Momentum and Impulse. Objective. Theory. Investigate the relationship between impulse and momentum.

Size: px

Start display at page:

Download "Momentum and Impulse. Objective. Theory. Investigate the relationship between impulse and momentum."

Dale Hill
5 years ago
Views:

1 [For International Campus Lab ONLY] Objective Investigate the relationship between impulse and momentum. Theory Reference Young & Freedman, University Physics (14 th ed.), Pearson, (p.262~266) Because, we can write Newton s second law for a particle with constant mass as (1) Many situations such as collision or explosion involve little known forces exerted on bodies for a short time. Thus, they cannot be analyzed by directly applying Newton s second law,. Two new concepts, momentum and impulse, and a new conservation law, conservation of momentum, enable us to analyze these situations. We call the momentum, or linear momentum, of the particle. Using the symbol for momentum, we have (2) If we now substitute the definition of momentum, Eq. (2), into Eq. (1), we get (3) This is the form in which Newton originally stated his second law: The net force acting on a particle equals the time rate of change of momentum of the particle (although he called momentum the quantity of motion ). Fig. 1 The velocity and momentum vectors of a particle From Eqs. (2) and (3), the greater the mass and speed of a particle, the greater is its magnitude of momentum, and a rapid change in momentum requires a large net force. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 1 / 17

If a constant net force acts on a particle during a time interval Δ from to, the impulse of the net force, denoted by, is defined to be the product of the net force and the time interval: Δ (4) If is

2 If a constant net force acts on a particle during a time interval Δ from to, the impulse of the net force, denoted by, is defined to be the product of the net force and the time interval: Δ (4) If is constant, then is also constant as in Eq. (3). In this case, is equal to the total change in momentum during the time interval, divided by the interval: Fig. 2 shows the net force as a function of time during a collision. Impulse during the time interval is represented by the area under the curve. Note that a large force acting for a short time can have the same impulse as a smaller force acting for a longer time if the areas under the force-time curves are the same, as in Fig. 2(b). In other language, an automobile airbag provides the same impulse to the driver as would the steering wheel by applying a weaker and less injurious force for a longer time. (5) (6) From Eqs. (4) and (6), we end up with a result called the impulse-momentum theorem: (7) Eq. (7) shows that the change in momentum of a particle during a time interval equals the impulse of the net force that acts on the particle during that interval. The impulse-momentum theorem also holds when forces are not constant. To see this, we integrate both sides of Eq. (3) over time between and : (8) The integral on the left is defined to be the impulse during this interval: (9) Fig. 2 The meaning of the area under a graph of versus 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 2 / 17

analyzes the data measured by various sensors.

including power supplies which provide up to 15 watts of power.

03N Motion Sensor 1 Measures linear position, velocity and acceleration.

Linear Track (2 adjustable feet included) 1 Includes two groves to guide the

both sides for attaching various accessories.

Contains hook-and-loop bumpers on one end and magnetic bumpers on the other

3 Equipment 1. List Item(s) Qty. Description PC / Software Data Analysis: Capstone 1 Records, displays and analyzes the data measured by various sensors. Interface 1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power. Force Sensor 1 Measures the magnitude of force. Range: 50N ~ 50N Resolution: 0.03N Motion Sensor 1 Measures linear position, velocity and acceleration. Photogate (Cable included) 1 Measures high-speed or short-duration events. Linear Track (2 adjustable feet included) 1 Includes two groves to guide the wheels of carts, a metric scale for measuring cart positions, and T-slots on both sides for attaching various accessories. Cart 1 Runs along the track on lower-friction wheels. Contains hook-and-loop bumpers on one end and magnetic bumpers on the other end, for elastic and inelastic collisions End Stops 2 Allow the Force Sensor to mount on the Track. Universal Bracket 1 Allows any accessory to mount on the Track. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 3 / 17

Item(s) Qty. Description Photogate Bracket 1 Allows the Photogate to mount on the Track.

Allows the Photogate to attach to the Bracket.

