good.

[0022] Furthermore, the anti-deformation reinforcement component 17 includes an outer reinforcing rib 18 and an inner reinforcing rib 19. The outer reinforcing rib 18 and the inner reinforcing rib 19 are made of metal with good mechanical properties. The outer reinforcing rib 18 and the inner reinforcing rib 19 are respectively fixedly installed on the inner and outer sides of the water-containing bucket 13. The end of the hook 16 passes through the outer reinforcing rib 18, the water-containing bucket 13, and the inner reinforcing rib 19 in sequence. The entire water basket 4 is the same as the outer cavity basket used for density measurement by the drainage method. Existing equipment can be purchased and simply modified. Some baskets do not have handles and need to be equipped with hooks 16. It should be noted that in the density measurement by the drainage method, a single bucket is used and the weight is measured during use. Therefore, the deformation of the bucket does not affect the density measurement. In this teaching experiment demonstration, the deformation of the bucket has a great impact on the experimental results. Therefore, it is necessary to add the anti-deformation reinforcement component 17 to prevent the bucket from deforming due to the uneven tension of the hook 16.

[0023] Structural Description: Horizontal base plate 1: The basic support structure of the device, which is a flat plate used to place water containers and ensure the horizontal stability of the entire device; Main support rod 2: a rigid column structure, vertically fixed to the horizontal base plate 1, used to support the balance beam 3 and ensure its stability and verticality; Balance beam 3: a strip-shaped rigid square bar, which is hinged to the main support bar 2 in the middle and used to suspend the water basket 4 at both ends, forming a lever system; Water basket 4: It consists of a water tank 13, a drain pipe 14, a single-leaf valve 15 and a hanging 16, and is used to hold clean water and discharge the water replaced by the sediment. Teaching demonstration block 5: It includes multiple pairs of cylindrical and conical blocks with the same base and height. The density is greater than that of water. It is used to replace water in experiments to demonstrate the volume relationship. Hinged rod 6: It is set on the upper part of the main support rod 2 and passes through the hinge hole in the middle of the balance beam 3, so that the balance beam 3 can rotate around it. Hook 7: Fixed to the left and right arms at the bottom of the balance beam 3, used to suspend the water basket 4, so that the water basket 4 can be stably hung on both sides of the balance beam 3; Leveling component 8: Located between hook 7 and hinge rod 6 on the balance beam 3, it is used for the initial leveling of the balance beam 3 to ensure experimental accuracy; Counterweight slider 9: It can be slidably attached to the balance beam 3. By adjusting its position, the lever arm length can be changed to achieve the leveling of the balance beam 3. Balance pointer 10: Set at the corresponding hinge rod 6 on the balance beam 3, used to indicate whether the balance beam 3 is in a horizontal balance state; Slider mark 11: Set on the balance beam 3 at the corresponding counterweight slider 9, used to record the balance position of the counterweight slider 9 for easy and quick leveling; Pointer mark 12: Set on the main support rod 2 at the position corresponding to the balance pointer 10, used to help determine the position of the balance pointer 10 and whether it is leveled; Water tank 13: A cylindrical hollow barrel-shaped structure with an opening on the top side, used to hold clean water, and is the main part of the water basket 4; Drain pipe 14: horizontally installed at the waist of the water tank 13, used to drain the clean water displaced by the teaching demonstration sediment block 5; Single-leaf opening and closing valve 15: installed on the drain pipe 14, used to control the opening and closing of the drain pipe 14, to facilitate drainage and water stoppage; Hanging 16: Located at the top of the water tank 13, used to suspend the water basket 4 on the hook 7 of the balance beam 3; Deformation-resistant reinforcement component 17: installed at the connection between the hook 16 and the water tank 13, to prevent the water tank 13 from deforming due to force; Outer reinforcing rib 18: made of metal, fixedly installed on the outside of the water tank 13, together with the inner reinforcing rib 19 to reinforce the water tank 13; Inner ring reinforcing rib 19: made of metal, fixedly installed on the inside of the water tank 13, and works with the outer ring reinforcing rib 18 to prevent deformation of the tank.

