PHYS1146 Lab 5

Report for Experiment #7 Work and Energy on an Air Track 11/10/23 Abstract The goal of this experiment was to verify the work-energy theorem and acceleration due to gravity was found by using a motion sensor and linear air track with a glider and air pulley along with PASCO programming and then manipulating the work-energy theorem equation. Two investigations were done, and one was done with the air track at an angle and the second was done while the air track was linear, but the glider had a hanging mass attached. The gravities calculated were averaged for both investigation one and two and then compared to the actual value of gravity. The value from investigation one was 10.240 with a percent difference of 4%. In investigation two the value of gravity was found to be 9.312 with a percentage difference of 5.08%. Both calculated values of gravity were close to the actual value of gravity, the work-energy theorem was verified. Introduction

The work-energy theorem states that the work done by all forces acting on an object is also equal to the change in kinetic energy. The purpose of this lab was to verify the work-energy theorem and to measure acceleration due to gravity. The change in kinetic energy is equal to kinetic energy final minus kinetic energy initial. Kinetic energy (K) = . Work done to an object that travels in a straight line along the x-axis is equal to . Therefore, the work-energy equation is equal to: However, when the motion of an object is caused by gravity acting on an angled air track the force is substituted in the following equation to: This equation changes again when an object is moving horizontally while connected to a hanging weight. This is because the change in position of the object is caused by the force of gravity acting on the hanging weight. The work-energy equation then becomes: In investigation one the motion along an inclined plane was investigated. A linear air track, positioned at an angle, with a motion sensor, a glider, and the PASCO Capstone program were used to plot data points for a position vs time graph. These data values could be organized into two different series, then velocity and velocity squared could be calculated. The data values were organized by when the glider was moving towards a motion sensor and when the object was moving away from the motion sensor. By graphing velocity 2 vs average position the slope of the line could be used to calculate the gravity value for when the glider was moving up or down the air track. In investigation two the same air track with a motion sensor, glider, and PASCO Capstone program were used. However, in investigation two the track was not at an angle, but a weight was connected to the glider and hung off the end of the air track. The PASCO Capstone program was again used to plot data points for a position vs time graph. These values were also organized by when the glider was moving towards a motion sensor and when the object was moving away from the motion sensor. By graphing velocity 2 vs average position the slope of the line could be used to calculate the gravity value for when the glider was moving up or down the air track. By doing these two setups the velocities could be compared because in investigation one the velocity was independent of the mass of the glider. While in investigation two the velocity depended on the mass of the glider as well as the mass of the hanging weight. The velocity values were used in derivations for gravity therefore, the velocities could also be used to compare gravity. Investigation 1 The experimental setup consisted of a linear air track with glider and air pulley, a PASCO PASPort USB Link and Motion Sensor, a small block, and a ruler. The air track was attached to an air source which was used to prevent frictional force from acting on the glider. The air source was turned on and the air track was leveled and a screw, which was responsible for adjusting the position of the air track, was adjusted until the glider stayed in place in the middle of the air track. The block was placed under the screw once the air track was leveled. PASCO Capstone was the program used for recording the position 1 2 mv 2 W = F x ∆ x F x ∆ x = 1 2 mv 2 f − 1 2 mv 2 i mgsin θ ( x − x i ) = 1 2 mv 2 f − 1 2 mv 2 i m ′ g ( x − x i ) = 1 2 (m + m ′ )v 2 f − 1 2 (m + m ′ )v 2 i

(2) The average positions between adjacent points were calculated by using: (3) The error in average position was calculated by using: (4) The points were then organized into series where the glider was moving towards the motion sensor and points where the glider was moving away from the motion sensor. A velocity squared vs average position plot was made with the two series. A set of data points of velocity squared vs average position when the glider was moving towards the motion sensor was plotted and a set of data points when the glider was moving away from the motion sensor was plotted. The slopes from these data sets were different because when the glider was moving away from the sensor it was accelerating due to gravity whereas when it was moving towards the sensor it was decelerating due to gravity. The slope values were represented by: (5) The equation of the graph in slope intercept from (v 2 = Bx + C) is equal to: (6) Fig. 3: Position vs Velocity 2 on an incline graph. For each series a best fit line was added and equations for both of the lines were added. The slopes of these lines were related to gravity and sin and was the angle of incline of the air track. δ v 2 = 8v 2 ( δ x ∆ t ) x avg = x n+1 + x n 2 δ x avg = ( δ x n x n ) 2 + ( δ x n+1 x n+1 ) 2 × x avg v 2 = 2gsin θ ( x − x 0 ) v 2 = 2gsin θ x − 2gsin θ x 0 θ θ θ  