Electronic Balance Measures mass of an object with a precision to 0.01g. 2.

on the photogate goes on. When the beam is not blocked, the output signal becomes 1 and the LED goes off.

4 Item(s) Qty. Description Photogate Bracket 1 Allows the Photogate to mount on the Track. Thumbscrew (Force Sensor) Thumbscrew (Photogate) 1 1 Allows the Force Sensor to attach to the End Stop. Allows the Photogate to attach to the Bracket. Spring Bumper Set 1 set Includes two spring bumpers with different spring constant. Electronic Balance Measures mass of an object with a precision to 0.01g. 2. Details (1) Photogate When the infrared beam is blocked, the output signal of the photogate becomes 0 and the LED lamp on the photogate goes on. When the beam is not blocked, the output signal becomes 1 and the LED goes off. This transition of signal can be used to calculate quantities such as the period of a pendulum, the velocity of an object, etc. The photogate sensor is an optical timing device used for very precise measurements of high-speed or short-duration events. It consists of a light source (infrared LED) and a light detector (photodiode). When an object moves through and blocks the infrared beam between the source and the detector, a signal is produced which can be detected by the interface. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 4 / 17

(2) Force Sensor (3) Motion Sensor The Force Sensor measures both pulling and pushing forces in the range of 50N to 50N. The sensor uses a strain gauge attached to an aluminum beam.

The gauge is wired to form a full-bridge circuit that is driven by a constant voltage source. The voltage across the bridge circuit is proportional to the applied force.

The sensor uses an electrostatic transducer as both a speaker and a microphone. For each sample, the transducer transmits a burst of 16 ultrasonic pluses with a frequency of about 49 khz.

5 (2) Force Sensor (3) Motion Sensor The Force Sensor measures both pulling and pushing forces in the range of 50N to 50N. The sensor uses a strain gauge attached to an aluminum beam. The gauge consists of an insulating flexible backing which supports a metallic foil pattern. As the aluminum beam is deformed, the foil is deformed, causing its electrical resistance to change. The gauge is wired to form a full-bridge circuit that is driven by a constant voltage source. The voltage across the bridge circuit is proportional to the applied force. The Motion Sensor measures position, velocity, and acceleration of a target. It produces a series of ultrasonic pulses and detects the sound reflecting back from an object in front of it. The sensor uses an electrostatic transducer as both a speaker and a microphone. For each sample, the transducer transmits a burst of 16 ultrasonic pluses with a frequency of about 49 khz. This burst of pulses can be heard as a single click. The ultrasonic pulses reflect off an object and return to the sensor. The target indicator on the sensor flashes when the transducer detects an echo. Sound intensity decreases with distance; to compensate, the sensor increases the gain of the receiver amplifier as it waits for the echo. The increased gain allows the sensor to detect an object up to 8m away. The lower gain at the beginning of the cycle reduces the circuit s sensitivity to echoes from false targets. The sensor measures the time between the rigger rising edge and the echo rising edge. It uses this time and the speed of sound to calculate the distance to the object. To determine velocity, it uses consecutive position measurements to calculate the rate of change of position. Similarly, it determines acceleration using consecutive velocity measurements. Range: Short Range Setting 0.15 ~ 2 m (noise rejection) Long Range Setting 0.15 ~ 8 m 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 5 / 17

(1-1) Install Universal Bracket at the left end of the track.

) 1 Slide the tap and square nut of the accessory into the T-slot of the track.

To demount it, 1 Loosen the thumbscrew (counterclockwise). 2 Remove it with care.

6 Procedure Experiment 1. Impulse-Momentum Theorem NOTE To mount any accessory to the track, (1) Set up equipment. (1-1) Install Universal Bracket at the left end of the track. (You will use this accessory in experiment 2.) 1 Slide the tap and square nut of the accessory into the T-slot of the track. 2 Tighten thumbscrew (clockwise) to secure it. To demount it, 1 Loosen the thumbscrew (counterclockwise). 2 Remove it with care. When you loosen thumbscrew, NEVER try to completely unscrew it. (It won t be reassembled again!) (1-2) Install Photogate Bracket at the right end of the track. (You will use this accessory in experiment 2.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 6 / 17

(1-3) Mount the Motion Sensor. 1 Mount the Motion Sensor on the left end of the track.