[0024] Working Principle: In the experimental preparation stage, the horizontal base plate 1 is first placed stably on the operating table. Its planar plate structure ensures the overall stability of the device. Two sets of water containers are placed on the horizontal base plate 1. Users can choose between graduated cylinders or beakers according to experimental needs. For rapid reading, graduated cylinders can be used; for high-precision measurement, beakers are used in conjunction with high-precision scales. The main support rod 2 of the device is vertically fixed to the horizontal base plate 1, and its rigid structure provides stable support for the subsequent balancing system. The hinge rod 6 of the device connects the middle of the balancing beam 3 to the top of the main support rod 2, allowing the balancing beam 3 to rotate flexibly around the hinge point. Initial calibration of the balance beam 3 is performed using the leveling component 8: two pre-filled water baskets 4 are hung on the hooks 7 at both ends of the balance beam 3, and the relative positions of the balance pointer 10 and the pointer mark 12 are observed. The position of the counterweight slider 9 on the balance beam 3 is adjusted by sliding the counterweight slider 9. The lever arm length is changed according to the lever principle until the balance pointer 10 and the pointer mark 12 coincide, ensuring that the balance beam 3 is in a horizontal balance state. At this time, the slider mark 11 records the current balance position of the counterweight slider 9, which is convenient for subsequent quick leveling.

[0025] In the experimental operation phase, a set of cylindrical and conical blocks with equal bases and heights were selected as demonstration sinking blocks 5. Since the density of the demonstration sinking blocks 5 is greater than that of water, when they are slowly placed into the two water baskets 4 on either side, the blocks will sink to the bottom, squeezing the clean water in the water baskets 4 and causing it to drain out through the drain pipe 14. The structural design of the water baskets 4 plays a crucial role in this process: the water-holding tank 13 is used to hold the clean water, and the top hook 16 is connected to the hook 7 of the balance beam 3; the drain pipe 14 at the waist is initially closed under the control of the single-leaf valve 15 to maintain a stable water level in the water baskets 4; when the sinking blocks are placed in, causing the water level to rise, the single-leaf valve 15 is opened, and the clean water displaced by the sinking blocks can flow out through the drain pipe 14. To prevent the force on the hook 16 from causing deformation of the water-holding tank 13 and affecting the experimental accuracy, the outer reinforcing ribs 18 and inner reinforcing ribs 19 in the anti-deformation reinforcement component 17 reinforce the tank from both the inside and outside, ensuring that the drainage volume accurately reflects the volume of the sinking blocks.

[0026] After drainage is complete, close the single-leaf valve 15 and collect the displacement water discharged from the water baskets 4 on both sides. If a graduated cylinder is used as the water container, the volume of the displacement water can be obtained by directly reading the graduated cylinder scale; if a beaker is used, the weight of the displacement water can be obtained by weighing it with a scale (water density is fixed, so weight is equal to volume). At this time, the volumes of the displacement water on both sides correspond to the volumes of the cylindrical block and the conical block, respectively.

[0027] Before the experiment, the balancing beam 3 was in equilibrium under the action of equal amounts of purified water in the water baskets 4 on both sides. When a cylindrical block and a conical block with the same base and height were placed in the basket (the ratio of the volume of the cylinder to the volume of the cone is 3:1), according to Archimedes' principle, the amount of water displaced by the cylindrical block should be three times that of the conical block. Combining this with the collected data on the volume of displaced water, it can be precisely proven from a numerical perspective that the volume of the cylindrical block with the same base and height is three times that of the conical block.

[0028] Throughout the teaching demonstration experiment, the device transforms abstract volume relationships into observable drainage volume data through structural design. The operation steps are simple and easy to understand, allowing students to participate in the dropping of sink blocks, drainage collection, and data measurement. This effectively enhances the teaching effect and helps students deeply grasp the essence of the volume formulas for cylinders and cones.

