Research Question:
How does the acceleration of an object depend on the mass of it?
Variables & Controls of the ExperimentVariables
Constant: net force (mass of the hanger) Controls
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MethodsTo Control the Variables:
First, set up a smooth, flat track for the cart to travel on. Use the same track to ensure that the movement of the cart is not impeded by extraneous objects/ surfaces that may significantly affect its motion, such as different textured surfaces that may increase the friction on the wheels and slow the cart down on the track. Also, use the same cart as the movement of different carts differ. Next, select a set of weights that can be taken off of the cart and placed aside to change the mass of the system (the independent variable). In addition, release the cart by letting it go without giving it an initial push or pull so that the initial velocity of the cart is 0 m/s, and its acceleration is majorly dependent on the tension force from the hanger. To Collect the Data:
Mark down exactly how much mass was on the cart to calculate the mass of the system as the mass of the cart changes due to different weights being placed on it. Place a motion detector at the beginning of the track, and align it with the path the cart will take to properly determine its position according to time. After releasing the cart and collecting its position with the motion detector, place the data into Logger Pro and use the data to create a position-time graph. Use the position-time data to create a velocity time graph, then use the slope of the proportional line of best fit of the velocity time graph to get the acceleration, which is our dependent variable. |
Procedure
We used the same cart and the same hanger, which had masses of 495 g and 50 g, respectively. Then, we put the cart onto the magnetic area of the ramp so it does not move as we repeatedly moved weights off of the cart and placed and recorded these changes in the separate weights. We placed the hanger onto the string that was attached to the cart and pulley system. Afterward, we released the cart 100 cm from the edge that is closer to the hanger. We used Logger Pro to obtain the acceleration (refer to "Methods", "To Collect the Data"). We did a total of 19 trials with different masses of the system.
Raw Data - Table
Mass of Cart: 0.495 kg
Mass of Hanger: 0.05 kg The measurement for acceleration was generated by the Logger Pro application, in which position was measured by the motion detector, converted into a position time graph, was used to create a velocity time graph, and had a line of best fit with the slope as acceleration. |
Processed Raw Data - Table
Total Mass of System (kg) = mass of Cart (0.495) + [1 kg weight] + mass of hanger (0.05 kg) + [mass of moved weights (0.05 kg + 0.05 kg + 0.02 kg + 0.01 kg)] = 1.675 kg
[ ] indicates that these masses were moved off of the cart to chang the total mass of the system Net Force (N) = mass on hanger (0.05 kg)* gravitational field strength/ constant (9.8 N/kg) = 0.49 N Acceleration (m/s^2) = obtained from Logger Pro (see Raw Data) All of these calculations were made with Google Sheets by typing out the equation used to calculate them and dragging the equation down to automatically be applied to other rows.
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Processed Raw Data - Graphical Representation
The image pictured on the left is a graph generated by the Logger Pro application. The independent variable (x-value) is Total Mass of the System (kg) and the dependent variable (y-value) is Acceleration (m/s^2). We can see that the graph is concave upward with a decreasing acceleration as the total mass of the system increases.
A nonlinear, inverse model (y = A/x) was the best fit for this graph as it made better predictions. We did not choose a power model (y = Ax^b) because the mathematical calculations were increasingly complex due to the b value being less than 1. The A value of this graph was 0.4441, and the equation itself was y = 0.4441/x, which can also be written as y=0.4441x^-1. There is no x-intercept because as the total mass nears 0 kg, the acceleration will get faster and faster but a system with a total mass of exactly 0 kg has 0 m/s^2 of acceleration. There is also no y-intercept because there is 0 m/s^2 acceleration when total mass is 0 kg. |