CN104021709A - Method for measuring gravitational acceleration through rolling of small ball - Google Patents
Method for measuring gravitational acceleration through rolling of small ball Download PDFInfo
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- CN104021709A CN104021709A CN201410264404.1A CN201410264404A CN104021709A CN 104021709 A CN104021709 A CN 104021709A CN 201410264404 A CN201410264404 A CN 201410264404A CN 104021709 A CN104021709 A CN 104021709A
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- pendulum
- gravitational acceleration
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Abstract
The invention discloses a method for measuring the gravitational acceleration through rolling of a small ball and relates to the field of physical parameter measurement, in particular to the field of gravitational acceleration measurement. The method for measuring the gravitational acceleration through rolling of the small ball aims to overcome the shortages that a simple pendulum cannot effectively constraint a pendulum ball and conic pendulum overlaying is prone to occurrence. According to the technical scheme, the method comprises the steps that the small ball rolls down from the upper portion of one side of a circular-arc groove with the curvature radius being R and conducts reciprocated swing under mutual conversion of potential energy and kinetic energy; a pair of light holes are formed in the side faces of the circular-arc groove, a light-emitting diode and a receiving diode of a photogate are arranged in the light holes, and the swing period T of the small ball is measured by the photogate, wherein the gravitational acceleration g=(2pi/T)*2*1.4R. The method has the advantages that the small ball is limited by the circular-arc groove and conic pendulum overlaying like that of the simple pendulum is avoided; the method for measuring the gravitational acceleration expands thoughts of students; compared with a method for measuring the gravitational acceleration through a compound pendulum, the measurement process and the data processing process are as simple as those of a method for measuring the gravitational acceleration through the simple pendulum.
Description
Technical field
The present invention relates to the measurement in the mensuration field, particularly acceleration of gravity of physical parameter.
Background technology
In Experiment of College Physics chamber, the conventional single pendulum of the measurement of acceleration of gravity and physical pendulum.The swing ball of single pendulum is only subject to the control of hitch point and cycloid, and cycloid is soft, in the time doing experiment with simple pendulum, if do not carried out, often just to make circular cone and put, circular cone pendulum is equivalent to the single pendulum that pendulum length shortens, cause the cycle of measuring to shorten, thereby cause the acceleration of gravity of measurement bigger than normal; Certainly, be also unlikely to most of the time to reach circular cone pendulum extreme like this, a taper swinging momvement normally superposes in simple harmonic motion.
Also have people's subjectivity to think that to-and-fro movement and the single pendulum of bead on smooth circular arc is equal to completely, as long as pivot angle is enough little, this swing is exactly simple harmonic motion.At this moment the pendulum length in Period Formula should be circular arc radius-of-curvature
rwith the little radius of a ball
rpoor.But the motion of single pendulum is the translation being subject in the circular arc path of bracing wire constraint, the motion of bead on curve is to roll, instead of translation, therefore, has the essential distinction with single pendulum.
Summary of the invention
Can not realize operative constraint easily occur the superposeing problem of circular cone pendulum for overcoming single pendulum to swing ball, the present invention proposes a kind of method of bead tolling measurement acceleration of gravity, is a kind of method that alternative single pendulum is measured acceleration of gravity.
The technical scheme that the present invention realizes goal of the invention employing is: the method for utilizing bead tolling measurement acceleration of gravity, it is characterized in that: one section of circular arc, the central angle that this section of circular arc opened its center of circle is less than or equal to 180 degree, described circular arc is R apart from circular arc bottom around one, and fixed point turns and forms one section of circular groove at a plane internal rotation, its radius-of-curvature is R, and this circular groove utilizes mould to adopt casting to make or adopt milling machine milling cutter to make or other technique that can reach this parameter attribute completes; A bead rolls down from the top of circular groove one side, under the mutual conversion of potential energy and kinetic energy, does reciprocally swinging; For avoiding bead to roll out circular groove, circular arc both sides of the edge are at least greater than the radius of bead to the height of circular arc bottom; There is pair of light-transmissive hole toroidal cavity side, arranges light emitting diode and the reception diode of photoelectricity door at this light hole, the cycle T that photoelectricity door swings for measuring bead, gravity acceleration g=(2 π/T)
2* 1.4R.
The invention has the beneficial effects as follows: single pendulum only has a hitch point, swing ball end is movable, swing ball and cycloid easily occur not at a flat in-plane moving, there is the simple harmonic motion of a circular cone pendulum of stack, even develop into taper swinging momvement, cause equivalent pendulum length to shorten, the present invention is subject to the restriction of circular groove, and the situation of circular cone pendulum can not appear in bead as single pendulum; The present invention proposes the method that another measures acceleration of gravity, expanded student's thinking; Physical pendulum is measured acceleration of gravity need to measure numerous data points, and data handling procedure is also more loaded down with trivial details, more consuming time, measures acceleration of gravity with respect to physical pendulum, and measuring process is simple as single pendulum is measured acceleration of gravity with data handling procedure.
