Product Instructions: Linear Air Track

Transcription

1 FO060 The Linear Air Track facilitates the study of linear motion under conditions of low friction. Air track blower This pressure is released through the series of drilled holes along the track, creating a cushion of air between the track and anything mounted on it. Setup and adjustment The inlet on the end of the air track for connection to the blower. For best operation, the air track should be used on a horizontal, flat surface. The air track is supported by three feet that screw into the legs on the bottom of the air track. The feet can be adjusted individually, by being screwed in and out, to compensate for tilt in two dimensions. First, set the track up so that it looks level to the eye. Attach the blower and turn on. Place a vehicle on the track, hold it still, then gently release it. The single foot should be adjusted so that the track is level along its length. Then, look carefully at the vehicle end-on. Both sides should be floating at an equal distance above the air track. If not, adjust the pair of levelling screws until the vehicle floats level. You may need to adjust the single screw again after this step. The levelling screws are packed separately for transport, but can be left in the air track for storage. The pair of feet should be adjusted so that the track is level along its width

2 Accessories The track includes three vehicles: two larger weighing 400g, and one smaller weighing 00g, so the ratio of masses is ::. When these are placed on the air track with a suitable blower attached, they will glide along experiencing almost no friction. A number of accessories are included for studies of different types of collision. The accessories simply plug into the ends of the vehicles. Accessory Magnets Horns with elastic bands Plastic buffers/counterweights Bronze buffers/counterweights Velcro pads Purpose Elastic collisions (C R ) Inelastic collisions or counterweights (0 <C R <) Perfectly inelastic collisions (C R 0) Horns with elastic bands These are suitable for producing elastic collisions between vehicles. The horn arms are flexible, and can be spread to ensure the elastic bands are taut between them, as illustrated. Horns with elastic bands stretched across. The horns can be bent apart to ensure the elastic bands are taut. They are very lightweight but, when a horn with elastic band is used, a plastic counterweight should be attached to the other end of the vehicle. When two horns are to be used, they should be mounted at 45 to the vehicle, so the elastic bands are perpendicular to each other. This ensures the bands have plenty of room to stretch. Horns with elastic bands stretched across. The horns can be bent apart to ensure the elastic bands are taut

3 Magnetic Buffers The magnets on the buffers are identically polarised so they will repel. As they act over a distance, no physical contact is made, so no energy can be lost to producing heat or sound from an impact. Therefore, they are an alternative way of producing elastic collisions between two vehicles. It is a good idea to test the strength of the repulsive force by attempting to push the magnets together by hand. These buffers are quite heavy, so the larger bronze counterweight should be used. A large vehicle with magnet accessory (right) and bronze counterweight (left) Do not allow the magnetic buffers to crash into each other at high speed. Magnetic materials are naturally fragile, so they are at risk of shattering. The collision will not be elastic if the magnetic buffers make contact. Plastic and Brass Buffers The main purpose is to act as counterweights for the other buffers, but they can also be used as buffers in a collision. The two materials do not interact with each other at a distance like the magnetic buffers, nor do they deform significantly like the elastic bands. When they hit each other, some energy is lost as a shockwave along the vehicle and eventually becomes heat and sound. Therefore, they are suitable for creating inelastic collisions. When the brass buffers are used, the magnets should be mounted on the other side of the vehicle as counterweights. When then plastic buffers are used, the Velcro pads should be used as counterweights. Plastic buffers Brass buffers

4 Velcro Buffers The pair consists of a Velcro hooked buffer and a loop buffer, which will stick together. The purpose of this is to prevent the vehicles from separating after a collision, resulting in a perfectly inelastic collision, as discussed in section 5. If the collision happens at too high a speed, then the Velcro may not be Velcro buffers able to stick effectively, but damage is unlikely to result. The plastic buffers should be used as counterweights when the Velcro buffers are mounted. After an experiment, the Velcro buffers can be separated simply by pulling them apart gently. The Velcro material is very strongly glued to the plastic buffer. Interrupter Cards These are for interrupting light gates for timing applications. The two larger cards are for the larger vehicles, and the smaller card for the smaller vehicle. They fit in the slots on the top of the vehicles as illustrated. Mounting the interrupter cards on the large vehicles (left) and small vehicle (right). Spring Buffers The springs should be mounted at the end of the air track to reflect vehicles back along the track. They need to be placed at an angle in order to fit on. The material is very springy, and will retain its elasticity throughout the life of the air track. At times, they may need to be bent back into shape. This can be done by hand. These can also launch a vehicle in a controlled way, as explained on the next page. Phosphor bronze spring attached to end of the air track

