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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]

An experiment is described where a metal nut was allowed to slide from the top to the bottom of a vertical wire loop. An interesting feature is that the normal reaction force on the nut decreases to zero and reverses direction before the nut reaches the bottom of the loop. [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]

The transition from sliding (where I v i / I i > I R i ) to rolling (where I v i / I i = I R i ) shown in Fig. By that time, I v i was no longer zero since the ball started to slide just before the finger holding the ball was lifted gently off the ball. If I D i / I R i = 0.1, then the ball will roll to a stop if I i < 6° and if the ball is given a push to get it started, since I a i is then negative.[6] Graph: Fig. In order to determine whether the ball rolls without sliding, it is necessary to measure both I v i and I i and to compare the I v i / I i ratio with I R i . [Extracted from the article]

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]

Objects that slide or roll along curved tracks in a vertical plane are investigated in two separate experiments. One concerns a smooth metal nut that slides along a semicircular track made from coat hanger wire. The other concerns a billiard ball that slides and rolls along a curved track. A simple theoretical model is provided to account for the results and to extend the discussion to consider shortest descent times down circular tracks. [ABSTRACT FROM AUTHOR]

(1) and (2) need to be solved numerically to find I y i and I F i vs. time, for given values of I k i SB 1 sb , I k i SB 2 sb , I c i , I m i , and I v i SB 0 sb . An interesting question is whether these and other balls store all the initial impact energy as elastic energy when they compress, or whether they lose some of that elastic energy while they are compressing as well as during the expansion stage. The maximum total elastic energy is 0.82 I E i SB 0 sb , so not all of the initial kinetic energy is stored as elastic energy; 82% of the initial kinetic energy is stored as elastic energy at maximum ball compression, and then 82% of the stored elastic energy is returned as kinetic energy when the ball bounces off the surface. The area under the curve while I y i increases represents the energy required to compress the ball, and the area under the curve while I y i decreases represents the energy recovered while the ball expands. [Extracted from the article]

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]

When two balls collide multiple times in a simplified Newton's cradle, the impact point gradually moves away from the lowest point of the swing. The effect was examined using two billiard balls, and was traced to the fact that the upper support points were not separated by exactly one ball diameter. [ABSTRACT FROM AUTHOR]

The trajectory of a ball rolling across an inclined plane was recorded on video film to determine if it followed a parabolic path as others have suggested. The challenges in this case were (a) to determine the magnitude and direction of the friction force on the ball, (b) to determine the effect of the friction force on the trajectory and (c) to explain why the rotation axis of the ball rotates as it rolls up and down the incline. The experiment is a simple extension of standard inclined plane experiments but with the extra pedagogical feature that angular velocity sometimes needs to be treated as a vector quantity. [ABSTRACT FROM AUTHOR]

A collision of one object with two or more objects is relatively complicated in general, but a simple example is provided by Newton’s cradle since all the objects are identical and in line. In the present paper, an experiment is described where a heavy mallet collides head-on with two billiard balls. The two conservation equations indicate that many different outcomes are possible. The experimental results are surprising but can be explained in terms of the relative mass and stiffness of the mallet and the balls. [ABSTRACT FROM AUTHOR]

Measurements are presented on oblique angle collisions between two identical balls, each suspended as pendulum bobs. If one ball is at rest and is struck by the other ball incident without spin at an oblique angle, then both balls spin in the same direction after the collision. The results can be explained in terms of friction between the balls. At glancing angles, the balls slide throughout the collision. At other angles of incidence, the balls initially slide, then grip, and then slide again when the balls release their grip. The collision can be described in terms of two different coefficients of restitution: one involving the normal incident and post-collision speeds and the other involving the tangential speeds of the contact points before and after the collision. The coefficient of restitution and the collision time vary with the collision speed but remain relatively constant if the speed changes by a factor less than about two. [ABSTRACT FROM AUTHOR]

The trajectory of a ball in air is affected by aerodynamic drag and lift. In general, the trajectory needs to be calculated numerically since the acceleration varies with time in both the horizontal and vertical directions. If the trajectory remains approximately parabolic, then simple analytical solutions can be found, giving useful insights into the behavior of the ball as it travels through the air. [ABSTRACT FROM AUTHOR]

