Experimental Study of Gravitational Field Strength with A Rotary Motion Sensor Mechanics Laboratory

Experimental Study of Gravitational Field Strength with A Rotary Motion Sensor Mechanics Laboratory

H. Nery

Physics Department, Presidential School in Namangan, Islom Karimov Street, Namangan City, Namangan, Uzbekistan

Abstract. This experiment aims to determine the value of g using the torque of a rotating system. Data was collected using a Vernier Rotary Motion sensor and Vernier Graphical Analysis software. Analysis of the data was carried out by linearising the relationship between angular acceleration and the distance between two rotating identical masses, and the value of g was determined.

Keywords: angular acceleration, rotating system, gravitational field strength

1. INTRODUCTION

   Gravitational field strength, g, on the Earths surface is a fundamental constant used in many calculations in physics. There are multitudes of methods to experimentally determine the value of g, such as using a simple pendulum [1], free fall motion [2] to name a few. Its value is universally accepted to be 9.81 m s-2, but its precise value does vary slightly depending on several experimental conditions and location.

   Rotational motion is one of the basic motions any system in the universe can be in, from the movement of galaxies to even the motion of subatomic particles in the atom [3]. We study the objects moving in straight lines in linear motions, while two- dimensional motions are established by projectile motion. Objects or systems involving rotation are much more complex because we have to consider which forces cause the turning effect of such systems.

   In this experiment, the idea of a falling object attached to a pulley accelerating downwards due to gravity was utilised. A rotary sensor records the angular motion of the pulley, allowing for an investigation into the relationship between the systems rotational and linear motion. The method is adapted from an experiment on rotational motion published by Purdue University [4].

   With this method timing-based error from human reaction time is reduced and provides continuous data collection for several sets of measurements.

   THEORETICAL BACKGROUND

   The motion of a falling object attached to a pulley system can be described by Newtonian mechanics principles. When a hanging mass is attached to a string wound around a disk pulley, the system experiences both linear and rotational motion [4].

   As the mass is released from a certain height from the ground, the disk pulley experiences a torque T given in Eq. 1.

   where m = mass of the hanging mass g = gravitational field strength r = radius of the disk pulley

   This torque causes the angular acceleration of the rotating system given by equation 2.

   where IT is the total moment of inertia of the rotating system. The rotating system consists of identical masses M separated by distance 2D from their centres connected by a rigid rod and the disk pulley. D is taken as the distance between the centre of one of the masses to the centre of the disk pulley.

   We can then express IT as

   Substituting equations 3 and 1 to equation 2 we get the relationship between the angular acceleration and the D as follows.

2. METHODOLOGY

   This experiment uses a Vernier Rotary Motion Sensor attached to a rotating system consisting of a disk pulley, a light string wound around the disk pulley and onto a second pulley, and a mass hanger attached to the free end of the string to apply a torque [5]. Figure 1 shows the experimental setup used.

   Figure 1 Experimental setup

   The rotary sensor is connected via Bluetooth to Vernier Graphical Analysis software for data collection of angular acceleration.

   Measurements of constants are taken prior to mounting the rotating system to the sensor. Measurements for masses were taken in a balance with a precision of 0.1 g, and the inner radius of the disk pulley using a vernier caliper of precision 0.01 mm.

   During each run, the string was wound until the hanging mass was at a fixed initial height, then the collection was started in the software, and the mass was released. The gradient of the angular velocity time graph generated by the software was taken for each set. This represents the value of angular acceleration, . The experiment was repeated until six readings of D and were taken.

3. RESULT & DISCUSSION

   Measurements of the variables are recorded as follows. Table 1 shows the measurements of the variables that have been kept constant throughout the experiment.

   Table 1

   Measurements of variables kept constant

   mass of identical masses, M	(80.5 0.1) g
   mass of hanging mass, m	(17.0 0.1) g
   inner radius of disk pulley, r	(25.00 0.01) mm

   To find the experimental value of g, equation 4 has been linearised and expressions for the gradient and subsequently for the constant g have been found as follows.

   From equation 5, we see that a linearised equation requires values of D2 and 1/. Table 2 shows the six sets of readings of D and , and values for D2 and 1/ have been included. Figure 2 is the graphical representation of the linear relationship of D2 and 1/. The black line represents the best-fit line, while the blue line is the worst-acceptable line of the graph.

   Table 2

   Measurements of angular acceleration as distance D varies

   D / cm	/ rad s-2	D2 / cm2	1	2	-1
   16.8 0.1	1.018	282 3		0.9823
   12.0 0.1	1.355	144 2		0.7380
   11.0 0.1	1.518	121 2		0.6588
   8.0 0.1	2.107	64 2		0.4746
   5.0 0.1	3.465	25 1		0.2886
   3.0 0.1	4.822	9.0 0.6		0.2074

   We can estimate the absolute uncertainty of the experiment by considering the absolute uncertainties of all measurements and the gradient. Error bars have been plotted for values of D2 to plot the worst acceptable line of the raph. To estimate the uncertainty of the gradient, a worst-acceptable line is drawn, and the gradient of the line is determined to be 0.00383. To find the uncertainty of the gradient, the gradient of the worst acceptable line is subtracted from the gradient of the best fit line, as shown below. Propagating the uncertainties of all other measurements and the gradients, we get the absolute uncertainty of g.

   absolute uncertaintygradient = 0.00390 – 0.00383

4. CONCLUSION

The experiment suggested that equation 4 is valid and successfully determined a value for the gravitational field strength constant g using a rotary motion sensor. The experimental value for g was quite close to the accepted value of 9.81 m s-2. The small uncertainty observed from the experiment may be attributed to measurement errors in finding D as there are quite a few uneven surfaces in the rotating system (i.e. screws of the masses and the ridges of the disk pulley). The pulleys may also have friction along the bearings and the sides. Air resistance, while theoretically very small, may also account for the error.

While there are still some drawbacks, it is undeniable that with the utilisation of the sensor, random and systematic errors have been significantly reduced in this experiment.

REFERENCES

1. Wahab D.S. et al., Experimental Study of Gravity Measurement with a Video-Based Laboratory Pendulum with Tracker Software: Comparison of Weighted and Unweighted Tests Journal of Novel Engineering Science and Technology, 4(01), 1-11. https://doi.org/10.56741/jnest.v4i01.644
2. H. Young & R. Freedman. University Physics with Modern Physics, Pearson Education Inc., United States of America, 2016 (p. 273)
3. Walid, I. H. B. I., & Umar, M. F. B. (2022). Development of a Free Fall Motion Experiment based on Smart Phone using Phyphox application. Journal of Physics Conference Series, 2309(1), 012085. https://doi.org/10.1088/1742-6596/2309/1/012085
4. PASCO Scientific. (1996). Experiment M7 Purdue University – Physics 22000. In Rotational Motion (pp. 316). https://www.physics.purdue.edu/academic-programs/lab-materials/Physics%20220%20lab%20files/m7—theory-and-procedure.pdf
5. Vernier Rotary Motion Sensor. (n.d.). https://www.cvkk.com.ua/files/rmv-btd.pdf

DOCUMENTATIONS

Figure 3 Experimental setup (close-up)

Figure 4 Sample angular velocity time graph for one value of

(Vernier Graphical Analysis software)