11.3 Motion on accelerated incline plane(application)

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Hints on solving problems

The incline plane has two contact or interface surfaces. One is the incline surface, where the block is placed and the other is the base of the incline, which is in contact with the surface underneath. The motion of the block, therefore, may depend on the motion of the incline itself.

We must understand here that there are two ways to look at such situation, which involves motion of the incline itself. We can analyze the motion of the elements of "block and incline" system in either the inertial frame (ground reference) or in the accelerated frame. The two techniques have relative merits in particular situations. As pointed out earlier in the course, we should stick to the analysis from inertial frame to the extent possible unless analysis in accelerated frame gives distinct advantage.

In this module, however, our objective is to learn analysis of motion in accelerated frame. As such, we have selected questions which are better suited to analysis in non-inertial frame of reference.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the motion on accelerated incline plane. The questions are categorized in terms of the characterizing features of the subject matter :

• Coefficient of friction in accelerated frame
• Incline on a smooth horizontal surface
• Incline in an accelerated lift

Coefficient of friction in accelerated frame

Problem 1 : A block is placed on a rough incline plane placed on the floor of a lift, which is moving up with acceleration “a”. The incline plane is raised at one end to increase the angle of incline slowly till the block is about to move down. Find angle of repose (this is the angle of incline for which block begins to slide).

Solution : Let the angle of incline, when block is about to move down, is “θ”. To analyze the forces, we select a coordinate system with axes in horizontal and vertical directions. This orientation of axes takes advantage of the fact that acceleration of the lift is in vertical direction.

The context of motion here is peculiar, in which inertial or non-inertial considerations for force analysis do not make any difference. The reason is that angle of repose is obtained by considering motion in horizontal direction only (as we shall see soon). It does not matter whether we analyze forces without pseudo force (inertial ground reference) or with pseudo force (non-inertial lift reference). Note that if we consider non-inertial analysis, then the pseudo force would be in vertically downward direction and as such would not come in consideration for force analysis in horizontal direction.