This article discusses the concept of the quality factor, Q, in damped oscillatory motion. The quality factor is a measure of the energy loss per cycle and can be calculated by plotting oscillator amplitude vs. time or by counting the number of observed cycles. The article provides examples of simple harmonic motion, such as a pendulum and a mass on a spring, and explains the equation that describes their motion. It also explains how the amplitude of the oscillation decreases exponentially with time due to damping and how the quality factor is related to the energy loss. The article suggests that counting the number of cycles can be a simple and quick method to estimate the quality factor of a damped oscillation. [Extracted from the article]

This article discusses a claim made in a previous journal article about kinetic energy conservation in elastic collisions. The authors of the current article argue that the coefficient of restitution (COR) should be defined in terms of the normal components of the incoming and outgoing speeds of the objects at the point of impact, rather than just the translational motion. They argue that this definition would result in a COR of unity in a perfectly elastic collision, regardless of the point of impact. The issue of impact point dependence is recognized by U.S. Major League Baseball, which specifies the performance of baseball bats in terms of the COR at the point of impact. [Extracted from the article]

When a short rod collides end-on with a long rod, kinetic energy is not conserved since the longer rod vibrates after the collision. In theory, the fractional energy loss is equal to 1 âˆ' L 1/ L 2 where L 1 is the length of the short rod, L 2 is the length of the long rod, and the long rod is initially at rest. Measurements of the energy loss are presented for a set of steel rods and a set of wood dowels. The steel rods were shorter than the dowels and behaved more like steel balls, with only a small decrease in kinetic energy. A larger fractional energy loss was observed when using the dowels since a larger fraction of the initial kinetic energy was converted to vibrational energy in the dowels. In neither case was the energy loss equal to 1 âˆ' L 1/ L 2. [ABSTRACT FROM AUTHOR]

If a pencil is held upright on a smooth horizontal surface and allowed to fall, then the bottom end first slides backward and then forward across the surface. If the top end of the pencil is given a firm horizontal push then the bottom end lifts off the surface and the whole pencil becomes airborne. [ABSTRACT FROM AUTHOR]

If an egg rolls end over end along a horizontal surface, its centre of mass rises and falls. Four different rolling modes are described, namely pendulum oscillations, falling from rest, rolling without slipping and jumping in the air. Supplementary videos are provided for each type of motion, together with a theoretical analysis. The results in each case depend on the initial conditions and the coefficient of sliding friction. The motion of a rolling egg is analogous to the motion of a loaded hoop, and falling motion is analogous to the fall of a rod or a pencil. [ABSTRACT FROM AUTHOR]

A driven pendulum has been widely studied in the large amplitude, chaotic regime. In the present paper, observations are reported on a driven pendulum at frequencies near the resonance frequency, for relatively small drive amplitudes. Strong beating is observed between the transient and steady state response. Even at low drive amplitudes, the resonance is affected by the decrease in oscillation frequency with increasing oscillation amplitude. [ABSTRACT FROM AUTHOR]

If a rod is held almost vertically with one end at rest on a horizontal surface then it will fall to the surface when released. The rod might pivot about its bottom end or it might slide forwards or backwards on the surface, depending on the coefficient of friction. Experimental and theoretical results are presented for a rod with a spherical bottom end. In that case the rod rolls on its bottom end before sliding backwards and/or forwards. Results for a 1.0 m ruler are also presented. [ABSTRACT FROM AUTHOR]

Points near the top end of a falling rod hinged at its bottom end can fall faster than g. If a rod falls on a frictionless surface then the bottom end slides backwards and the centre of mass falls vertically with acceleration less than g. The effect was simulated by mounting a rod on wheels and filming the result with a video camera. The experiment provides an example of accelerated motion where the acceleration increases with time. [ABSTRACT FROM AUTHOR]

Experimental results are presented concerning a bead sliding on a rotating, horizontal rod. If the rod rotates at constant angular velocity then the sliding speed in the radial direction increases exponentially with time, provided that the rotation speed is sufficient to overcome friction. Otherwise the bead rotates in a circular path of constant radius. [ABSTRACT FROM AUTHOR]

If a smooth ball is dropped vertically without spin on a smooth horizontal surface then one might expect the ball to bounce vertically without spin. If it does not then the centre of mass of the ball does not coincide with its geometric centre. An experiment is described where a billiard ball and a superball are deliberately biassed by adding a small steel ball on one side. The results agree well with theoretical calculations. [ABSTRACT FROM AUTHOR]

