scholarly journals Launch Velocities in Successful Golf Putting: An Analytical Analysis

2017 ◽  
Vol 5 (2) ◽  
pp. 24
Author(s):  
John F. Mahoney ◽  
Daniel P. Connaughton
Keyword(s):  
Critical Value ◽  
Theoretical Studies ◽  
Newtonian Mechanics ◽  
Ambient Conditions ◽  
Analytical Analysis ◽  
Bouncing Ball ◽  
Golf Putting ◽  
A Value ◽  
Special Case

Background: This study is concerned with the special case of a putted ball intersecting a standard golf hole at its diameter. The velocity of the ball at the initial rim of the hole is termed the launch velocity and depending upon its value the ball may either be captured or it may escape capture by jumping over the hole. The critical value of the launch velocity (V) is such that lesser values result in capture while greater values produce escape. Purpose: Since the value of the V entered prominently in some theoretical studies of putting, the aim of the current study is to provide an original re-evaluation of V and to contrast our results with existing results. Method: This analytical analysis relies on trigonometry in conjunction with Newtonian mechanics and the mathematics of projectiles. The results of a recent study into the mathematics of a bouncing ball which included the notions of restitution and friction were also employed in the analysis. Results: If bouncing and slipping do not occur when the ball hits the far rim of the hole our analysis produces a value of V of 1.356 m/s. When bouncing and slipping are present we find that V is at least 1.609 m/s but increases beyond this value as slipping and friction become greater. Useful relations which relate the dynamics and geometry of the ball to V are provided. Conclusion: Since ambient conditions may influence the extent of bounce and slippage we conjecture that the value of V is not unique.

The Stability of an Axially–Loaded Continuous Beam

1955 ◽  
Vol 59 (535) ◽  
pp. 506-509
Author(s):  
A. M. Dobson
Keyword(s):  
Axial Load ◽  
Classical Method ◽  
Complete Solution ◽  
Critical Value ◽  
Continuous Beam ◽  
Independent Variables ◽  
A Value ◽  
Method Of Solution ◽  
Axially Loaded ◽  
The Stability

The Classical method of solution of the stability of an axially–loaded continuous beam is by means of the three moments equation, using the Berry Functions, which are functions of the axial load. As the axial load approaches a value equal to the critical value for a pin–jointed beam, the Berry Functions tend to infinity, and the use of the three moments equations —(i. e. treating the end fixing moments as the independent variables)—leads to certain difficulties in the complete solution of the problem.The major difficulty lies in the question of stability. The critical value is determined by the vanishing of the determinant of the coefficients of the fixing moments in the three moments equations. This value could be found by plotting the determinant against end load (c. f. Pippard and Pritchard). However, in a problem involving a large number of bays, the calculation necessary to do this is likely to be considerable, for there may be many branches to the curve.

scholarly journals XVI.—On the Path of a Rotating Spherical Projectile. II.

1900 ◽  
Vol 39 (2) ◽  
pp. 491-506 ◽  
Author(s):  
Tait
Keyword(s):  
Great Part ◽  
Numerical Data ◽  
Terminal Velocity ◽  
Considerable Improvement ◽  
Golf Ball ◽  
Initial Speed ◽  
Interesting Application ◽  
A Value ◽  
Special Case

The first instalment of this paper was devoted in great part to the general subject involved in its title, but many of the illustrations were derived from the special case of the flight of a golf-ball. Since it was read I have endeavoured, alike by observation and by experiment, to improve my numerical data for this interesting application, particularly as regards the important question of the coefficient of resistance of the air. As will be seen, I now find a value intermediate to those derived (by taking average estimates of the mass and diameter of a golf-ball) from the results of Robins and of Bashforth. This has been obtained indirectly by means of a considerable improvement in the apparatus by which I had attempted to measure the initial speed of a golf-ball. I have, still, little doubt that the speed may, occasionally, amount to the 300, or perhaps even the 350, foot-seconds which I assumed provisionally in my former paper:—but even the first of these is a somewhat extravagant estimate; and I am now of opinion that, even with very good driving, an initial speed of about 240 is not often an underestimate, at least in careful play. From this, and the fact that six seconds at least are required for a long carry (say 180 yards), I reckon the “terminal velocity” at about 108, giving v2/360 as the resistance-acceleration.

