A wooden wedge of mass M and inclination angle αrests on a smooth floor. A block of mass m is ke

step 1 Hi friends, here it is given. A wedge is kept over the smooth horizontal surface, having a mass

A wooden wedge of mass M and inclination angle αrests on a...

A wooden wedge of mass $M$ and inclination angle $\alpha$ rests on a smooth floor. A block of mass $m$ is kept on wedge. A force $P$ is applied on the wedge as shown in figure, such that a block remains stationary with respeet to wedge. The magnitude of force $\mathrm{P}$ is (a) $(M+m) g \tan \alpha$ (b) $g \tan \alpha$ (c) $m g \cos \alpha$ (d) $(M+m) g \operatorname{cosec} \alpha$

A wooden wedge of mass $M$ and inclination angle $\alpha$ rests on a smooth floor. A block of mass $m$ is kept on wedge. A force $P$ is applied on the wedge as shown in figure, such that a block remains stationary with respeet to wedge. The magnitude of force $\mathrm{P}$ is
(a) $(M+m) g \tan \alpha$
(b) $g \tan \alpha$
(c) $m g \cos \alpha$
(d) $(M+m) g \operatorname{cosec} \alpha$

Key Concepts

Inclined Plane Dynamics

This concept deals with the analysis of objects on inclined planes, where forces such as gravity are resolved into components parallel and perpendicular to the plane. It is essential in determining how forces such as friction, normal force, and applied forces interact to produce motion or equilibrium on an incline.

Non-inertial Reference Frame and Pseudo Forces

When a system is accelerating, such as a wedge being pushed on a frictionless surface, the frame attached to the system becomes non-inertial. In such frames, pseudo forces (also known as fictitious forces) must be introduced to account for the acceleration when applying Newton's laws, ensuring that the analysis of equilibrium or motion is correct relative to the accelerating frame.

Free-Body Diagrams

Free-body diagrams are graphical representations that depict all the forces acting on an object. They are crucial in problems like these for breaking down the gravitational force, applied force, normal force, and any pseudo forces into components, thereby allowing the application of Newton's laws to solve for unknown quantities.

Force Components and Trigonometry

Solving for forces on inclined planes often requires the use of trigonometric relationships to decompose forces into components along directions parallel and perpendicular to the plane. This technique enables the establishment of equilibrium conditions or net force equations in each direction, which is essential for determining applied forces or resulting accelerations.

Newton's Second Law

Newton’s Second Law, which states that the net force acting on an object is equal to the mass of the object times its acceleration (F=ma), serves as the foundation for analyzing the dynamics of the system. It is applied in both inertial and non-inertial frames (with pseudo forces included) to set up the equations needed to solve for the parameters such as the magnitude of an applied force needed to maintain equilibrium.

Please give Ace some feedback