Physics Lab Report 4

Experiment #4

“Force Evaluation”

ENGR 216 Section 503

Lab Team No.5

Ayman Karaki

Zaina Aloudeh

Kholoud Al-Dosari

Mariam Hassan

Turn-In Date: 03-08-2020

Instructor: Primal Vivin Pinto

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Objective The purpose of this experiment was to evaluate the coefficient of both static friction "𝜇𝜇𝑠𝑠" and kinetic friction "𝜇𝜇𝑘𝑘" using 4 different surfaces made of different materials, wood or rubber, and with different surface areas sliding on an inclined wooden plane. This was done after calibrating the equipment used to convert from Pixels to centimeters.

Theory At the beginning, the equipment was calibrated to convert from Pixels to SI-units. Then,

expressions for both coefficients "𝜇𝜇𝑠𝑠" and "𝜇𝜇𝑘𝑘" were derived using Newton’s second law and Work-Energy Theorem. Also, a movable wooden piece was used to create an inclined plane in order to find "𝜇𝜇𝑠𝑠". While, static friction is a force that prevent the body from moving when its fixed, kinetic friction occurs when the body is moving, whether there is acceleration or at fixed speed, and it is always in the opposite direction of movement. Moreover, its important to mention that surface area does not play any role in determining friction force, as the friction force only depends on "𝜇𝜇 " which is a property of the surface and "𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝐹𝐹𝑁𝑁𝑁𝑁𝐹𝐹𝐹𝐹" that depends on both the weight and inclination angle. Hence, changing the surface area of an object will not affect the static or kinetic coefficients of friction. Finally, to evaluate the accuracy and precision of this procedure, percentage of difference and mean were calculated, and the following equations were derived and used:

• Converting from (Pixel) to (centimeter): 1 cm = 12.2484 pxl • Converting from (centimeter) to (meter): 1 cm = 0.01 m

• Standard deviation (STD): 𝑆𝑆𝑆𝑆𝑆𝑆 = � � (𝑥𝑥𝑖𝑖−�̅�𝑥)2

𝑛𝑛 𝑖𝑖=1 𝑛𝑛−1

• Percentage of difference: % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑛𝑛𝐷𝐷𝐷𝐷 𝑏𝑏𝐷𝐷𝑏𝑏𝑏𝑏𝐷𝐷𝐷𝐷𝑛𝑛 𝐷𝐷𝐷𝐷𝑠𝑠𝑟𝑟𝑟𝑟𝑏𝑏𝑠𝑠 𝐴𝐴𝐴𝐴𝐷𝐷𝐷𝐷𝐴𝐴𝐴𝐴𝐷𝐷 𝐷𝐷𝐷𝐷𝑠𝑠𝑟𝑟𝑟𝑟𝑏𝑏

× 100% • Friction force: 𝑑𝑑 = 𝜇𝜇 × 𝑁𝑁 • Calculating acceleration: 𝑁𝑁 = 𝑑𝑑

2𝐷𝐷 𝑑𝑑𝑏𝑏2

• Calculating coefficients of friction: 1- Using Newton’s second law:

∑𝐹𝐹𝑦𝑦 = 0  𝑁𝑁 = 𝑁𝑁𝑚𝑚𝐹𝐹𝑁𝑁𝑚𝑚𝑚𝑚 a- Static friction: ∑𝐹𝐹𝑥𝑥 = 0  𝑁𝑁𝑚𝑚𝑚𝑚𝑑𝑑𝑠𝑠 𝑚𝑚 = 𝜇𝜇𝑠𝑠𝑁𝑁𝑚𝑚 𝐹𝐹𝑁𝑁𝑚𝑚 𝑚𝑚

𝜇𝜇𝑠𝑠 = 𝑡𝑡𝑁𝑁𝑠𝑠 𝑚𝑚

b- Kinetic friction:

∑𝐹𝐹 = 𝑁𝑁𝑁𝑁  𝑁𝑁𝑚𝑚 sin 𝑚𝑚 − 𝜇𝜇𝑘𝑘𝑁𝑁𝑚𝑚 cos 𝑚𝑚 = 𝑁𝑁𝑁𝑁

𝜇𝜇𝑘𝑘 = 𝑡𝑡𝑁𝑁𝑠𝑠 𝑚𝑚 − 𝑁𝑁

𝑚𝑚 𝐹𝐹𝑁𝑁𝑚𝑚 𝑚𝑚

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2- Using work energy theorem (only "𝜇𝜇𝑘𝑘" can be calculated):