5 Putting A Vehicle In Motion The vehicle is pushed into the spring by distance x, then released. The energy stored in the spring is used to put the vehicle in motion. The energy stored in a spring is: The kinetic energy of an object is: k: spring constant kx x: displacement m: mass mv v: velocity So assuming all the energy in the spring is converted into kinetic energy: mv = kx And because the spring constant and mass are constant: v α x Therefore: v α x So, for example, if you wish one vehicle to be launched at half the speed of another, the spring should be compressed by half the distance. To launch twice the mass at the same speed requires more energy, so the spring must be compressed further: kx m = mv x m = x x m = x.5x

6 Timing Equipment There are several timing devices on the market that use light gates to time interruptions and calculate speed and acceleration. They will allow you to enter the size of the interrupter, typically in cm or mm, before starting the experiment. From the interrupter size, and the duration of the interruption, the device calculates the speed of the object attached to the interrupter: v: velocity l v = l: length of interrupter t t: duration Some devices only allow lengths in whole cm, so you may need to cut the interrupter cards to a suitable length. It is important to leave the bottom part of the interrupter card at its original length so it can be fitted into the slots on the vehicle. ~4mm The large interrupter fits heavier vehicles. The narrower top part interrupts the light gate and can be cut to any desired length, as long as the wider bottom part is not altered, so it fits in the vehicle. ~6mm The small interrupter fits the light, shorter vehicle. The wider top part interrupts the light gate and can be cut to any length, as long as the narrower bottom part is not altered, so it fits in the vehicle

7 Good Experimental Practice Regular checks on the track to ensure it is level are essential to the achievement of good results. With the air blower running, place a vehicle at the centre of the track and wait for any motion along the track. Adjust the levelling screws as necessary. It should be possible to have a vehicle sitting at rest for several seconds, while it awaits an incoming vehicle. Before taking any measurements, it is advisable to perform the intended experiment and simply observe the reaction. This way, a qualitative appreciation can be gained, and the result of future experiments can be predicted through intuition. Having a qualitative understanding of the results of the experiment provides a good foundation for deriving and understanding a quantitative description, thus leading to laws that describe any collision. When the time comes to take measurements, the experiment should be repeated a number of times, and averages taken to reduce errors and identify anomalies. Typically three good runs should be a minimum. As in most scientific investigations, it is good practice to control the variables! In any experiment, try to keep as many factors as possible the same, e.g. same masses, same launch speed while you try different buffers. Similarly, when investigating collisions at different speeds, the buffers and masses should be kept the same

8 Elastic Collisions An elastic collision is one in which no momentum or kinetic energy is lost, although it can be redistributed among the objects involved. Consider two masses m and m, which are moving relative to each other. Before they collide, they have velocity and u, and after they collide they have velocity and v. m m u Masses m and m, moving at velocity and u respectively, before collision. m m v The same masses moving at velocity and v respectively, after collision. Conservation of energy Rearranging (m + m u ) = (m v + m v ) m (v ) = m (u v ) Expanding m ( u )(v + u ) = m (u v )(u + ) Equation.0 Conservation of momentum m ( u ) = m (u ) Equation. Dividing.0 by. + = u + v Equation. Rearranging u = ( v ) Equation.3 Equation. shows that the average velocity of the masses before and after the collision is the same. Equation.3 shows that the relative velocity of one mass to the other is reversed. Assuming we can measure the masses and the initial velocities, we can use these simultaneous equations to calculate the speeds after the collision. When elastic is stretched then released, it will return to its original shape with almost no loss of energy. Therefore, the horns with elastic bands can be used to create elastic collisions. The magnetic buffers also give elastic collisions