could be found by taking the inverse sin of the height of the gliders starting position divided by distance the glider traveled, . The height of the air track was 0.036±0.0005 m and was found using a ruler and the error in height was found by halving the smallest value of measurement on the ruler. The distance of the air track was given in the lab procedure and was 1.000±0.002 m. The uncertainty in the value of sin was calculated by using: (7) Sin was 0.036 and the uncertainty in sin was calculated to be 0.0140. The slope of each line was labeled as B1 and B2 with B1 being the slope from the positive velocity values and B2 being the negative velocity values and B = 2 (8). B1 was found to be 0.680 and B2 was found to be 0.798 m/ s 2 . Gravity values for each series could then be calculated by manipulating (8) where (9). The value of g1 was found to be 9.400±0.669 m/s 2 the value of g2 was found to be 11.080±0.605 m/s 2 . The average of both gravity values was found by adding the values and dividing the calculated number by two. The average value of gravity was 10.24 m/s 2 . The percentage between experimental gravity and the known value of gravity was calculated by using: % difference = (10) Table 1: Experimental values and gravity calculations. θ = sin − 1 ( 0.036 1 ) = 2.063 θ   δ sin θ sin θ = ( δ h h ) 2 + ( δ d d ) 2 θ   g   sin θ g = B 2sin θ g known − g calculated g known × 100 B1 (pos) 0.680 B2 (neg) 0.798 g1 9.400 g2 11.080 g(avg) 10.240 %dif 4% ϭ sin 0.014 ϭ g1 0.669 ϭ B 0.369 B avg 0.739 ϭ g2 0.605 ϭ g avg 0.902

Fig. 4: Position vs time graph from investigation two. The points in which the glider collided with the end of the track were found by seeing where the direction of the glider changed on the graph. The points of collision were discarded and the points between the collision were analyzed. Velocities and velocities squared were then calculated. The velocity computed was average velocity between subsequent points and was calculated by using (1). Velocity squared was calculated by squaring each calculated value of velocity. The error in velocity was calculated by using (2). The average positions between adjacent points were calculated by using (3) and the error in average position was calculated by using (4). The points were then organized into series where the glider was moving towards the motion sensor and points where the glider was moving away from the motion sensor. A velocity squared vs average position plot was made with the two series. A set of data points of velocity squared vs average position when the glider was moving towards the motion sensor was plotted and a set of data points when the glider was moving away from the motion sensor was plotted. Fig. 5: Position vs Velocity 2 from investigation two. For each series a best fit line was added and equations for both of the lines were added, and the slopes could be found through the equation displayed. The slope of each of the lines was again labeled B1

and B2. B1 was equal to 1.131 m/s 2 and B2 was equal to 1.702 m/s 2 . The slope values were represented by: (11) In this equation m’ represented the mass of the hanging weight (0.0309±0.0005 kg) and m represented the mass of the glider (0.375±0.0005 kg). Error in mass was found by halving the smallest division on the digital scale. In slope intercept form the equation was: (12) The experimental value of gravity for both series could be found by using: (13) The average of the gravity values was found by adding the values and dividing the calculated number by two and found to be 9.312 m/s 2 with an uncertainty of 0.500. The uncertainty in the value of average gravity was calculated by using: (14) The percentage difference could be found by using (10) and was found to be 5.08%. The work energy theorem was validated because the measured value of gravity is close to the true value. Table 2: Experimental values and gravity calculations. v 2 = 2m ′ g ( x − x 0 ) m + m ′ v 2 = 2m ′ g(x) m + m ′ − 2m ′ g ( x 0 ) m + m ′ g = ( m + m ′ 2m ′ ) × B δ g g = ( δ B B ) 2 + ( δ m m ) 2 + ( δ m ′ m ′ ) 2 B1 (pos) 1.1308 B2 (neg) 1.7019 g1 7.434 g2 11.189 g(avg) 9.312 %dif 5.080% ϭ g/g 0.500 ϭ B 0.708 B avg 1.416 h (m) ϭ h (m) 0.0005 d(m) 1

a. Friction reduces the total energy of the glider when it moves both upwards and downwards. 3. For the configuration of Investigation 1, draw force diagrams for all the forces on the glider including friction, for both the case of upwards and downwards motion. Away represents when the glider is moving away from the motion sensor and towards represents when the glider is moving towards the motion sensor. While the glider is moving away from the motion sensor the direction of movement would be down the track, when the glider is moving toward the motion sensor the direction of movement would be up the air track. The air flow prevented a high amount of kinetic friction, but the friction could not be fully prevented, and it would act in the opposite direction of movement. 4. For the configuration of Investigation 2, what is the acceleration of the glider if m' →∞ ? a. The acceleration would become closer to gravity as the effect of friction would be reduced. The acceleration is provided by gravity which is independent of mass. 5. For the configuration of Investigation 2, what is the change in potential energy from the moment of release to the moment of collision with the bumper? (Hint: Look at the change in x in the data list. Remember that only the weight is changing potential energy.) Considering the kinetic energy of the system just before it crashes into the bumper, what is the change in total energy of the system? Is the change in energy positive or negative? Explain whether your result makes sense. a. There is a negative change in total energy. This is because friction takes away energy and therefore less kinetic energy than potential energy. Because the energy is not conserved the glider does not get as close to the motion sensor each time it bounces back up.

Acknowledgments References [1] Hyde, Batishchev, and Altunkaynak, Introductory Physics Laboratory, pp 51-57, Macmillan Higher Education, 2022. [2] IPL Lab Report Guide, https://web.northeastern.edu/ipl/wp-content/uploads/2017/09/IPL-Lab-Report- Guide.pdf [3] Giancoli, Physics Principles with Applications, Pearson, 2021.