An experiment is described where a billiard ball and a golf ball were projected with backspin on a horizontal surface to measure the transition from sliding to rolling. During both phases, the torque on the ball is affected by an offset in the line of action of the normal reaction force. [ABSTRACT FROM AUTHOR]

A common procedure when conducting physics experiments is to repeat a measurement several times to calculate the mean and standard deviation. That might be the only instruction we give to students as a means to minimize random errors. However, that technique does not guarantee that the answer will be correct. It might give the same wrong answer every time. How then can we teach students to avoid systematic errors? We don't focus on that problem enough when teaching physics. Part of the problem is that we and the students usually know the correct answers in advance, since student experiments have usually been repeated thousands of times by others. [ABSTRACT FROM AUTHOR]

The normal coefficient of restitution (COR) for a sphere that bounces obliquely or at right angles on a horizontal surface usually has an approximately constant value between 0 and 1, which are the limits for a completely inelastic or a perfectly elastic collision respectively. Different results are obtained for the bounce of a non-spherical object. Measurements are presented for a double sphere consisting of two golf balls joined together, incident obliquely on a smooth horizontal surface. The COR varied from 0.15 to 1.0 depending on the alignment of the two balls and the incident spin. The results are explained in terms of a simple bounce model where the lower ball slides on the horizontal surface, and in terms of three different values of the COR defined in terms of (a) the bounce height, (b) the vertical speed of the contact point and (c) the ratio of the final to the initial kinetic energy. [ABSTRACT FROM AUTHOR]

An experiment is described using double and triple conical pendulums to simulate the behaviour of a rotating chain. The first few rotation modes are calculated theoretically and provide good agreement with experimental results. By contrast, a chain containing many links is difficult to analyse, even though the first few nonlinear modes of a vertically suspended chain are easy to excite and are easily observed using a video camera. The experiment would be suitable as a student project since it is easy to set up and the physics is easily understood, unlike a conventional double pendulum. [ABSTRACT FROM AUTHOR]

A standard problem for physics students is to calculate or measure the acceleration of an object down an inclined plane. Additional information can be obtained by measuring the angular acceleration as well as the linear acceleration. An experiment is described where a billiard ball was filmed as it rolled down an inclined plane at different inclination angles. The transition from rolling without slipping to rolling with slipping was identified, and the relevant friction coefficients were measured independently from the linear and angular accelerations. [ABSTRACT FROM AUTHOR]

The physics of a hula hoop is determined primarily by the effect of the centripetal force acting on the hoop. However, the friction force on the hoop also plays a role. A simple experiment is described showing how an increase in the friction coefficient can help to prevent the hoop falling. [ABSTRACT FROM AUTHOR]

A simple experiment is described concerning the vertical bounce of a nonspherical object. The object consisted of two golf balls joined together. The balls were dropped onto a heavy, rigid horizontal surface and were filmed with a video camera to measure the coefficient of restitution (COR). For spherical objects, the COR is usually defined in terms of the vertical speed of the centre of mass and can vary between zero and unity depending on the elastic properties of the object and the surface on which it bounces. For nonspherical objects, the COR is best defined in terms of the vertical speed of the contact point. For the two golf balls, the centre of mass COR was found to vary between −4 and +2 depending on the incident spin and the angle of inclination of the long axis. The contact point COR was found to be 0.8 ± 0.2, regardless of the incident spin and angle of inclination. [ABSTRACT FROM AUTHOR]

A Newton's cradle with three steel balls was filmed with a slow motion video camera to examine the collision dynamics. The initial collision with two stationary balls in contact was as expected, with the incident ball bouncing backwards slowly, the middle ball moving forwards slowly and the ball at the far end emerging at a speed slightly less than the incident ball. A short time later, a different outcome was observed when one ball collided with two balls visually in contact. In that case the incident ball came to rest and remained in contact with the middle ball, as if the two balls in contact were actually separated before the collision. The second observation can be explained if the impact force has the form F = kxn where n is larger than the n = 1.5 exponent usually used to describe collisions between solid spheres. Alternatively, the result can be explained with an n = 1.5 exponent if there is a very small gap between the balls, not visible on video film. The latter explanation was confirmed by measuring electrical contact between the balls. [ABSTRACT FROM AUTHOR]

A hula hoop rotating around a person's waist is an example of circular motion where the centripetal force on the hoop is directed towards the centre of the circle rather than the centre of the hoop. The friction force on the hoop is relatively small by comparison since the hoop rolls around the waist. A slight complication is that the hoop rotates in an elliptical path if the waist rotates back and forth in a straight line or if the waist rotates in an elliptical path. Three simple experiments are described to illustrate the physics involved. [ABSTRACT FROM AUTHOR]

