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Conservation of Energy on an Inclined TrackIntroduction272224592710Figure 1: Inclined Track SetupFigure 1: Inclined Track SetupAs the cart rolls freely up and then back down the track, mechanical energy changes form from kinetic energy to gravitational energy, and then back again. Position and velocity of the cart are both graphed vs. time. Potential energy, kinetic energy, and total mechanical energy are then calculated and graphed for the entire motion.Equipment Qty Items Part Number 1 Smart Cart (Blue) ME-12411 250g Cart Mass (Pair) ME-6757A 1 Dynamics Track End Stops (Only 1 needed) ME-89711 Track Rod Clamp ME-98361 1.2m Starter Dyn Track ME-94931 Large Rod Base ME-87351 Rod, 45 cm ME-87361 Elastic Bumper (Elastic & 1 pair brackets needed) ME-8998 Required, but not included:1 Meter Stick (Only 1 needed) SE-88271 Balance SE-87231 PASCO Capstone software SetupOn a level tabletop, set up the track as shown in Figure 1 using the rod base and 45-cm rod. If the tabletop is not level, it will have an effect on your results. Note that the lower end of the track rests directly on the table, and that the track feet are NOT used in this experiment. If needed, you can modify the setup to include track feet to get the track level before lifting one end. The square nut on the Track Rod Clamp slides into the T-Slot on the track, and allows the track to be secured at various angles. 424542971252Figure 2: Elastic BumperFigure 2: Elastic BumperInstall the fixed End Stop at the top of the incline (see Figure 1), and an Elastic Bumper at the bottom, using at least two pieces of elastic (see Figure 2).Place the Smart Cart on the track so that its +x-direction, printed on top of the cart, is pointing up the track.Turn on the Smart Cart, and connect it wirelessly in PASCO Capstone.In the lower toolbar, select Smart Cart Position Sensor, and set the sample rate to 50 Hz.Initial Measurements4209415106680Figure 3: Finding sin θFigure 3: Finding sin θIf we know the height H at the upper end of the track, and length L of the track (see Figure 3), then we can find sin θ using Equation 1: sin θ = H/L(1)Carefully measure the length L of the track. In Capstone, open the Calculator in the left toolbar, and create a new “calculation” for L. For example, if L = 1.22 m, in Calculator, you would type:L = 1.22and enter m for the units.3804920107315Figure 4: Measuring HFigure 4: Measuring HAdjust the height H of the upper end of the track to about 20 cm. The height should be measured to the underside of the track as shown in Figure 4. Carefully measure H and record the value as a “calculation” for H in the Calculator, with units of m.Measure the combined mass of the cart plus two cart masses. Record in the Calculator as “m” in units of kg.Measuring Position and VelocityIn Capstone, create a graph of Position vs. Time. Add a new plot with Velocity on the vertical axis.Hold the cart at rest on the inclined track near the elastic bumper at the lower end. Make sure the +x-direction printed on top of the cart is pointing up the incline.Add the two cart masses to the cart.Start recording data and give the cart a rapid but short-distance (about 0.10 m) push up the track. After you stop pushing, the cart should go fairly high up the incline, near the upper end stop, without hitting it. Repeat if needed until you get a good graph. You can delete unwanted runs using the Delete feature in the lower toolbar.Stop the cart and stop recording when the cart returns to the lower end of the track.Click on the velocity graph to select it. On the graph toolbar, create an annotation. Drag the cursor for the annotation to the point where your push ends, and type “Push ends” in the annotation box. Repeat for the moment where you begin to stop the cart when it returns, and type “Stop begins” in this annotation box.Calculating Energies375260320774Figure 5: Finding hFigure 5: Finding hThe vertical height h of the cart will have an origin (h = 0) at the lowest position of the cart, where you began your initial push. The Position x of the cart is measured along the track from where your push began. From Figure 5 comes the equation: sin θ = h/xShow that combining this equation with Equation 1 results in Equation 2 for the cart’s height h at any moment:h = xH/L(2)Enter Equation 2 in the Calculator in Capstone as:?h?=[Position, Blue (m)?]*[H (m)?]/[L (m)?]with units of m. Note that “Red” can replace “Blue” here if using the red cart.The potential energy (PE) of the cart can be found from: PE = mgh(3)where m is the mass and g (9.8 m/s2) is the gravitational acceleration. Enter Equation 3 in the Calculator as:?PE?=[m (kg)?]*9.8*[h (m)?]with units of J (joules).The kinetic energy (KE) of the cart can be found from: KE = 0.5mv2(4)where v is the cart’s speed (magnitude of velocity) at that moment. Enter Equation 4 in the Calculator as:??KE?=0.5*[m (kg)?]*[Velocity, Blue (m/s)?]^2with units of J (joules). Note that “Red” can replace “Blue” here if using the red cart.The total mechanical energy (E) of the cart can be found from: E = PE + KE(5)Enter Equation 5 in the Calculator as:??E?=[PE (J)]+[KE (J)?]with units of J (joules).Graphing EnergiesIn Capstone, create a graph of PE vs. Time. Add a new plot area with KE on the vertical axis. Add a third plot area with E on the vertical axis.On the E vs. Time plot, right-click on the vertical axis values and choose to auto-scale from zero.From the graph toolbar, add a multi-coordinate tool, to show the value of all three energies at the same moment. If needed, use the tool properties to change the numerical format of the vertical coordinate, so that it shows enough decimal places. Drag the multi-coordinate tool a time just after your initial push ends (the first peak on the KE graph). For this initial time, record the PE, KE, and E.Then repeat the PE, KE, and E measurements for the following four times: About halfway between the end of your push, and the highest point (where the cart momentarily stops)The time of the highest pointAbout halfway between the highest point, and the time you begin to stop the cart (when the KE hits its second peak).The final time, just before you begin to stop the cart.As the car moves UP the track: Does PE increase, decrease, or remain approximately constant?Does KE increase, decrease, or remain approximately constant?Does E increase, decrease, or remain approximately constant?Repeat, but: As the car moves DOWN the track: ................
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