Introduction
In this laboratory work, the dependence of the centripetal force on the radius, the mass of the sample, and the angular velocity at which the sample rotates around the vertical axis are studied. The primary purpose of the work is to study the nature of the dependence between the variables by drawing scatter plots.
Analysis
The present laboratory work examined the effect of various independent variables on centripetal force, such as mass, radius, or velocity. The sample was placed on an experimental setup that rotated around a vertical axis — the force value was measured using an automatic force transducer attached by wire to the sample. Table 1 below shows the measured force values (marked in green) in three configurations, each with different values for specimen mass, radius, and angular force.
At first glance, it is evident from Table 1 that there is an increase in the corresponding centripetal force affecting the sample as each independent variable increases. To verify the nature of the relationship between the three pairs of variables, it was necessary to plot the corresponding points on the scatter plots, as shown in Figure 1, Figure 2, and Figure 3. Figure 1 clearly shows that the relationship between the variables is almost linear, as the corresponding coefficient of determination is defined as 0.9982 (Turney, 2022). In other words, there is a consistent linear increase in the force of 121.9 units as the sample mass increases for each. In this case, the slope determines the acceleration (m/s2 ) of the sample, and it turns out that it is constant throughout the circular motion. It is noteworthy that a centripetal force of 0.327 N acts on the sample at the zero-mass point, which is impossible within the framework of this experiment, which shows that the y-intercept has no physical sense.
A linear dependence (R2 = 0.9772) was also determined for the relationship between the centripetal force and the radius at which the sample was located. The data in Fig. 2 show that the further the sample was from the center of rotation (vertical axis), the greater the centripetal force was acting on it. In this case, the physical meaning of tilt can be defined as surface tension (N/m), but this is unlikely given the experimental conditions. In other words, the tilt may have no physical meaning and only determines the change in the centripetal force of the sample as the radius grows. It is noteworthy that when the radius is zero, that is, the sample is on the rotation axis, its centripetal force is equal to the y-intercept, which also does not make physical sense.
The dependence of force on the angular velocity at which the sample moved was also obtained. As in the previous cases, it is noticeable that an increase in angular velocity corresponds to an increase in the centripetal force, and the linear trend is less accurate than in the previous cases. The linearization obtained by squaring the angular velocity values provides a more accurate linear trend. This, in turn, is entirely consistent with the theoretical dependence of the centripetal force on the square of angular velocity. Again, the y-intercept, in this case, did not make sense since the force did not exist for zero rotation speed.
Conclusion
To summarize, the dependencies of centripetal velocity on sample mass, radius, and angular velocity have been studied. As the mass and radius of the sample increased, the centripetal velocity increased linearly; the same was true for the dependence of centripetal force on the square of angular velocity. The accuracy of the linear trend was determined by high values of the coefficient of determination, while this accuracy was affected by uncertainties of direct measurements. The results are entirely consistent with the theoretical model, in which Fc = mw2r.
References
Turney, S. (2022). Coefficient of determination (r²) | calculation & interpretation. Scribbr.