3 Set the range switch to the short range ( ) setting.

Treat it with care. CAUTION (1-5) Connect the sensors to the interface.

Deformation of the cover could cause the sensor to fail. (1-4) Install the Force Sensor.

2 Using FS thumbscrew, attach the Force sensor.

Use track feet to raise the Motion Sensor end of the track.

7 (1-3) Mount the Motion Sensor. 1 Mount the Motion Sensor on the left end of the track. 2 Aim the sensor s transducer at the cart (slightly up to avoid detecting the track-top). 3 Set the range switch to the short range ( ) setting. CAUTION Do not tighten the bumper spring too hard. It could cause the Force sensor to fail. Treat it with care. CAUTION (1-5) Connect the sensors to the interface. Do not touch the mesh cover of the Motion Sensor. Deformation of the cover could cause the sensor to fail. (1-4) Install the Force Sensor. 1 Install the End Stop at the right end of the track. 2 Using FS thumbscrew, attach the Force sensor. 3 Screw the spring bumper into the Force Sensor. (1-6) Adjust the inclination of the track. Use track feet to raise the Motion Sensor end of the track. Do not raise the Motion Sensor end of the track too high. The faster the cart moves, the more likely that it may move to one side or the other during the collision. A smooth but slow collision is better than a fast, jerky one. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 7 / 17

(2) Set up Capstone software. (2-4) Add graph displays. (2-1) Add [Motion Sensor] and [Force Sensor]. The interface automatically recognizes the Motion Sensor and the Force Sensor.

In the [Properties] window, Uncheck [Change Sign]. The sign of collected data of the Force sensor is initially positive for the pushing force.

8 (2) Set up Capstone software. (2-4) Add graph displays. (2-1) Add [Motion Sensor] and [Force Sensor]. The interface automatically recognizes the Motion Sensor and the Force Sensor. (2-2) Configure the Force Sensor. Click the Force Sensor icon in the [Hardware Setup] panel and then click the properties button ( ) in the lower right corner. In the [Properties] window, Uncheck [Change Sign]. The sign of collected data of the Force sensor is initially positive for the pushing force. 1 Click and drag the [Graph] icon from the [Displays] palette into the workbook page. 2 Click [Add new plot area ] of the tool bar to add a axis synchronized graph. 3 Select [Time(s)] for the -axis, and [Force(N)] and [Velocity(m/s)] for each -axis. (3) Measure the mass of the Cart. kg (2-3) Adjust the sample rate of measurements. Select [1.00 khz] for [High Resolution Force Sensor], and [20.00 Hz] for [Motion Sensor] in the [Controls] palette. (4) Mount the Cart on the Track. Place the cart in front of the Motion Sensor at least 15cm away. (The sensor cannot measure the distance closer than 15cm.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 8 / 17

9 (5) Zero the Force Sensor. (8) Stop data collection. Press [ZERO] button on the sensor. Wait until the cart stops after collision. When the cart stops, click [Stop]. NOTE You should zero the sensor prior to each data run. (6) Begin recording data. Click the [Record] button at the left end of the [Controls] palette to begin collecting data. The Motion Sensor starts clicking. If a target is in range, the target indicator flashes with each click. (7) Release the cart. (9) Analyze the data. NOTE 1 Scaling and Panning graphs. The Motion Sensor uses an electrostatic transducer as both a speaker and a microphone. For each sample, the transducer transmits a burst of 16 ultrasonic pluses. The ultrasonic pulses reflect off an object and return to the sensor. The sensor measures the time between the trigger rising edge and the echo rising edge. It uses this time and the speed of sound to calculate the distance to the object. You should remove objects that may interfere with the measurement. These include objects, and also your hand, between the sensor and target object, either directly in front of the sensor or to the sides. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 9 / 17

2 Area under a curve 4 Fit Function Click [Select range(s) ] icon and then drag the data range of interest.

Click [Display area ] icon to measure the area under the curve. 3 Data point Use [Show coordinates ] icon to read off the data point.

