Search

If a ball is incident obliquely on a horizontal surface and is allowed to bounce more than once, then it is likely to bounce many times before it starts rolling along the surface. The number of bounces before rolling commences depends on the initial vertical speed and the normal coefficient of restitution. The transition from bouncing to rolling is examined using a simple theoretical model and is compared with experimental data obtained by filming the process with a video camera. We find that the final rolling speed is proportional to the initial horizontal speed of the ball and depends on the initial ball spin, but is independent of the tangential coefficient of restitution. Representative videos for different balls are included as supplementary material, including a superball thrown with a backspin that creates a back and forth motion. Instructors could use the experiment and/or analysis for an advanced undergraduate lab or use a simplified observational exercise for non-majors. Editor's Note: When a ball is dropped on a horizontal surface with no initial spin, previous studies have found that its bouncing behavior can be simply described using a coefficient of restitution, which gives the ratio of the velocity after the bounce to before the bounce. The value of this coefficient is −1 for a perfectly elastic ball/surface, with a smaller magnitude for any real ball. In many sports, like golf, basketball, or bowling, balls are thrown at an angle and are often given some initial spin by the player. Depending on initial conditions, these balls can bounce, roll, or start by bouncing and then transition to rolling. Here, the transition from bouncing to rolling is shown to be described by using both a vertical and horizontal coefficient of restitution, with the horizontal velocity defined at the spinning edge of the ball rather than its center. Videos of undergraduate level experiments are included, with results used to validate the model. [ABSTRACT FROM AUTHOR]

Oblique angle collisions between two balls or two disks have been addressed by many authors. This paper describes oblique angle collisions between three or four billiard balls. Measurements and calculations are presented for cases where one ball is incident obliquely on two or three identical balls, which are in contact and initially at rest. Conservation laws alone are not sufficient to predict the outcome. Adding information about the impact forces on each ball (and ignoring friction forces) provides good theoretical agreement with experimental results. [ABSTRACT FROM AUTHOR]

An oblique collision of an object with a rigid surface involves an initial sliding phase that persists throughout the collision at glancing angles of incidence but which involves a subsequent grip phase at higher angles of incidence. The grip phase itself terminates towards the end of the collision if the contact region starts to slide backwards. Experimental evidence of the three separate stages of the collision process is presented using a high speed video camera to film the impact of a rubber disk on a rigid horizontal surface. A simplified model of the process is presented, providing analytical solutions that are at least qualitatively consistent with the experimental observations. [ABSTRACT FROM AUTHOR]

The coefficient of sliding friction is usually regarded as a constant for any given surfaces although it is sometimes known to vary with sliding speed. An experiment is described where the coefficient of friction for a billiard ball decreases to a very small value as the sliding speed decreases to zero, after which the ball starts rolling without sliding. A simple theoretical explanation is provided. [ABSTRACT FROM AUTHOR]

Li et al. conducted an experiment on the bounce of a capsule dropped vertically onto a horizontal surface. They found that if the capsule has spin, it can bounce to a slightly greater height than the drop height. This is because rotational kinetic energy is converted to vertical translational kinetic energy during the bounce. The effect is more pronounced when the capsule is incident obliquely with spin. Similar effects have been observed with other elongated objects, such as a spinning plastic egg or two golf balls joined together. This phenomenon can be visually appealing and is often used in lecture demonstrations. [Extracted from the article]

A simple model is presented to explain how the spin of a ball can be enhanced when the ball is incident obliquely on a flexible surface. The mechanism involves tangential distortion of the surface and a return of the elastic energy stored in the surface via the action of the static friction force on the ball. As an example, we consider the enhancement of the spin that is possible using an appropriate type of tennis racquet string. [DOI: 10.1119/1.3313455]

Measurements on the swing of a baseball bat are analyzed to extract the basic mechanics of the swing. The force acting on the bat is determined from the velocity of the center of mass, and the angular velocity of the bat provides additional information on the couple exerted by the two bands. The motion of the bat was calculated for other force-couple combinations to determine their effects on the swing of the bat. It was found that a couple is needed to start the swing, and a large opposing couple is required near the end of the swing to prevent the bat rotating through an excessive angle before it impacts with the ball. [DOI: 10.1119/1.2983146]