9 Experiment : Elastic Collision, Two Equal Masses, One Stationary Select the two heavy vehicles, and attach the magnets and counterweight accessories, or the horns with elastic bands. Place one vehicle in the centre of the track, and make sure it is stationary. Place the other vehicle at the end, so that the magnets attached to the two vehicles are facing each other. Position one light gate roughly half way between the two vehicles, and place the other light gate on the other side of the stationary vehicle. Set up your timing device so that it will capture the speed at each light gate. Light Gate Light Gate Experimental setup for the elastic collision of moving and stationary vehicles. Start your timing device, and launch the vehicle at the end of the air track towards the stationary vehicle. A good way to do this is to push it against the bronze spring at the end of the air track, then carefully release it. When the two vehicles collide, most, if not all, of the speed should be transferred to the stationary vehicle. When the vehicle has passed through the second light gate, stop your experiment, and compare the speeds measured at the two light gates. Conservation of energy v u = u v (u = 0) v u = v Equation.4 Conservation of momentum = u v (u = 0) = v Substitute.4 into.5 u v + v u = v Equation.5 v = Substituting v into.5 u = = 0 These values obey the rules derived on the previous page: The average velocity of the masses before and after the collision is the same The relative velocity of one mass to the other is reversed

10 Experiment : Elastic Collision, Two Equal Masses, Moving In Opposite Directions Set up the apparatus as in experiment, but this time placing the vehicles at either end of the track. This time, the vehicles are going to be projected towards each other, at approximately the same speed. Your timing device should be set up to measure two speeds at each light gate. Start your timing device, and release the two vehicles simultaneously. The two vehicles should travel through the light gates before colliding, and then travel back through their respective light gates in the opposite direction. At this point, the experiment can be stopped. Compare the velocities measured at the light gates before and after the collision. Conservation of momentum: = u v Substituting u = : = v = v Equation.6 Consider the two rules: The average velocity of the masses before and after the collision is the same The relative velocity of one mass to the other is reversed v = (u u ) Substitute v = and u = ( ) = ( ( u )) = = = u Substituting = u in to 3.6 v = = u =

11 Experiment 3: Elastic Collision, Two Equal Masses, Moving In The Same Direction This is very similar to experiment, but both vehicles will be moving, at different speeds, in the same direction. It is difficult to execute, and some practice is required for successful implementation. Light Gate Light Gate u Experimental setup for the elastic collision of two moving vehicles. Both vehicles must move in the same direction, but one must collide with the other. Therefore, obviously the rear vehicle must be moving faster than the forward vehicle. They must collide after travelling through light gate, but before travelling through light gate. Your timing device should measure two speeds at light gate, then measure two speeds at light gate. Start your timing device, and simultaneously launch both vehicles in the same direction, the forward one at half the speed of the other. The faster vehicle should travel through light gate, then collide with the slower vehicle, transferring some of its speed to it. That vehicle should then travel through light gate, followed by the first vehicle. Compare the speeds of the vehicles before and after the collision. Conservation of momentum = u v Substituting u = u = v + v = 3 Equation.7 Remember the rules of elastic collisions: The average velocity of the masses before and after the collision is the same The relative velocity of one mass to the other is reversed v = ( u ) v = ( ) = Equation.8 Adding 3.4 and 3.5: = = = u Therefore: v =

12 Experiment 4: Elastic Collision, Two Different Masses Setup the apparatus as in experiment, but replace the centre vehicle with the lighter one. Your timing device should be set to record the speed at each light gate. Start your timer, and put the heavier vehicle in motion towards the lighter vehicle. It should travel through the first light gate before colliding. The lighter vehicle should then travel through the second light gate, and the heavier vehicle should follow it at reduced speed Stop the experiment, and compare the speeds before and after the collision. You probably observed that the smaller vehicle left the point of collision at high speed, while the heavier vehicle lost most of its speed but carried on in the same direction. Conservation of energy (v u ) = u v Substituting u = 0 v = u v Equation.9 Conservation of momentum Substituting u = 0 = v Equation.0 Substitute 3.7 into 3.6 u v + 4 v = u 3 4 v = v v v = 4 3 Equation. Substitute 3.8 into 3.7 =.4 3 = 3 So, after the collision, the lighter vehicle moves away ~.3 times faster than the heavy vehicle before the collision, whereas the heavy vehicle slows to one third of its original speed. Notice the laws of elastic collisions are also obeyed: The average velocity of the masses before and after the collision is the same The relative velocity of one mass to the other is reversed

