Question

Classical Mechanics:

A block of wood is projected up an inclined plane with initial speed V₀. If the inclination of the plane is θ, and the coefficient of the sliding friction is μ_k,

a) Draw a force diagram and a free body diagram.
b) Choose the x and y coordinate system, and mark it on the free body diagram.
c) Apply Newton's Second Law in the y-direction.
d) Apply Newton's Second Law in the x-direction.
e) Use parts c) and d) to find the time it takes to reach the maximum height on the incline (use integrations, not kinematic equations)
f) How would you calculate the total time for the block to return to the point of projection?

Note: you don't have to solve for time, just provide the steps in point form and argue about the facts (i.e. direction of forces, boundary conditions, etc).

Answer 1

The force diagram will indicate the friction force acting downwards along the slope,

the component mgsin also acting downwards along the slope and the normal force R = mgcos acting perpendicular to the direction of the slope.

Let the xy axis be chosen such that x axis is oriented along the direction of the slope (upwards) and y axis is oriented perpendicular to the direction of the slope.

Along x direction:

along y direction:

Using the expression for acceleration,

taking t_o = 0,

at the maximum height, dx/dt = v = 0

therefore,

this is the time it takes to reach the maximum height.

When the block is traveling downwards, the acceleration will be:

use this to obtain t' which is the time it takes from the highest point to reach the bottom (x = 0). The total time will be: T = t+t'.