When one hard steel ball collides with another, kinetic energy is conserved, even if the balls have different diameters. Why is kinetic energy conserved in such a collision, given that kinetic energy is not conserved when two unequal length steel springs or rods collide? Experimental results with bouncing balls, springs, and rods are presented, which reveal the answer. For colliding springs and rods a significant fraction of the initial kinetic energy is retained after the collision as vibrational energy in the longer spring and rod. When two hard balls collide, a negligible fraction of the initial energy is converted to vibrational energy because the collision time is much longer than the transit time of an acoustic wave across each ball due to the fact that the contact region of a hard spherical ball is much softer than the rest of the ball. [DOI: 10.1119/1.2948778]

The physics of falling from a height, a topic that could be included in a course on forensic physics or in an undergraduate class as an example of Newton's laws, is applied to a common forensic problem. [DOI: 10.1119/1.2919736]

A billiard ball struck by a cue travels in the same direction as the cue unless the ball is struck toward one side in order to impart sidespin. In that case the ball deflects or 'squirts' away from the line of approach of the cue, typically by a few degrees. Measurements and calculations are presented showing how a cue tip slides across the ball if it is unchalked, resulting in a large squirt angle, and how it grips the ball when it is chalked, resulting in a smaller squirt angle. [DOI: 10.1119/1.2825392]

When a tennis ball rests on top of a basketball and both drop to the floor together, the tennis ball is projected vertically at high speed. A mass-spring model of the impact, as well as air track data, suggest that the tennis ball should be projected at relatively low speed. Measurements of the forces on each ball and the bounce of vertically aligned superballs are used to resolve the discrepancy. [DOI: 10.1119/1.2772286]

The gear effect arises when a ball collides with an object and the object recoils with a component of its acceleration in a direction parallel to the two contacting surfaces. Two experiments are described that show how the surfaces of the ball and the object can engage without slip similar to two gears. The effect is used in golf to correct for off-center impacts and it plays a minor role in baseball. [DOI: 10.1119/1.2713788]

A ball can be hit faster if it is projected without spin, but it can be hit farther if it is projected with backspin. Measurements of the tradeoff between the speed and spin for a baseball impacting a baseball bat are presented. The results are inconsistent with a collision model in which the ball rolls off the bat and instead imply tangential compliance in the ball, the bat, or both. If the results are extrapolated to the higher speeds that are typical of the game of baseball, they suggest that a curveball can be hit with greater backspin than a fastball, but by an amount that is less than would be the case in the absence of tangential compliance. [DOI: 10.1119/1.2209246]

(Received 23 May 2005; accepted 16 September 2005) If an inclined pencil is released from rest with its bottom end resting on a table, the bottom end will slide forward or backward or it will remain at rest at the start of the fall, then slide backward for a short period before sliding forward. The magnitude and direction of the displacement of the bottom end of the pencil depends on the initial angle of inclination, the coefficient of friction, and on the length and mass distribution of the pencil. The same ground reaction forces play a similar role in the fall of trees and chimneys, the bounce of a football and any other elongated object, and in activities such as walking and running. When an elongated object is thrown obliquely to the ground, the object can bounce either forward or backward depending on the angle of inclination at impact. Spherical objects bounce away from the thrower. The difference arises because the horizontal friction force is determined not only by the normal reaction force, but also by the line of action of the normal reaction force relative to the center of mass.

A spinning ball incident at right angles to a surface bounces at an oblique angle and with reduced spin. Consequently, a spinning ball struck head-on does not rebound along its incident path, which presents a control problem in bat and racquet sports. The rebound angle and spin depend in a nontrivial manner on the coefficient of friction between the ball and the surface and on the elastic properties of the ball and the surface. Values of the normal and tangential coefficients of restitution coefficients are presented for a tennis ball impacting a smooth and a rough surface and on strings of a tennis racquet. The implications for spin generation in tennis are briefly described. [DOI: 10.1119/1.2008299]

A block sliding down an inclined plane normally accelerates. However, if the friction force increases with speed, then the block can slide at a constant terminal speed in a manner similar to the fall of an object through a fluid. Measurements of the increase in the coefficient of friction for tennis ball cloth sliding on a smooth surface are described over speeds varying by a factor of 9000. For the low speed measurements, the ball cloth was attached to the bottom of a weighted box and pulled along a horizontal surface by a constant horizontal force. Results at higher speeds were obtained by bouncing a tennis ball off the surface. (c) 2005 American Association of Physics Teachers. [DOI: 10.1119/1.1891174]

Experimental results on the large-amplitude motion of a double pendulum are presented, with emphasis on the first half cycle. The initial part of the swing is reproducible and is of interest in modeling various human movement activities such as running, throwing, kicking, and the swing of a bat or racquet. Beyond this time, the motion is chaotic. The forces and torques acting on each pendulum segment are analyzed to explain its motion. The results show how a 'perfect' bat could be designed where all the kinetic energy from the player's arms is transferred to the ball and none is retained in the arms or the bat after the impact.

