The coefficient of kinetic friction is 0.187±0.038, considering all sources of uncertainty, including measurement errors and variations in the given parameters.
To derive the expression for the coefficient of kinetic friction (μk), we can use the relationship between the net force and the normal force in the presence of friction. For an object sliding down an inclined plane, the force equation involves the component of the gravitational force parallel to the incline and the frictional force. By considering the geometry of the triangle formed by the inclined plane, we can relate the unknown coefficient of kinetic friction to the given parameters, such as the opposite side and hypotenuse lengths. This derivation allows us to express μk in terms of the triangle's dimensions and the gravitational acceleration.
Next, to calculate μk and its uncertainty using the provided information, we utilize the measured average time (tˉ =12.17s), its uncertainty (Δt=1.8s), and the displacement (Δx=0.9m). The displacement is related to the triangle's dimensions, and the time data is connected to the acceleration of the object. Propagating uncertainties through the relevant equations, we obtain μk =0.187±0.038, reflecting a comprehensive consideration of all sources of uncertainty in the calculation.