Brief description of the drawings
Fig. 1 is circular cone pendulum schematic diagram; Fig. 2 is the method schematic diagram that a kind of bead that rolls is measured acceleration of gravity.
Wherein, 1, circular groove, 2, light hole, 3, bead.
Embodiment
Circular cone pendulum:
F
draw* cos θ=mg
F
draw* sin θ=F
to=m ω
2r=m ω
2* Lsin θ
Two formulas are divided by, and obtain
G=ω
2lcos θ=(2pi/T)
2lcos θ wherein, pi=3.1415926
And the formula of single pendulum measurement acceleration of gravity is g=ω
2l=(2pi/T)
2l, be equivalent to pendulum length and shortened to by L the single pendulum of L*cos θ for circular cone pendulum, for the experiment with simple pendulum of stack circumference pendulum, because the numerical value of acceleration of gravity is constant, equivalent pendulum length in molecule shortens, cycle T in denominator also can shorten, if also calculated according to the original pendulum length of single pendulum, can cause acceleration of gravity measured value to become large.In experiment, generally experiment with simple pendulum can not made to a pure circular cone pendulum, because so too obvious; A taper swinging momvement often superposes in simple harmonic motion, cause the measured value of the acceleration of gravity of measuring to have change to a certain degree large, and, very unfortunately, do not know that this circular cone pendulum has great radius-of-curvature on earth, to such an extent as to be unable to estimate the impact that it brings.
The method of bead tolling measurement acceleration of gravity:
Bead is mg (R-Rcos (θ in reference position with respect to the potential energy of toroidal cavity extreme lower position
0)), in the time that bead rolls, potential energy is mg (R-Rcos (θ)); Its kinetic energy comprises two parts: the firstth, and the translational kinetic energy of barycenter, mv
2the/2, the secondth, around the rotational kinetic energy of barycenter, I ω
2/ 2, the moment of inertia that wherein I is spheroid, ω is angular velocity, is (2/5) mr for the moment of inertia I of solid sphere
2, the moment of inertia of empty spherical shell is (2/3) mr
2if consider solid sphere, I ω
2/ 2=(2/5) mr
2ω
2/ 2=(1/5) mr
2ω
2, r ω is the speed of bead contact circular groove, is slightly greater than the speed of barycenter, in the time of r<<R, can be similar to and think r ω=v; Therefore kinetic energy is mv
2/ 2+ I ω
2/ 2=(7/5) mv
2/ 2=(7/10) mv
2.
Bead is in the process of motion, and conversion mutually occurs for kinetic energy and potential energy, if negligible friction, gross energy remains unchanged, and bead rollback is in the future moving, mg { R-Rcos (θ
0)=mg { R-Rcos (θ) }+(7/10) mv
2, i.e. mgR { cos (θ)-cos (θ
0)=(7/10) mv
2.
V=ds/dt=d (R θ)/dt=R d θ/dt as can be seen from Figure 1, wherein s is arc length, therefore
mgR(cos(θ
0)-cos(θ))+(7/10)m?(Rdθ/dt)
2=0
Both members is to time t differentiate,
mgRsin(θ)*dθ/dt+(7/10)m?2R
2dθ/dt?*d
2θ/d
2t=0
Then extract common factor, and remove this common factor in both members, obtain
gsin(θ)+(7/5)?Rd
2θ/d
2t=0
When angle is smaller, equally with single pendulum be less than 5 degree at pendulum angle, think sin (θ)=θ,
G θ+(7/5) Rd
2θ/d
2t=0 or g θ+1.4Rd
2θ/d
2t=0 or d
2θ/d
2t+(5g)/(7R) * θ=0
The solution of its equation is:
θ(t)=C
1Cos[(5g/(7R))
0.5?t]+?C
2Sin[(5g/(7R))
0.5?t]
So cycle T meets (5g/ (7R))
0.5t=2p, cycle T
T=2p(1.4R/g)
0.5；g=?(2π/?T)
?2*1.4R
According to the formula contrast T=2 π (l/g) of single pendulum
0.5, wherein l is single pendulum pendulum; This rule that the reciprocating rolling of bead on smooth circular arc be described with there is significant difference, character is completely different, has some approximation but really also have, and can measure acceleration of gravity.