The center of percussion is commonly regarded as a sweet spot when referring to a baseball bat or a tennis racquet because it is assumed that there will be no sudden motion of the handle with respect to the hand if the corresponding axis of rotation passes through the hand. A problem with this interpretation is that the hand extends over a finite length of the handle and exerts an opposing reaction force on the handle. The hand also changes the total mass and moment of inertia of the system, while the arm restricts free motion of the hand. Experimental results are presented showing that the axis of rotation passes through the hand or the wrist for all the usual impact points on a hand-held implement. As a result, the impact point that feels best is usually the node of the fundamental vibration mode, not the center of percussion.

Measurements are presented of the speed at which objects of different mass can be projected by an overarm throw. Light objects can be thrown faster than heavy objects, although the difference in speed is not as large as one might expect. For a factor of 60 increase in the thrown mass, there was a decrease of only 2.4 in the throw speed. The relatively small change in throw speed is due to the fact that the force that can be applied to a thrown object increases with object mass. Estimates of the muscle forces involved indicate that the increase in force with mass is primarily an inertial rather than a physiological effect. The total kinetic energy of the mass, hand, and the forearm was found to be almost independent of the object mass, and the throw speed is almost independent of the mass of the upper arm.

Measurements of the normal reaction force and the friction force acting on an obliquely bouncing ball were made to determine whether the friction force acting on the ball is due to sliding, rolling, or static friction. At low angles of incidence to the horizontal, a ball incident without spin will slide throughout the bounce. At higher angles of incidence, elementary bounce models predict that the ball will start to slide, but will then commence to roll if the point of contact on the circumference of the ball momentarily comes to rest on the surface. Measurements of the friction force and ball spin show that real balls do not roll when they bounce. Instead, the deformation of the contact region allows a ball to grip the surface when the bottom of the ball comes to rest on the surface. As a result the ball vibrates in the horizontal direction causing the friction force to reverse direction during the bounce. The spin of the ball was found to be larger than that due to the friction force alone, a result that can be explained if the normal reaction force acts vertically through a point behind the center of the ball.

When a ball is incident obliquely on a flat surface, the rebound spin, speed, and angle generally differ from the corresponding incident values. Measurements of all three quantities were made using a digital video camera to film the bounce of a tennis ball incident with zero spin at various angles on several different surfaces. The maximum spin rate of a spherical ball is determined by the condition that the ball commences to roll at the end of the impact. Under some conditions, the ball was found to spin faster than this limit. This result can be explained if the ball or the surface stores energy elastically due to deformation in a direction parallel to the surface. The latter effect was investigated by comparing the bounce of a tennis ball with that of a superball. Ideally, the coefficient of restitution (COR) of a superball is 1.0 in both the vertical and horizontal directions. The COR for the superball studied was found to be 0.76 in the horizontal direction, and the corresponding COR for a tennis ball was found to vary from -0.51 to + 0.24 depending on the incident angle and the coefficient of sliding friction. [DOI: 10.1119/1.1450571] (Received 9 July 2001; accepted 20 December 2001)

Details are given of an inexpensive force plate designed to measure ground reaction forces involved in human movement. Such measurements provide interesting demonstrations of relations between displacement, velocity, and acceleration, and illustrate aspects of mechanics that are not normally encountered in a conventional mechanics course, or that are more commonly associated with inanimate objects. When walking, the center of mass follows a curved path. The centripetal force is easily measured and it provides an upper limit to the speed at which a person can walk. When running, the legs behave like simple springs and the center of mass follows a path that is the same as that of a perfectly elastic bouncing ball.

