The box has a mass m and slides down the smooth chute having the shape of a parabola. If it has an initial velocity of v0 at the origin determine its velocity as a function of x. Also, what is the normal force on the box, and the tangential acceleration as a function of x?

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Explanation:

An expression for the velocity can be derived by applying the Principle of Energy Conservation:

Let assume that chute is frictionless, so that box is moved because of gravity. The equations of equilibrium for the box are:

The expression for the radius of curvature and trigonometric functions are, respectively:

Let assume that chute has the following form:

First and second derivatives of the function are, respectively:

Trigonometric functions and radius of curvature are:

The velocity as a function of x is:

The normal force on the box as a function of x is:

The tangential acceleration on the box as a function of x is:

molecules are free to separate from each other and will take on the shape of the container. molecules move completely at random and high speeds. they vibrate in a fixed position and do not move around much. molecules expand to the edge of their container.