[0029] Although embodiments of the present invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the present invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. An experimental apparatus for determining the volume ratio of a cylinder to a cone using water displacement, comprising: A horizontal base plate (1) is a planar plate structure, and two sets of water containers are provided on the horizontal base plate (1). Main support rod (2), the main support rod (2) is a rigid column structure, the main support rod (2) is set on the horizontal base plate (1), and the main support rod (2) is perpendicular to the horizontal base plate (1). Balance beam (3), the balance beam (3) is a strip-shaped rigid square bar, the balance beam (3) is set above the main support rod (2), and the middle part of the balance beam (3) is hinged to the main support rod (2); Two sets of water baskets (4) are respectively set at both ends of the balance beam (3), and the two sets of water baskets (4) contain an equal amount of clean water; Multiple sets of teaching demonstration sink blocks (5), the multiple sets of teaching demonstration sink blocks (5) include multiple pairs of cylindrical blocks and conical blocks of equal height and equal depth, the density of the teaching demonstration sink blocks (5) is greater than the density of water.

2. The experimental apparatus for determining the volume ratio of a cylinder and a cone using water displacement as described in claim 1, characterized in that, The balance beam (3) has a hinge hole in the middle, and the main support rod (2) has a hinge rod (6) on the upper part. The hinge rod (6) passes through the hinge hole and is rotatably connected to the balance beam (3). The balance beam (3) is hinged to the main support rod (2) through the hinge rod (6).

3. The experimental apparatus for the volume ratio of a cylinder and a cone using water displacement according to claim 2, characterized in that, The bottom of the balance beam (3) is fixedly connected with two sets of hooks (7). The two sets of hooks (7) are respectively set on the left and right arms of the balance beam (3). A leveling component (8) is set on the balance beam (3) between the hooks (7) and the hinge rod (6). A balance pointer (10) is set on the balance beam (3) at the location corresponding to the hinge rod (6).

4. The experimental apparatus for the volume ratio of a cylinder and a cone using water displacement according to claim 3, characterized in that, The leveling component (8) includes a counterweight slider (9) and a slider mark (11). The counterweight slider (9) is slidably engaged on the balance beam (3), and the balance beam (3) is provided with a slider mark (11) corresponding to the counterweight slider (9).

5. The experimental apparatus for the volume ratio of a cylinder and a cone using water displacement according to claim 3, characterized in that, A pointer mark (12) is provided on the main support rod (2) at the position corresponding to the balance pointer (10).

6. The experimental apparatus for the volume ratio of a cylinder and a cone using water displacement according to claim 3, characterized in that, The water basket (4) includes a water tank (13), a drain pipe (14), a single-leaf valve (15), and a hook (16). The water tank (13) is a cylindrical hollow barrel structure with an opening on the top side. A horizontal drain pipe (14) is provided at the waist of the water tank (13). A single-leaf valve (15) is provided on the drain pipe (14). A hook (16) is provided at the top of the water tank (13). The two sets of water baskets (4) are suspended on both sides of the balance beam (3) by the hook (16) and the hook (7). A deformation-resistant reinforcing component (17) is provided at the connection between the hook (16) and the water tank (13).

7. The experimental apparatus for the volume ratio of a cylinder and a cone using water displacement according to claim 6, characterized in that, The anti-deformation reinforcement component (17) includes an outer ring reinforcement rib (18) and an inner ring reinforcement rib (19). The outer ring reinforcement rib (18) and the inner ring reinforcement rib (19) are made of metal with good mechanical properties. The outer ring reinforcement rib (18) and the inner ring reinforcement rib (19) are respectively fixedly installed on the inner and outer sides of the water tank (13). The end of the hook (16) passes through the outer ring reinforcement rib (18), the water tank (13) and the inner ring reinforcement rib (19) in sequence.

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