13 Inelastic Collisions An inelastic collision in one in which some kinetic energy is lost, by conversion to other forms of energy. The ratio of the speed of an object before and after a collision is called the coefficient of restitution: C R : coefficient of restitution C v R = v: speed after impact u u: speed before impact For elastic collisions, where the speed is the same before and after the collision, the coefficient of restitution is. For inelastic collisions, the coefficient is less than. Consider a bouncing ball. The ball loses height after each bounce, until it stops bouncing completely. In this case, the energy is lost mostly to compressing the air inside the ball, causing it to heat up, and some is lost to the sound made by the ball bouncing. h 0 h The solid line is the path of the centre of mass of the ball as it bounces along. The height of the bounce decreases each time because the collision between the ball and the ground is inelastic. If on one bounce, the ball reached height h 0, then upon hitting the ground at speed u, all the gravitational potential energy will have become kinetic energy. mu = mgh 0 On the next bounce, the kinetic energy of the ball, now travelling at speed v, becomes gravitational potential energy again, taking it to height h. mv = mgh So the ratio of the heights: h = v h 0 u = C R In the case of two vehicles colliding, the coefficient of restitution gives the ratio of the relative velocities before and after the collision. C R ( u ) = v Equation

14 Experiment 5: Inelastic Collisions There are several combinations of experiments involving inelastic collisions, using different buffers, vehicle masses and launch speeds. The greatest value can probably be obtained by comparing elastic and inelastic collisions under the same conditions, only with different buffers. The simplest experiment is to have one vehicle pass through a light gate before colliding into a stationary vehicle, similarly to experiment. Use the brass buffers, plastic buffers or one Velcro buffer with a different buffer. Consider equation.0. Substitute u = 0: C R = v Equation. Conservation of momentum: m ( ) = m (u v ) Substituting m = m and u = 0: = v v = Equation. Substituting equation. into.: C R = = ( C R ) v = ( + C R ) So, if the coefficient of restitution is between 0 and (i.e. an inelastic collision) then the velocity of the moving vehicle after impact will be a fraction of the velocity before, in the same direction. This is unlike the elastic collision, where all the speed was transferred to the stationary vehicle. From measuring the velocities of the vehicles before and after impact, it should be possible to estimate the coefficient of restitution of that collision using equation.. Try different buffers and vehicle masses and velocities to observe what effect this has on the coefficient of restitution. Remember to alter one factor at a time, so you can clearly relate your results to the factor you altered

15 Perfectly Inelastic Collisions A perfectly inelastic collision is defined as an inelastic collision whose coefficient of restitution is zero: v u = 0 This implies that the velocity after the collision is zero. If this were a ball hitting a floor, it would simply land and not rebound. In effect, it is sticking to the floor. Now consider two objects colliding with a zero coefficient of restitution: v u = 0 v = 0 = v If the two vehicles are now moving at exactly the same velocity immediately after the collision, they are effectively stuck together. m m u Masses m and m, moving at velocity and u respectively, before collision. m m v The masses m and m moving at the same velocity v after the collision

16 Consider the momentum: m + m u = m + m v = v = v: m + m u = v(m + m ) Rearranging: v = m + m u m + m Equation 3.0 Consider a perfectly elastic collision with two equal masses. Using equation 5.0: v = m + mu m + m (m = m = m) v = + u Equation 3. Experiment 6: Perfectly Inelastic Collision, Two Equal Masses, One Stationary The setup is exactly the same as in experiment, but with the Velcro buffers. Launch one vehicle towards the other. The vehicles should stick together, and continue travelling. Stop the experiment once the stuck together vehicle has moved through a light gate. Compare the speeds before and after the collision. Experiment 7: Perfectly Inelastic Collision, Two Equal Masses, Moving In Opposite Directions This is the same as experiment, but with Velcro buffers. Launch the vehicles towards each other simultaneously, at the same speed, from opposite ends of the air track. They should collide and stick together in the middle of the track. You can stop the experiment after the collision. Using equation 3., complete the table below, and compare with your experimental results. Experiment u v 6 7 Extend this investigation further by using different launch speeds and vehicle sizes, and try to predict results by substituting different mass values into equation