LEP Newton s 2 nd Law / Air track with Cobra3. Related topics Linear motion, velocity, acceleration, conservation of energy.

Transcription

1 Related topics Linear motion, velocity, acceleration, conservation of energy. Principle According to Newton s 2 nd law of motion for a mass point, the relationship between mass, acceleration and force are investigated. Tasks The distance-time law, the velocity-time law and the relationship between mass, acceleration and force are determined. The conservation of energy can be investigated. Portable balance, CS Connecting cord, l = 100 cm, red Connecting cord, l = 100 cm, blue Connecting cord, l = 100 cm, yellow PC, WINDOWS 95 or higher Fig. 2. Connection of the compact light barrier with the Cobra3 BASIC UNIT Equipment Cobra3 Basic Unit Power supply, 12 V RS232 cable Cobra3 Translation/Rotation Software Light barrier, compact Air track Blower Pressure tube, l = 1.5 m Glider for air track Slotted weight, 1 g, polished Slotted weight, 10 g, black Slotted weight, 50 g, silver bronze Stop, adjustable Starter system, mechanical, with trigger Magnet with plug for starter system Tube with plug Plasticine Hook with plug Silk thread, l = 200 m Weight holder, 1 g Bench clamp -PASS Right-angle clamp Support rod, l = 250 mm Support rod, l = 10 cm Measuring tape, l = 2 m yellow blue red Fig.1: Experimental set-up to measure linear accelerated movements. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

2 Alternative experimental set-ups are located at the end of this experimental description. Set-up In accordance with Figs. 1 and 2. Ensure that the thread runs parallel to the track. Procedure Set the measuring parameters in accordance with Fig. 3. Place the glider in the starting position and affix it to the starter system with the magnet. The weight holder has to be located directly adjacent to the light barrier s wheel. Release the glider; it starts accelerated motion. At the end of the movtion it collides with the stopper. Recording of the measured values begins as soon as the value of the measured velocity exceeds 0.01 m/s. During the course of the motion, the glider s velocity is continuously actualized. At the end of the measurement recording, the velocity again falls below 0.01 m/s and the recording process ends. The mass of the glider can be altered by adding slotted weights. Always place weights having the same mass on the glider s weight-bearing pins, as optimum gliding properties are provided only with symmetrically loading. The accelerating force acting on the glider can be varied by changing the number of weights (on the weight holder) acting via the silk thread and the precision pulley. Determine the mass of the glider without the supplementary slotted weights by weighing it. Position the adjustable stop on the track in such a manner that the glider is gently braked just before the accelerating weight touches the floor. To determine the acceleration as a function of the mass, increase the mass of the glider progressively by 20 g increments (10 g on each side). In determining the acceleration as a function of force, the total mass remains constant. Successively transfer 2 g (1 g from each side) from the glider to the weight holder. The accelerated mass must not exceed 20 g. Fig. 3. Measurement parameters (Light barrier) Before beginning with the measurements, it is advisable to check the track`s adjustement. Remarks If the values (50 ms) in the Get value every (50) ms dialog box are too high or too low, noisy or non-uniform measurements can occur. In this case, adjust the measurement sampling rate appropriately. At excessively low velocities and exceedingly high sampling rates (short ms times) the velocity 0 m/s can be intermittently measured at irregular intervals. In this case increase the sampling time. Theory and evaluation Newton s equation of motion for a mass point of mass m to S which a force F is applied is given by the following: where is the acceleration., The velocity v obtained by application of a constant force is given as a function of the time t by the expression for S S m a F S d 2 S r a dt 2 Assuming that the position of S F S v 1t2 m t S v S S v 102 0; r S r of the mass point is S 1 F S r 1t2 (0) 2 m t2. In the present case the motion is onedimensional and the force produce by a weight of m 1 is 0 F S 0 m 1 0 g S 0 m 1 g where g is the acceleration of gravity. If the total mass of the glider is m 2 the equation of motion is given by 1m 2 m a S 0 m 1 g; The velocity is 0 S v 1t2 0 v m 1 g t m 1 m 2 and the position is 0 r S 1t2 0 s 1t2 1 2 m 1 g m 1 m 2 t 2. (1) (2) (3) PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