On the origin of black hole evaporation radiation

1976 ◽  
Vol 351 (1664) ◽  
pp. 129-139 ◽  
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Static Vacuum ◽  
Black Hole Evaporation ◽  
A Value ◽  
Special Case ◽  
Collapse Process

The physical basis underlying the black hole evaporation process is clarified by a calculation of the expectation value of the energy-momentum tensor for a massless scalar field in a completely general two dimensional collapse scenario. It is found that radiation is produced inside the collapsing matter which propagates both inwards and outwards. The ingoing com­ponent eventually emerges from the star after travelling through the centre. The outgoing energy flux appears at infinity as the evaporation radiation discovered by Hawking. At late times, outside the star, the former component fades out exponentially, and the latter component approaches a value which is independent of the details of the collapse process. In the special case of a collapsing hollow, thin shell of matter, all the radiation is produced at the shell. These results are independent of regularization ambiguities, which enter only the static vacuum polariza­tion terms in the energy-momentum tensor. The significance of an earlier remark about black hole explosions is discussed in the light of these results.

Invariant problems in discounted dynamic programming

1978 ◽  
Vol 10 (2) ◽  
pp. 472-490 ◽  
Author(s):  
David Assaf
Keyword(s):  
Dynamic Programming ◽  
Arbitrary State ◽  
Transition Mechanism ◽  
Current State ◽  
A Value ◽  
Discounted Dynamic Programming ◽  
Optimal Policies ◽  
Finite Action ◽  
Value Determination ◽  
Special Case

Discounted dynamic programming problems whose transition mechanism depends only on the action taken and does not depend on the current state are considered. A value determination operation and method of obtaining optimal policies for the case of finite action space (and arbitrary state space) are presented.The solution of other problems is reduced to this special case by a suitable transformation. Results are illustrated by examples.

Numerical Simulation of Localized Bulging in an Inflated Hyperelastic Tube with Fixed Ends

2020 ◽  
pp. 2050118
Author(s):  
Zehui Lin ◽  
Linan Li ◽  
Yang Ye
Keyword(s):  
Axial Stress ◽  
Necessary Condition ◽  
Theoretical Studies ◽  
Inflation Pressure ◽  
Material Models ◽  
Rubber Material ◽  
Bifurcation Phenomenon ◽  
A Value ◽  
Pressure Maximum ◽  
Almost All

When a hyperelastic tube is inflated, the inflation pressure has a maximum for almost all rubber material models, but has no maximum for commonly used arterial models. It is generally believed that the pressure having a maximum is a necessary condition for localized bulging to occur, and therefore aneurysms cannot be modeled as a mechanical bifurcation phenomenon. However, recent theoretical studies have shown that if the axial stretch is fixed during inflation, localized bulging may still occur even if a pressure maximum does not exist in uniform inflation. In this paper, numerical simulations are conducted to confirm this theoretical prediction. It is also demonstrated that if the axial pre-stretch is not sufficiently large, unloading near the two ends can reduce the axial stress to a value close to zero and Euler-type buckling then occurs.

scholarly journals Measurement of the Permeability of Biological Membranes Application to the glomerular wall

1973 ◽  
Vol 62 (4) ◽  
pp. 489-507 ◽  
Author(s):  
A. Verniory ◽  
R. Du Bois ◽  
P. Decoodt ◽  
J. P. Gassee ◽  
P. P. Lambert
Keyword(s):  
Poiseuille Flow ◽  
Path Length ◽  
Pore Radius ◽  
Frictional Resistance ◽  
Theoretical Studies ◽  
Equivalent Radius ◽  
A Value ◽  
Anesthetized Dogs ◽  
The Mean ◽  
Diffusion And Convection

The transport equation describing the flow of solute across a membrane has been modified on the basis of theoretical studies calculating the drag of a sphere moving in a viscous liquid undergoing Poiseuille flow inside a cylinder. It is shown that different frictional resistance terms should be introduced to calculate the contributions of diffusion and convection. New sieving equations are derived to calculate r and Ap/Δx (respectively, the pore radius and the total area of the pores per unit of path length). These equations provide a better agreement than the older formulas between the calculated and the experimental glomerular sieving coefficients for [125I]polyvinylpyrrolidone (PVP) fractions with a mean equivalent radius between 19 and 37 Å. From r and Ap/Δx, the mean effective glomerular filtration pressure has been calculated, applying Poiseuille's law. A value of 15.4 mm Hg has been derived from the mean sieving curve obtained from 23 experiments performed on normal anesthetized dogs.

