Download Determining the Coefficient of Kinetic Friction between a Wooden Box and a Glass Plate and more Study notes Physics in PDF only on Docsity!
Measuring the Coefficient of Kinetic Friction
Introduction
When two surfaces in contact move relative to one another in a direction tangent to the surfaces, each surface exerts a shearing force on the other. This force referred to as kinetic friction, although complicated in origin, satisfies a simple mathematical relationship, fk = μkN , (1)
where fk, the kinetic friction force, is exerted tangent to the surface and opposite to its direction of motion. Here, N is the force that one surface exerts normal to the other surface, and μk, the coefficient of kinetic friction, depends only on the intrinsic properties of the two surfaces in contact. [1] The purpose of this experiment is to determine the coefficient of kinetic friction be- tween the bottom surface of a wooden box, lined with felt, and a glass plate positioned on an inclined plane, as shown in Figure 1. Given the block positioned on the plane
Figure 1: Box and Inclined Plane
inclined at angle θ, let M be the amount of mass suspended from the string, such that when the box, initially at rest, is nudged up the incline, it moves with constant velocity. Correspondingly, let m be the amount of mass attached to the string, such that when
the box is nudged down the incline, it moves with constant velocity down the incline. It can be shown that M − m M + m
= μk cot(θ). (2)
Procedure
Insert a 500 gram mass and two 100 gram masses in the box, as shown in Fig. 2.
- Without the mass holder attached, determine the angle of elevation of the plane such that when the box is nudged down the incline, it moves with constant ve- locity (See Appendix A.). Since there is no mass attached to the string, this corresponds to m = 0 described in the introduction. Attach the mass holder to the string, and add a sufficient amount of mass so that when the box is nudged up the incline, it moves with constant velocity, as described in the introduction.^1 Record these data in Table 1 as Case one.
- Raise the plane approximately 5 ◦^ above the angle obtained in step one, and deter- mine M and m, as described in the introduction. Report these values in Table 1, as Case two.
- Do the same for three additional, approximately, 5 ◦^ increments, and report the values in Table 1 accordingly. Do not exceed an angle of elevation of 45 ◦.
- Plot a graph of cot(θ) vs. MM^ −+mm , where the former is the abscissa and the latter is the ordinate. Note: since case one corresponds to the situation when m = 0, M −m M +m evaluates to one independent of the value of^ M^.
- Perform a linear regression of the data, forcing the intercept to be zero. Report the values of the estimated regression parameters and the value of R^2.
- Calculate the standard error, δM , of the slope using equation 7.
- According to Eq. 2, μk is equal to the slope of the regression line. In Table 2 report μk, i.e. report the experimental value of the slope in the form Mest ± δM , where Mest is the slope of the regression line.
A Appendix
In performing the experiment you are instructed to nudge the box so that it moves with constant velocity. In order to obtain acceptable experimental results, this proce- dure should be performed in a consistent way. The following is a suggestive, but not necessarily prescriptive, way of satisfying this condition:
(^1) When adding mass to the mass holder, do so in increments of 20 grams or larger throughout the experi- ment.
and the standard error of the intercept δB is
δB = σy
1 − r^2 N − 2
N − 1
N
x¯^2 σ x^2
where r is the estimated correlation coefficient between the yi and xi, σy and σx are the estimated standard deviations of the yi and xi, and x¯ is the estimated average of the xi.^2 The quantities x¯, σx, and σy can straightforwardly be calculated using function keys on a scientific calculator or defined functions in Excel. If, in the regression equation, Eq. 3, the quantity B, the y-intercept, is required to be zero, i.e. the regression equation is now yi = M xi + i , (6)
then the degrees of freedom in the random errors are N − 1 rather than N − 2 so that
δM =
1 − r^2 N − 1
σy σx
To reflect the statistical uncertainty in a quantity Q, where Q is either the slope M or y-intercept B, the quantity Q is typically reported as
Qest ± δQ , (8)
which can be understood informally to mean that, assuming the experimental results are consistent with theory, then the value of Q, predicted by theory, is likely to lie within the limits defined by Equation 8.
References
[1] Wikipedia. Friction. http://en.wikipedia.org/wiki/Friction,
- [Online; accessed 21-March-2012].
(^2) Note: in Equation 5 σx and σy are unbiased estimates.
Case θ◦^ M (gms) m (gms) cot(θ) MM^ −+mm 1 2 3 4 5
Table 1: Data and Computations
Table 2: The experimental value of the coefficient of kinetic friction, μk