3 Typical measurement data are displayed in a v-t (or s-t) diagram (Fig. 4). In addition to the measured points of interest (the rising branch of the v (t) curve), the collision phase of the glider with the stopper is also recorded. These latter measured points must be deleted before continuing with the evaluation. At the left image margin, at low velocities, the velocities have been only inadequately registered due to the slow rotation of the light barrier s wheel and show a nearly constant velocity course. Fig. 5 shows the velocity-time curve, a straight line, which conforms to the relationship v = a t. A regression line has been fitted to the measured points; the slope m supplies the acceleration a, in this case m/s 2. The time course of acceleration a (t) is shown in Fig. 6. Initially, the acceleration increases. This is due to the light barrier s measurement principle. The light beam interruptions by the spoked wheel are counted for equal time intervals, in this case Fig. 4. Measurement data before the deletion of the unnecessary measured points Fig. 7. Path-time law Fig. 5. Velocity-time law with linear regression curve Fig. 8. Path-time law with the square of time on the x axis Fig. 6. Acceleration-time law Fig. 9. Kinetic energy PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

4 50 ms. At very low velocities even with accelerated movement only one interruption per time interval occurs, i.e. the velocity is measured as a constant value. Subsequently, measured points of constant acceleration are recorded. Ultimately, the acceleration again decreases, i.e. when the glider reaches the end of the track or the stop. If the Measure icon is chosen, a horizontal cursor line can be positioned. It approximately passes through the centres of the points that represent constant acceleration. The measured average acceleration value a agrees well with the value determined using regression in Fig. 5. Now the acceleration of gravity can be calculated. The following relationship applies (see equation (1)). (m 1 + m 2 ) a = m 2 g and thus (m 1 + m 2 ) a / m 2 = g Since m 1 (the mass of the glider) and m 2 (the accelerated mass) are known (their weights can be checked), the acceleration of gravity g can be calculated with the calculator in Utilities in WINDOWS when a has been measured. For a typical measurement, where m 1 = g, m 2 = 5 g and a = m/s 2 are measured, it follows that g = 9.4 m/s 2. According to equation (3) the curve of the path-time law exhibits a parabolic course (Fig. 7). This fact can be verified as follows: The time axis is squared to obtain a linearized curve course (Fig. 8). Using the Measurement / Channel Manager, the time is placed on both the x and the y axes. This is necessary because only the y axes can be mathematically reworked. Using Analysis / Channel modification, the operation x := x * x is performed on the y axis. This new channel is exported into the original measurement (Export Measurement / Measuring Channel). Fig. 10. Potential energy Fig. 11. Sum of the respective kinetic and potential energies Fig. 12. Experimental set-up for measurement of the linear, accelerated movement (with movement sensor) PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

5 Finally, using Measurement / Channel Manager, the new squared time is assigned to the x axis and the path s (t), to the y axis. The regression line in Fig. 8 proves that the curve course is now linear and thus also the original quadratic dependence of the path on the time. In a very similar manner, other measured channels can also be mathematically transformed. To verify the energy balance, the kinetic and potential energy has to be calculated and displayed. Kinetic energy (Fig. 9): E kin (t) = 0.5 (m 1 + m 2 ) v 2, where m 1 + m 2 = g. Conversion using: Analysis / Channel modification / Operation x:= 0.5 * * x * x, where x = v (t). Potential energy (Fig. 10): E pot (t) = m 2 g (h s (t)), where h = m. Conversion using: Analysis / Channel modification / Operation x:= 5 * 9.81 * (0.932 x ), where x = s (t). The law of conservation on energy states that the sum of kinetic and potential energy in this closed system must be constant. This statement can easily be checked by the addition of potential and kinetic energy (Fig. 11). Remarks In order to achieve a higher path resolution, the movement sensor ( ) can be used instead of the compact light barrier ( ) (cf. Fig. 12 and Fig. 13). Please ask for this alternative set-up. Mount the movement sensor in such a manner that its housing is to the right of the track when the car moves toward the sensor. If the movement sensor is mounted the other way round, negative velocities will be measured. Pass the thread across the larger of the movement sensor s two cord grooves. Set the measurement parameters according to Fig. 14. Fig. 13: Connection of the movement sensor to the Cobra3 Basic Unit Fig.14: Measurement parameters (movement sensor) red black yellow BNC1 BNC2 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

6 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen