A block of mass m is at rest with respect to a rough incline...

A block of mass $m$ is at rest with respect to a rough incline kept in clevator moving up with acceleration $a$ (Fig. 7.364). Which of following statement is correct? a. The contact force between block and incline is parallel to the incline. b. The contact force between block and incline is of magnitude $m(g+a)$ c. The contact force between block and incline is perpendicular to the incline. d. The contact force is of magnitude $m g \cos \theta$.

A block of mass $m$ is at rest with respect to a rough incline kept in clevator moving up with acceleration $a$ (Fig. 7.364). Which of following statement is correct?
a. The contact force between block and incline is parallel to the incline.
b. The contact force between block and incline is of magnitude $m(g+a)$
c. The contact force between block and incline is perpendicular to the incline.
d. The contact force is of magnitude $m g \cos \theta$.

Key Concepts

Non-Inertial Reference Frame

When analyzing problems in accelerating systems such as an elevator or clevator, it is useful to adopt a non?inertial reference frame. In this frame, inertial or fictitious forces must be introduced to account for the acceleration of the system. This allows one to apply Newton’s laws of motion as if the system were at rest, simplifying the analysis while ensuring that the apparent forces acting on the object are accurately represented.

Effective Gravity

In a non?inertial frame, such as an accelerating elevator, the actual gravitational acceleration is modified by the elevator’s acceleration. The effective gravitational field becomes the vector sum of the real gravitational acceleration and the acceleration due to the elevator’s motion. In the case of upward acceleration, the effective weight of an object increases to m(g + a), and this effective weight is used in resolving forces acting on the object.

Force Resolution on Inclined Planes

When dealing with an object on an inclined surface, forces must be resolved into components parallel and perpendicular to the plane. The effective weight m(g + a) is resolved according to the incline angle, resulting in a perpendicular component that influences the normal force and a parallel component that may be balanced by friction. This standard method of resolving forces is essential for determining equilibrium conditions on an inclined plane.

Resultant Contact Force

The contact force exerted by a surface on a body in contact is the vector sum of the normal force (acting perpendicular to the surface) and the frictional force (acting parallel to the surface). In equilibrium, this resultant force counteracts the effective weight of the object. Its magnitude can be determined by combining the perpendicular and parallel components, and in many cases—especially when they match the components of m(g + a) resolved on the plane—the magnitude of the resultant contact force turns out to be m(g + a).

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