Conservation of Mechanical Energy View the following yideo, which demonstrates and explains the experiment in which a glider accelerates along an airtrack due to the slight tilt ofthe track. Theoretical Principles The motion of the glider can be analyzed in two different, equivalent ways. The first is using Newton's Law, i.e. the relationship between force and acceleration. The second is using energy conservation. Compared to our preyious discussions of forces causing acceleration, there is a small ney.r subtlety here, which is that there is a force acting on the object in a direction, which is neither along the direction of motion, nor perpendicular to it, but at some arbitrary angle. To figure out the acceleration, we mustfind the component ofthe force, which is along (Le. parallel to} the direction of motion. The force which is at an angle to the direction of motion ofthe glider, is the gravitational force ofthe Earth. The part ofthe grayitational force which is parallel to the track is giyen by Pg" = Mg sin El where B is the angle between the track and the horizontall. M is the glider's mass. If we assume there are no other forces along the direction of motion, we conclude that the acceleration is {g sin B]. Note also, that we can express the sine in terms of the ratio of height change to position change. sin B = {hi hxg XL} The velocities of the glider at the two photo gate locations are related by the familiar equation 2 amiss. x111 where all these quantities are measured in the direction parallel to the track. :22 is the yelocity at position x: of the second {lower} photogate. in is the yelocity at position XL ofthe first (higher) photogate. This equation can be used to get an alternative expression for the acceleration. Going back to the preyious equation, if we multiply it by DMZ and use the expression for sin El, we get i z 1 . . EMyg 5ij = Mgfhl _.ll!g) We can identify the following quantities in this equation. Kinetic energy: K = :Myz Grayitational potential energy: U = M gh ' Look at your textbook, if you neec a review The equation can then be written as K2 + U2 = Kj + U1 The sum of the kinetic energy and the gravitational potential energy is sometimes referred to as mechanical energy. This last equation says that the mechanical energy is conserved i.e. it remains constant in time. Note that we got this equation by assuming there were no forces other than gravity along the track. So, for example, the effect of friction is not taken into account here. In the above discussion, reformulating the physics in terms of kinetic and potential energy is just a matter of algebraic manipulation. However, the equation is actually valid in more general situations. For example, the equation would still hold if the track was curved and if the track ascended and descended to several different heights between the initial and final points. Assignment The experiment is performed for three different masses, in two trials, corresponding to two different tilts of the track. Use the data in the accompanying excel file and fill out the two tables. You are asked to show that the sum of the kinetic energy with the gravitational potential energy remains constant throughout the motion. You are also asked to find the acceleration of the glider from g and from the velocities. Answer the questions below. Questions Watch the video and fill out the excel file first. Then look at questions. 1. How is the effect of friction counteracted in the experiment? 2. How is the speed of the glider measured in the experiment? 3. In principle, one could think about triggering the photogate on the glider itself, instead of using a flag, i.e. record the time it takes the entire glider to pass the photogate. Would this make the measurement of the speed more or less accurate? Hint: the glider is accelerating everywhere along the track, including as it is passing through the photogate. . If two people choose two different levels as the 0 height i.e. one person chooses the lab floor and another the table top, would the potential energy values be different? Could this spoil agreement between the energy sums? 5. Comment on whether the experiment demonstrates conservation of mechanical energy. 6. Comment on whether the two different calculations of the acceleration agree.X AutoSave ( Off) conservation of energy with airtrack only data (v3) - Read-Only Search (Alt+Q) Rahe Khan RK File Home Insert Page Layout Formulas Data Review View Acrobat Power Pivot Comments Share X Cut Calibri 11 A 29 Wrap Text General AutoSum Normal Bad Good AY Paste Copy BIU -MAY Merge & Center ~ % 9 08 28 Conditional Format as Fill Neutral Calculation Check Cell Insert Delete Format Sort & Find & Analyze Format Painter Formatting Table Clear ~ Filter ~ Select v Data Undo Clipboard Font Alignment Number Styles Cells Editing Analysis N24 vi XVfx A B C D E F G H K M N O Flag length g meter / 2 meters second^2 0.049 9.804 first second first velocity at kinetic energy potential energy and potential timer at velocity at kinetic energy potential energy and potential difference of photogat photogat photogat first at first at first energy at first second second at second at second energy at second energy sum of 6 mass e photogate photogate photogate photogate photogate photogate photogate photogate photogate the photogates (flag K1= V2 = K2 = formul length) / (1/2)*mass*v1^ (flag (1/2)*mass*v2^ (E2-E1)*200 / h1 h2 M t1 t1 2 U1=mass*g*h1 E1=K1+U1 t2 length) / t2 2 U2=mass*g*h2 E2=K2+U2 (E2+E1) kilogram meters / meters / 8 units meters meters S seconds seconds Joules Joules Joules seconds seconds Joules Joules Joules percent 9 0.103 0.078 0.289 0.0953 0.0564 10 0.103 0.078 0.309 0.0962 0.0569 11 trial 1 0.103 0.078 0.339 0.0968 0.0573 12 13 0.116 0.083 0.289 0.0875 0.049 14 0.116 0.083 0.309 0.0844 0.0489 15 trial 2 0.116 0.083 0.339 0.0799 0.0467 16 17 Note that excel's trig functions use radians by default. The formula for the calculation of the track angle inlcudes conversion to degrees. In the 18 calculation of the acceleration from g, make sure to use the sine from column I. calculation and measurement +X AutoSave Off ) conservation of energy with airtrack only data (v3) - Read-Only Search (Alt+Q) Rahe Khan RK File Home Insert Page Layout Formulas Data Review View Help Acrobat Power Pivot Comments Share X Cut Calibri 11 ab Wrap Text Genera Normal Bad Good FIX AutoSum ~ Paste Copy Fill BIULA = Merge & Center % Conditional Format as Calculation Check Cell it Delete Format Sort & Find & Analyze Format Painter Formatting Table v Neutral Clear Filter ~ Select v Data Undo Clipboard Font Alignment Number Styles Cells Editing Analysis N24 vi XVfx A B C D E F G H K M N O 16 17 Note that excel's trig functions use radians by default. The formula for the calculation of the track angle inlcudes conversion to degrees. In the 18 calculation of the acceleration from g, make sure to use the sine from column I. position position of distance height acceleration acceleration percent of first second between difference of sine of track track n along along track difference for 19 photogat photogate photogates photogates angle angle track from from velocities acceleration 0 = asin( (h1-h2) / percent sin (0) = (x2-x1) ) (v212-V1^2) / difference of 20 formula x1 x2 x2-x1 h2-h1 (h1-h2)/(x2-x1) *180/PI() g*sin(0) (2*(x2-x1)) column K and L meters / meters / units meters meters meters meters pure number degrees seconds^2 seconds^2 percent trial 1 0.35 1.19 0.35 1.19 0.35 1.19 trial 2 0.4 1.19 0.4 1.19 0.4 1.19 29 30 31 32 33 34 35 36 37 calculation and measurement +