𝑤𝑤𝑏𝑏𝑡𝑡𝑏𝑏𝐴𝐴𝑟𝑟 = 𝑘𝑘.𝐸𝐸2 − 𝑘𝑘.𝐸𝐸1  𝑊𝑊𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝐷𝐷𝑏𝑏𝑦𝑦 + 𝑊𝑊𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑏𝑏𝐷𝐷𝑡𝑡𝑛𝑛 = 1 2 𝑁𝑁(𝑣𝑣𝐷𝐷2 − 𝑣𝑣𝐷𝐷2)

 1 2 𝑁𝑁(𝑣𝑣𝐷𝐷2 − 𝑣𝑣𝐷𝐷2) = −𝜇𝜇𝑘𝑘𝑁𝑁𝑚𝑚𝑑𝑑 𝐹𝐹𝑁𝑁𝑚𝑚 𝑚𝑚 + 𝑁𝑁𝑚𝑚𝑑𝑑 𝑚𝑚𝑑𝑑𝑠𝑠 𝑚𝑚  𝜇𝜇𝑘𝑘 = tan(𝑚𝑚) −

(𝐴𝐴𝑓𝑓 2−𝐴𝐴𝑖𝑖

2)

2𝐴𝐴𝑑𝑑 𝐷𝐷𝑡𝑡𝑠𝑠(𝜃𝜃)

Methods Equipment:

• Air hockey table. • Pink tracking stickers. • Wooden cuboid. • Black rubber fabric. • Inclined plane (changeable angle). • Meter scale ruler. • Wooden cube.

• Angle indicator. • MobaXterm App. • Windows operating system. • Linux operating system. • Web camera. • Python script.

Procedure:

At the beginning, two surfaces, with different areas, of the wooden cuboid was covered with black rubber fabric creating 4 different surfaces on the cuboid (small wooden, large wooden, small rubber and large rubber). Also, each surface was provided with a pink sticker to be able to track it using the web cam. Then, an inclined plane was adjusted using two flat wooden pieces connected using an iron hinge to create an angle between the two pieces. After that, a connection between Windows and Linux operating systems was made using mobaXterm to use the web camera in recording the object.

As the cuboid was put on the inclined plane, the camera started recording and tracking the pink sticker by running a Python script. The angle of inclination was increased gradually and when the cuboid starts sliding down (overcomes the static friction) the angle was maintained fixed by supporting the inclined plane with a wooden cube. At the same time, the python script was recording and saving position and time values in an Excel file. Then, the angle was measured 7 times using angle indicator app, and the perpendicular height between the camera and the inclined plane was measured using a metric stick. Therefore, using the conversion equation (from 2nd lab), a calibration factor was calculated to convert from Pixels to Centimeters. Later, position versus time values were plotted, after converting Excel data to proper units, to calculate the acceleration along the plane. The process was done once for each surface of the cuboid. Then, values for both coefficients of static and kinetic frictions were calculated using derived formulas.

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Results While Tables (1 – 4) indicate the measured inclination angles in both degrees and radians,

graphs (1 – 4) illustrate corresponding position versus time plots for each surface. Moreover, all the tables include the mean, standard deviation and percentage of difference. Also, Table (5) shows calculated "𝜇𝜇𝑠𝑠" and"𝜇𝜇𝑘𝑘" for each trial. All the measurements were rounded to 4 significant figures.

Table (1): measured angles for small wooden surface

Trial Angle (deg) Angle (rad) 1 14.31 0.2497 2 14.03 0.2449 3 13.83 0.2414 4 13.65 0.2382 5 13.51 0.2358 6 13.29 0.2319 7 13.19 0.2302

Mean 13.69 0.2389 STD 0.4007 0.006993

% diff 6.137 6.137

Table (2): measured angles for large wooden surface

Trial angel (deg) angle (rad) 1 15.34 0.2677 2 15.35 0.2679 3 15.02 0.2621 4 14.83 0.2588 5 14.60 0.2548 6 14.31 0.249 7 13.93 0.2431