To find the velocity just before the collision and the velocity just after the collision, we need to find the fit function of the - graph.

10 2 Area under a curve 4 Fit Function Click [Select range(s) ] icon and then drag the data range of interest. Because the Force Sensor (1kHz) and the Motion Sensor (20Hz) sample at the different rate, as you set in step (2-3), the resultant time intervals of measurements are different, as shown below. Click [Display area ] icon to measure the area under the curve. 3 Data point Use [Show coordinates ] icon to read off the data point. The measured velocities are average velocities during each time interval so they could be quite different from the real instantaneous velocities in the region of sudden change. To find the velocity just before the collision and the velocity just after the collision, we need to find the fit function of the - graph. 1 Using - graph, find the time just before the collision and the time just after the collision. 2 Find the linear fits for the before-collision region and the after-collision region of - graph. 3 Calculate the instantaneous velocities by substituting 1 into 2. Follow the steps below to find the linear fit for - graph. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 10 / 17

11 Click [Select range(s) ] icon and then drag the data range of interest. (10) Verify impulse-momentum theorem. Find the experimental values of following equations by reading off the graphs, and verify impulse-momentum theorem. (2) (9) (7) (11) Repeat measurement. 1 Vary the collision speed by adjusting the inclination of the track or by adjusting the starting position of the cart. Click [Select curve fits ] and select [Linear: mt+b] to find linear fit for selected data points. 2 Change the bumper spring with different spring constant Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 11 / 17

Experiment 2. Impulse (1) Set up equipment. Follow the setup of experiment 1, and then, (1-1) Remove the Motion Sensor. (1-2) Install the Photogate.

NOTE In this experiment, the Photogate works as an optical switch so that you can automatically start or stop data collection.

However, if it looks complicated to use automatic measurement method, then you don t have to use the Photogate.

12 Experiment 2. Impulse (1) Set up equipment. Follow the setup of experiment 1, and then, (1-1) Remove the Motion Sensor. (1-2) Install the Photogate. Attach the photogate on the bracket using a thumbscrew. NOTE In this experiment, the Photogate works as an optical switch so that you can automatically start or stop data collection. The automatic measurement method is very helpful in easy analysis. However, if it looks complicated to use automatic measurement method, then you don t have to use the Photogate. In this case, skip steps (1-2)~(1-4) and (2-2)~(2-5). (1-3) Connect the sensors to the interface. (1-4) Adjust the position of the Photogate. Place the Photogate where the cart blocks the Photogate beam just before the cart collides the Force Sensor, as shown in the figure below. (See figure on next page.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 12 / 17

The interface automatically recognizes the

Click the input port which you plugged the

13 (2) Set up Capstone. (2-1) Add the Force Sensor. The interface automatically recognizes the Force Sensor. (2-2) Add the Photogate. Click the input port which you plugged the Photogate into and select [Photogate] from the list. (2-3) Create and configure a timer. Create a timer and configure automatic recording conditions. 1 Click [Timer Setup] in the [Tools] palette, and then select [Pre-Configured Timer]. 2 Check [Photogate, Ch1]. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 13 / 17

Select [Measurement Based] for [Condition Type] of [Start Condition].

The photogate generates 1 while it is blocked, and 0 while open.

5 Skip steps 5 to 6 and finish the timer setup.

: 0.5 (2-4) Configure automatic recording conditions.

The state value of the Photogate changes from 0 to 1.

