Jon Wallem Anundsen, 18 år, Kristiansand
Skole: Kristiansand Katedralskole, Gimle
Developing a theoretical model for the sliding motion of a frictionless object on a curve
The aim of this project was to develop a general theoretical model of the motion of a frictionless object sliding on a curved surface, where the shape of the surface is defined by a function of horizontal position.
Two approaches were used in attempting to find a model for the motion of the object. In the first approach, it was assumed that the instantaneous acceleration of the object at a point on the curve would be the same as the acceleration of an object placed on an inclined plane with the same slope as the curve at that point. From this assumption, horizontal acceleration was found as a function of horizontal position, and was used to find a function for the horizontal speed of the object.
The second approach was based on the assumption that the total mechanical energy of the object would be constant, and that any loss of gravitational potential energy would therefore cause an equivalent gain of linear kinetic energy, and vice versa. By considering the change in the vertical position of the object, its linear speed was found as a function of its horizontal position, and this function was modified to give the horizontal component of the speed. When the two functions for the horizontal speed of the object were compared, it was found that they produced different results. Therefore, it was concluded that the first approach is invalid, which indicates that the forces acting on an object sliding on a curved surface are not as simple as those acting on an object on an inclined plane.
The reciprocal of the object’s horizontal speed (as found from the second approach) was integrated with respect to horizontal position to find an expression for the time it takes for the object to slide to a given position. This expression was used to calculate the motion of an object sliding on the graph of the function f(x)=-ln|cos(x)|. Because the integral was difficult to evaluate analytically, a numerical approximation was used, and the results were visualised by graphing the horizontal and vertical positions of the object against time. Additionally, an expression for the horizontal acceleration of the object at any given point on the curve was found based on the function for horizontal speed found in the second approach, and this expression was compared with the one found in the first approach. The differences between the two expressions indicated that the normal force acting on an object on a curved surface depends on other factors than just the slope of the curve, and might be affected by the velocity of the object. For further study, it would be interesting to find out exactly what determines the normal force, as this knowledge might be useful in developing a more realistic model that accounts for the effects of friction.