The vertical bounce of a plastic egg was investigated by dropping the egg on a horizontal surface and filming the result with a video camera. If the egg is dropped on one end then it bounces just like a spherical ball. If the top end of the egg is pointing forwards or backwards when it lands on the surface, or if the egg is spinning when it lands, then the bounce is very different. The egg can bounce forwards or backwards and it can bounce to a large height or only a very small height. The experiment would be suitable for a student project. It involves both translational and rotational motion of an object subject to an impact force and a torque. [ABSTRACT FROM AUTHOR]

Newton's cradle is a well-known physics toy that is commonly used by teachers to demonstrate conservation laws in mechanics. It can also be used to investigate the physics of colliding objects, by recording motion of the balls on video film. Various experiments are described using 3-ball and 5-ball cradles, showing how different types of collision can be investigated depending on whether the balls are touching or separated. The sequence of collisions was examined as a function of time, with the surprising observation that very small gaps between the balls, invisible on video film, make a significant difference to the collision dynamics. Additional experiments illustrate the effects of changes in ball stiffness and the mass of the middle ball and provide an introduction for students and teachers into possible research projects. Each experiment is illustrated with a supplementary video. [ABSTRACT FROM AUTHOR]

Newton's cradle is often discussed in science classrooms as a clear example of the laws of conservation of momentum and energy although it has been shown that this use is somewhat misleading. Approaches to understanding the behaviour of this apparatus are often over-simplified and deficient or over-complex and with little impact among teachers. In this paper, basic principles of mechanics are used to explain the behaviour of ideal cradles of various lengths and to use a simple model by which to simulate its operation. The purpose is to demonstrate that the real behaviour of Newton's cradle provides a rich source of application of those principles in coming to terms with how the apparatus behaves and experiencing the process of modelling in physics. [ABSTRACT FROM AUTHOR]

An experiment is described where three different spinning balls were dropped vertically on a horizontal surface, and the bounce was filmed with a video camera. The change in spin and horizontal speed of each ball was estimated theoretically, using equations familiar to physics students. The experiment provides an extension of the usual student exercise where a non-spinning ball is dropped vertically on a surface to measure the coefficient of restitution. [ABSTRACT FROM AUTHOR]

A simple experiment for students is to measure the coefficient of restitution (COR) for a vertical bounce on a horizontal surface. In this paper, measurements are presented of the COR for a tennis ball bouncing at an oblique angle on a horizontal surface. Changes in the horizontal and rotation speeds were also measured, by filming the bounce with a video camera. A measurement of this type involves a more extensive examination of basic mechanics, since the friction force acting at the bottom of the ball acts to change both the horizontal speed of the ball and its rotational velocity. The COR for an oblique bounce was found to be larger than that for a vertical bounce and was also found to depend on the incident spin of the ball. [ABSTRACT FROM AUTHOR]

Carl Mungan's article[1] on pulling a spool predicts the directions of the friction force acting on the spool at different pull angles. The friction force, I f i , pointed down the incline since the torque I fR i about the center of mass of the spool was equal and opposite the torque I Tr i . 1(b) and (c), those spools would rotate clockwise on the axle, so the string rotates off each of those spools when the spools rotate clockwise on the horizontal surface. [Extracted from the article]

What happens when a perfectly elastic ball collides with a completely inelastic ball? It is shown that the outcome depends on the stiffness of each ball. A standard textbook problem in mechanics is to calculate the outcome of a head-on collision between two balls using conservation of momentum and kinetic energy. It is easily shown that the outcome depends on the relative mass of the balls and their initial speeds. In practice, there is always a net loss of kinetic energy during a collision, which can be expressed in terms of the coefficient of restitution (COR) for the collision. The COR is defined as the relative speed of the two balls after the collision divided by the relative speed before the collision. In a perfectly elastic collision the COR is unity. In a completely inelastic collision, the COR is zero. [ABSTRACT FROM AUTHOR]