14 3 Select [One Photogate (Single Flag)]. Click [Recording Conditions] in the [Controls] palette. Select [Measurement Based] for [Condition Type] of [Start Condition]. 4 Check [State], which outputs the value of state of the photogate. The photogate generates 1 while it is blocked, and 0 while open. Set the parameters of [Start Condition] as below. 5 Skip steps 5 to 6 and finish the timer setup. [Condition Type] : Measurement Based [Data Source] : State() [Condition] : Is Above [Value] : 0.5 (2-4) Configure automatic recording conditions. 1 Start Condition The cart approaches and blocks the IR of the photogate. The state value of the Photogate changes from 0 to 1. Start data collection Set the parameters of [Stop Condition] as below. [Condition Type] : Measurement Based [Data Source] : State() [Condition] : Is Below [Value] : Stop Condition The cart comes out of the Photogate after colliding with the Force Sensor. (IR unblocked) The state value of the Photogate changes from 1 to 0. Stop data collection. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 14 / 17

15 (2-5) Check the automatic recording configuration. (3) Mount the cart on the track. You can see the [Record] button at the beginning. Place the cart on the track at the starting position. Use the Universal Bracket so you can release the cart at the same position for all trials. If you click [Record], Capstone waits until the start condition is achieved. ([Record] toggles to [Stop] and recording status display indicates [Waiting].) Even if the timer is ticking, no data is recorded. (4) Zero the Force Sensor. Press the [Zero] button on the sensor (5) Begin recording data. If you block the photogate (start condition) using your finger, data recording starts automatically. The recording status display indicates [Recording]. (6) Release the cart. Re-adjust the angle of the track, if required. If you open the photogate (stop condition), data recording stops automatically. The status display indicates [Ready]. (7) Check the graph.. NOTE (2-6) Adjust the sample rate of measurements. Select [1.00 khz] for [High Resolution Force Sensor]. If the collision speed is too high, the bumper spring becomes over-compressed, i.e. impact is directly transferred to the Force Sensor, and as a result, the graph will show a sharp peak at the center as below. (2-7) Create a graph display. In this case, you should decrease the collision speed so the graph shows a smooth peak. Click and drag the [Graph] icon from the [Displays] palette into the workbook page. Select [Time(s)] for the -axis and [Force(N)] for the -axis. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 15 / 17

2 Area of the graph 3 Data point Refer to the step (9) of experiment 1. (9) Analyze the data. Compare the graphs of two cases.

16 (8) Repeat experiment using the other spring bumper. Change the spring bumper with a different spring constant and repeat steps (3)~(7). Make sure you do not change the inclination of the track and the starting position of the cart so the velocities just before collision are always same. 2 Area of the graph 3 Data point Refer to the step (9) of experiment 1. (9) Analyze the data. Compare the graphs of two cases. 1 Displaying multiple runs Click to depress [Allow simultaneous viewing ] icon. Click the pull-down arrow [ ] and select runs to display. (10) Repeat measurement. Vary the collision speed by adjusting the inclination of the NOTE track or the starting position of the cart. Repeat steps (3)~(9). Recorded data run has a default name Run#%1, where %1 is an automatically generated run number. You can change the name of each data run if required. 1 Click [Data Summary] in the [Tools] palette 2 Select [Show Sensor Data] tap. 3 Right-click on the run name of interest. 4 Select [Rename] from the pop-up list. (11) Analyze your result. You do not measure the speed of the cart in this experiment. However, it is reasonable that the change in momentum is same for each run if you release the cart in the same position. 1 st 2 nd 3 rd 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 16 / 17

Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below.

17 Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below. Delete your data files and empty the trash can from the lab computer. Turn off the Computer and the Interface. Keep the Track Feet attached to the track. Assemble the Universal Bracket and Spring Bumper as shown below. Tighten all thumbscrews in position. Do not try to completely unscrew the thumbscrew and nut assembly of Track Feet, End Stops, and Brackets. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( ) Page 17 / 17

Momentum and Impulse

General Physics Lab Department of PHYSICS YONSEI University Lab Manual (Lite) Momentum and Impulse Ver.20180328 NOTICE This LITE version of manual includes only experimental procedures for easier reading