In this paper, the dynamics of a bouncing ball is described for several common ball types having different bounce characteristics. Results are presented for a tennis ball, a baseball, a golf ball, a superball, a steel ball beating, a plasticene ball, and a silly putty ball. The plasticene ball was studied as an extreme case of a ball with a low coefficient of restitution (in fact zero, since the collision is totally inelastic) and the silly putty ball was studied because it has unusual elastic properties. The first three balls were studied because of their significance in the physics of sports. For each ball, a dynamic hysteresis curve is presented to show how energy is lost during and after the collision. The measurement technique is quite simple, it is suited for undergraduate laboratory experiments, and it may provide a useful method to test and approve balls for major sporting events.

The sweet spot of a baseball bat, like that of a tennis racket, can be defined either in terms of a vibration node or a centre of percussion. In order to determine how each of the sweet spots influences the 'feel' of the bat, measurements were made of the impact forces transmitted to the hands. Measurements of the bat velocity, and results for a freely suspended bat, were also obtained in order to assist in the interpretation of the force waveforms. The results show that both sweet spots contribute to the formation of a sweet spot zone where the impact forces on the hands are minimised. The free bat results are also of interest since they provided particularly elegant examples of wave excitation and propagation, suitable for a student demonstration or experiment.

The main objective in lawn bowls is to bowl a ball along a curved trajectory on a smooth grass surface so that it stops closer to the jack (a small white ball) than the opponent's ball. The curvature of the path is induced by shaping the ball so that one side is heavier than the other. Some of the properties of the trajectory are described to illustrate this interesting example of precessional motion.

It is shown that a tennis racket has a dead spot, but it does not have a well-defined sweet spot, when measured in terms of the rebound of a tennis ball. A ball dropped onto the center of the strings bounces to about 30% its original height. The bounce is much weaker near the tip of the racket, being almost zero at the dead spot. These effects are explained in terms of the effective mass and rotational inertia of the racket, and by reference to the behavior of other cantilevered beams. It is concluded, somewhat paradoxically, that the best place to hit a serve or smash is at the dead spot.

The frame of a string instrument is subject to a large tension force when the strings are installed. Intuitively, one might expect that the frame would be stiffened by the strings. Experimental data and a theoretical analysis are presented to show that this is not generally the case. An archer's bow is much stiffer when it is strung, but a tennis racquet is softened when the strings are added. As a result, the mode frequencies for transverse vibrations increase for a bow and decrease for a racquet when the strings are added. The effect of the strings depends on the extent to which the frame is bent at equilibrium. [DOI: 10.1119/1.1376378]

A perfectly happy ball is one that bounces to its original height when dropped on a massive, rigid surface. A completely unhappy ball does not bounce at all. In the former case, the coefficient of restitution (COR) is unity. In the latter case, the COR is zero. It is shown that when an unhappy ball collides with a happy ball, the COR increases from zero to unity as the stiffness of the happy ball decreases from infinity to zero. The COR is independent of the mass of each ball. The implication of reducing the COR of a tennis ball, as a possible means of slowing the serve in tennis, is also considered. It is shown that (a) the COR for a collision with a racket varies with the impact point and is a maximum at the vibration node near the center of the strings, and (b) the serve speed is reduced by only about 20% if the COR for a bounce on the court is reduced to zero. [C] 2000 American Association of Physics Teachers.

The centrifugal force can be utilized to analyze the phenomena in a rotating coordinate system, specifically vehicle rolls. A vehicle at rest on a horizontal road surface has a center of mass located at a horizontal distance from the wheels on the left and ride rides. When subjected to a horizontal forcer acting at a specific height above the road surface, the motion of the vehicle will be opposed by horizontal friction forces that are acting on the left and right wheels.

The collision of a ball with a baseball bat or a tennis racket is usually modeled in terms of rigid body dynamics, assuming that the hand exerts no impulsive reaction force on the handle during the collision. In this paper, a uniform aluminum beam was used as an idealized bat or racket, in order to examine both the rigid body approximation and the assumption that the hand force can be neglected. An aluminum beam was chosen so that its length and stiffness could easily be varied and so that the results could be compared with solutions for a flexible beam. It was found that rigid body models of beams, bats, or rackets are of limited use but the hand force can usually be neglected. The flexible beam model provides remarkably good agreement with experimental results and provides new insights into the dynamics of this type of collision, including the nature of the sweet spot.