Mean 14.77 0.2577 STD 0.5281 0.009217

% diff 9.615 9.615

Table (5): calculated values small wooden surface

"𝜇𝜇𝑠𝑠" acceleration (𝑁𝑁/𝑚𝑚2) "𝜇𝜇𝑘𝑘" 0.2435 0.8614 0.1531

large wooden surface "𝜇𝜇𝑠𝑠" a (𝑁𝑁/𝑚𝑚2) "𝜇𝜇𝑘𝑘"

0.2636 0.9893 0.1592 small rubber surface

"𝜇𝜇𝑠𝑠" a (𝑁𝑁/𝑚𝑚2) "𝜇𝜇𝑘𝑘" 0.5088 0.4528 0.4570

large rubber surface "𝜇𝜇𝑠𝑠" a (𝑁𝑁/𝑚𝑚2) "𝜇𝜇𝑘𝑘"

0.5359 0.5832 0.4684

y = 0.8614x2 - 1.898x + 1.0156 R² = 0.9982

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

1.2 1.4 1.6 1.8

Po st

isi on

(m )

Time (s)

Graph (1): position versus time for small wooden surface

y = 0.9893x2 - 2.0107x + 1.0101 R² = 0.998

0

0.1

0.2

0.3

1 1.2 1.4 1.6

Po sit

io n

(m )

Time (s)

Graph (2): position versus time for large wooden surface

y = 0.5832x2 - 1.7447x + 1.2785 R² = 0.9977

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4

1.6 1.8 2 2.2 2.4

Po sit

io n

(m )

Time (s)

Graph (4): position versus time for large rubber surface

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Discussion First, the calibration equation obtained from 2nd lab (𝑦𝑦 = 11451𝑥𝑥 − 1.9831) was used

after measuring the perpendicular height between the camera and the inclined plane (𝐻𝐻 = 62.5 𝐹𝐹𝑁𝑁) and by substituting (𝑥𝑥 = 1/62.5) and dividing by the measured length (10 cm), the calibration factor was found to be (1 𝐹𝐹𝑁𝑁 = 18.12329 𝑝𝑝𝑥𝑥𝑁𝑁). Also, the angle of inclination was measured 7 different times and the Mean, Standard Deviation and the percentage of difference were calculated for each trial. The percentage of difference measuring the angles (in degrees) had a range of (3.412% - 9.615%). After that, Python script was used to track the cuboid’s position versus time while sliding down the plane. The experiment was conducted several times until a motion along a straight line was obtained (1-D acceleration) to get more accurate results. Then, the acceleration along the plane can be calculated by differentiating the equation of motion twice after plotting position in (meters) and time in (seconds). It was found that the coefficient of static friction is always greater than the kinetic coefficient of friction. Also, it can be concluded that the coefficient of friction does not depend on the surface area of object (If the surface area increases, the pressure decreases resulting the same friction force) and it’s a property of the surface’s material. Hence, the friction force depends only on the normal force (weight of object and inclination angle) and the material of the surface. Meanwhile, it can be observed from the results that both wooden surfaces have almost the same coefficients of friction, and the same thing applies for the rubber surfaces.

During the experiment, several sources of error affected the accuracy of the obtained results. Some random errors were due to human error (reading and/or taking measurements, using equipment and the vibrations while inclining the plane) and environmental conditions, for example, Humidity (affecting friction). On the other hand, many systematic errors happened such as, limitations of used equipment (metric stick in measuring height, camera frames’ capturing limitation, measuring different angle, etc.). Also, the tracking sticker was not exactly placed in the center of the surface, which led the cuboid to slip while sliding (not moving in 1-D). Moreover, the height between the camera and the inclined plane was not fixed along the plane which affects the accuracy calculating the calibration factor and the obtained results (error in detecting position). While doing the experiment, a block was used to push the inclined plane (slower and more accurate changes in the angle and to support the plane without changing the angle) and several trials were done until a straight motion without slipping was obtained for each surface, in order to overcome and minimize the errors. Thus, although there are variety of errors, the results are acceptable compared to the accurate values of friction for (wood on wood) and (wood on rubber).

Team members contributions:

Ayman: wrote (Discussion).

Zaina: wrote (Method).

Mariam: wrote (Objective and Theory).

Kholoud: wrote (Results).