Claims (1)
1. utilize the method for bead tolling measurement acceleration of gravity, it is characterized in that: one section of circular arc, the central angle that this section of circular arc opened its center of circle is less than or equal to 180 degree, described circular arc is R apart from circular arc bottom around one, and fixed point turns and forms one section of circular groove (1) at a plane internal rotation, its radius-of-curvature is R, and this circular groove (1) utilizes mould to adopt casting to make or adopt milling machine milling cutter to make or other technique that can reach this parameter attribute completes; A bead (3) rolls down from the top of circular groove (1) one side, under the mutual conversion of potential energy and kinetic energy, does reciprocally swinging; For avoiding bead (3) to roll out circular groove (1), circular arc both sides of the edge are at least greater than the radius of bead (3) to the height of circular arc bottom; There is pair of light-transmissive hole (2) toroidal cavity side, arranges light emitting diode and the reception diode of photoelectricity door at this light hole (2), and photoelectricity door is used for measuring the cycle T that bead (3) swings, gravity acceleration g=(2 π/T)
2* 1.4R.
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| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201410264404.1A CN104021709A (en) | 2014-06-16 | 2014-06-16 | Method for measuring gravitational acceleration through rolling of small ball |
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|---|---|---|---|
| CN201410264404.1A CN104021709A (en) | 2014-06-16 | 2014-06-16 | Method for measuring gravitational acceleration through rolling of small ball |
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Family
ID=51438438
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|---|---|---|---|
| CN201410264404.1A Pending CN104021709A (en) | 2014-06-16 | 2014-06-16 | Method for measuring gravitational acceleration through rolling of small ball |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107146503A (en) * | 2017-06-22 | 2017-09-08 | 高国杰 | A kind of multi-functional physical demonstration apparatus and method |
| CN109389891A (en) * | 2018-12-28 | 2019-02-26 | 安徽工程大学 | A kind of rational mechanics comprehensive experimental device and its experimental method |
| CN113415102A (en) * | 2021-06-21 | 2021-09-21 | 李明渊 | Rolling research goods of furniture for display rather than for use |
Citations (5)
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|---|---|---|---|---|
| CN2046216U (en) * | 1989-03-29 | 1989-10-18 | 张凤纪 | Rolling/sliding frictional factor testing instrument of gravity and small ball |
| CN102236989A (en) * | 2010-05-07 | 2011-11-09 | 上海市格致中学 | Portable gravity acceleration tester |
| RU2461027C1 (en) * | 2011-04-11 | 2012-09-10 | Федеральное государственное бюджетное учреждение науки Тихоокеанский океанологический институт им. В.И. Ильичева Дальневосточного отделения Российской академии наук (ТОИ ДВО РАН) | Method and apparatus for measuring gravitational acceleration |
| CN102810276A (en) * | 2012-05-10 | 2012-12-05 | 许卫刚 | Method and system for measuring displacement, speed and acceleration |
| JP2013061158A (en) * | 2011-09-12 | 2013-04-04 | Yoichi Takagi | Apparatus and method for measuring gravity of earth |
-
2014
- 2014-06-16 CN CN201410264404.1A patent/CN104021709A/en active Pending
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN2046216U (en) * | 1989-03-29 | 1989-10-18 | 张凤纪 | Rolling/sliding frictional factor testing instrument of gravity and small ball |
| CN102236989A (en) * | 2010-05-07 | 2011-11-09 | 上海市格致中学 | Portable gravity acceleration tester |
| RU2461027C1 (en) * | 2011-04-11 | 2012-09-10 | Федеральное государственное бюджетное учреждение науки Тихоокеанский океанологический институт им. В.И. Ильичева Дальневосточного отделения Российской академии наук (ТОИ ДВО РАН) | Method and apparatus for measuring gravitational acceleration |
| JP2013061158A (en) * | 2011-09-12 | 2013-04-04 | Yoichi Takagi | Apparatus and method for measuring gravity of earth |
| CN102810276A (en) * | 2012-05-10 | 2012-12-05 | 许卫刚 | Method and system for measuring displacement, speed and acceleration |
Non-Patent Citations (1)
| Title |
|---|
| 胡巍: "《测量重力加速度的原理》", 《牡丹江师范学院学报》, no. 1, 31 December 2006 (2006-12-31), pages 20 - 21 * |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107146503A (en) * | 2017-06-22 | 2017-09-08 | 高国杰 | A kind of multi-functional physical demonstration apparatus and method |
| CN109389891A (en) * | 2018-12-28 | 2019-02-26 | 安徽工程大学 | A kind of rational mechanics comprehensive experimental device and its experimental method |
| CN113415102A (en) * | 2021-06-21 | 2021-09-21 | 李明渊 | Rolling research goods of furniture for display rather than for use |
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Application publication date: 20140903 |