The effects of speed humps and bumps in slowing vehicles were investigated using a mass suspended on a spring. An impulse was applied to the mass by lowering and raising the top end of the spring by hand. The impulse duration was varied to simulate different driving speeds, and the displacement of the mass vs time was recorded with a video camera. Maximum displacement of the mass was observed when the impulse duration was about equal to the vibration period of the mass–spring system, consistent with a simple model of the system. The additional effect of a shock absorber is also examined. The experiment could be undertaken in a student laboratory or used as a lecture demonstration to gain deeper insights into impulsive forces and the behaviour of mass–spring systems. [ABSTRACT FROM AUTHOR]

A mass attached to a spring can be set in motion by applying a force either to the mass or the spring. We consider a problem where a short impulse is applied to the spring in an attempt to project the mass at maximum speed. The problem arises when swinging a flexible golf club or a fishing rod, both of which vibrate at low frequency as a mass–spring system. It is shown that maximum speed of the mass at the end of a spring usually arises when the impulse duration is slightly greater than half of the vibration period of the mass–spring system. If that condition is maintained then the speed will increase if the amplitude of the impulse is increased and/or if the vibration frequency of the mass–spring system is increased. Experimental results are presented for a steel ball attached to a soft spring, and also for a short fishing rod, both being suitable for student experiments. [ABSTRACT FROM AUTHOR]

A Newton's cradle experiment is described where the end 'ball' was much heavier than the rest and acted as an end stop. The experiment was conducted to examine the elastic response of billiard balls to a measured impact force, as well as the behaviour of a billiard ball chain in a cradle. The results agree well with a Hertzian force law and are qualitatively similar to those for a conventional cradle. If the balls are initially touching, all balls recoil away from the end stop, at varying speeds. [ABSTRACT FROM AUTHOR]

The impact force and duration when one billiard ball collides head-on with another is estimated by assuming that the balls are connected by a spring. The results are compared with a measurement when a billiard ball collides with a heavy, rigid surface. [ABSTRACT FROM AUTHOR]

A standard physics experiment for students is to measure the terminal velocity of an object in air or in glycerine. A more difficult challenge is to measure the terminal velocity in water since the depth of the water needs to be large or the object needs to be dropped in the water at a speed close to its terminal velocity. Results obtained by filming the fall in a fish tank are presented for a range of different spheres, showing how the terminal velocity depends on the density of the sphere and its radius. The results provide a useful measurement of the drag force and the drag coefficient for a sphere in water. An additional experiment to measure the Magnus force is described where a ball was spinning when it entered the water. [ABSTRACT FROM AUTHOR]

Measurements are presented of the drag force on several different balls falling vertically through water in a fish tank, at speeds up to 4.5 m s−1. Experiments of this type are usually undertaken in a student laboratory by measuring the terminal velocity of an object falling either through air or a viscous liquid. At speeds slightly higher than the terminal velocity, additional effects can be observed visually in water, including flow separation, turbulent flow and the formation of air cavities. Changes in the drag coefficient were observed due to all three effects, although different changes were observed with different balls, depending on the Reynolds number. Cavity formation was found to depend strongly on the roughness of the ball surface as well as the incident ball speed. Results are presented at Reynolds numbers from about 20 000 up to about 250 000. [ABSTRACT FROM AUTHOR]

Two balls on an incline can remain at rest if friction between the balls prevents them rolling down the incline. Results with various balls are presented to show how ball mass, diameter and friction affects the outcome. A stack of four identical balls is also examined. [ABSTRACT FROM AUTHOR]

A well known physics problem is to calculate the friction forces required to support a ladder against a wall. A more tractable problem is to calculate the friction forces needed to support an inclined beam on a ball or a cylinder. In the latter case there are three rather than two points of sliding support. Measurements and calculations are presented to show that a ball can support an inclined beam without slipping, provided the incline angle is not too large. [ABSTRACT FROM AUTHOR]

The impact of a ball with a flexible beam is analysed both experimentally and theoretically. The impact generates a bending wave that may or may not return to the impact point before the ball bounces off the beam. If the ball rebounds before the bending wave returns, then the rebound speed of the ball is determined by localised bending of the beam in the vicinity of the impact point. For a short duration impact, the rebound speed is therefore independent of whether the beam is freely supported or clamped at one or both ends, unless the impact occurs close to an end of the beam. Two different values of the coefficient of restitution can be defined, depending on whether the speed of the beam is measured when the ball rebounds or at a later stage. Experimental results were obtained using a small rubber ball impacting light wood and steel beams. A video camera and two piezoelectric disks were used to capture the relevant ball speeds and vibration response of the beam. The experiment would be suitable for undergraduate students. [ABSTRACT FROM AUTHOR]