Instabilities of Coulomb phases and the Gribov picture of confinement

2016 ◽  
Vol 31 (28n29) ◽  
pp. 1645024
Author(s):  
Manuel Asorey ◽  
Alessandro Santagata
Keyword(s):  
First Principles ◽  
Critical Value ◽  
Coupling Constants ◽  
Quark Confinement ◽  
Abelian Gauge ◽  
Vacuum Decay ◽  
Coulomb Phase ◽  
A Value ◽  
Analytic Derivation ◽  
Phase Instabilities

A new picture of quark confinement based on the instability of Coulomb phase at low energy was introduced by Volodya Gribov in the early nineties. In QCD the effective [Formula: see text] coupling constant can reach very large values in the infrared regime what generates Coulomb phase instabilities. In the Gribov picture the instability leads to a vacuum decay into light quarks for coupling constants [Formula: see text] larger than a critical value [Formula: see text], for SU(N) gauge theories. The instability of Coulomb phase can be derived from first principles in any non-Abelian gauge theory for [Formula: see text], a value which is larger than the Gribov critical value. In this paper we review the analytic derivation of the Gribov mechanism from first principles and analyze the effects of dynamical quarks in the instability of the Coulomb phase. The instabilities associated to light quarks turn out to appear at larger values of [Formula: see text] than the ones induced from pure gluon dynamics, unlike it is expected in the standard Gribov scenario. The analytic results confirm the consistency of the picture where quark confinement is mainly driven by gluonic fluctuations.

scholarly journals A note on the flow induced by a line sink beneath a free surface

1991 ◽  
Vol 32 (3) ◽  
pp. 251-260 ◽  
Author(s):  
G. C. Hocking ◽  
L. K. Forbes
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Stagnation Point ◽  
Ideal Fluid ◽  
Critical Value ◽  
Homogeneous Fluid ◽  
Substitution Method ◽  
A Value ◽  
Line Sink ◽  
Low Froude Number

AbstractThe problem of withdrawing water through a line sink from a region containing an homogeneous fluid beneath a free surface is considered. Assuming steady, irrotational flow of an ideal fluid, solutions with low Froude number containing a stagnation point on the free surface above the sink are sought using a series substitution method. The solutions are shown to exist for a value of the Froude number up to a critical value of about 1.4. No solutions of this type are found for Froude numbers greater than this value.

scholarly journals Exact inbreeding coefficient and effective size of finite populations under partial sib mating.

Genetics ◽  
1995 ◽  
Vol 140 (1) ◽  
pp. 357-363
Author(s):  
J Wang
Keyword(s):  
Poisson Distribution ◽  
Family Size ◽  
Inbreeding Coefficient ◽  
Stochastic Simulations ◽  
Effective Size ◽  
Finite Populations ◽  
A Value ◽  
The Family ◽  
Sib Mating ◽  
Special Case

Abstract An exact recurrence equation for inbreeding coefficient is derived for a partially sib-mated population of N individuals mated in N/2 pairs. From the equation, a formula for effective size (Ne) taking second order terms of 1/N into consideration is derived. When the family sizes are Poisson or equally distributed, the formula reduces to Ne = [(4 - 3 beta) N/(4 - 2 beta)] + 1 or Ne = [(4 - 3 beta) N/(2 - 2 beta)] - 8/(4 - 3 beta), approximately. For the special case of sib-mating exclusion and Poisson distribution of family size, the formula simplifies to Ne = N + 1, which differs from the previous results derived by many authors by a value of one. Stochastic simulations are run to check our results where disagreements with others are involved.

The bifurcation and secondary bifurcation of capillary-gravity waves

1985 ◽  
Vol 399 (1817) ◽  
pp. 391-417 ◽  
Keyword(s):  
Gravity Waves ◽  
Phase Speed ◽  
Implicit Function ◽  
Existence And Multiplicity ◽  
A Value ◽  
Double Eigenvalue ◽  
Bifurcation Phenomena ◽  
Critical Phase ◽  
Secondary Bifurcation ◽  
Special Case

The bifurcation and secondary bifurcation of capillary-gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two. The existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurcation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.

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