Block A of mass m and block B of mass 2 m are placed on a...

Block $A$ of mass $m$ and block $B$ of mass $2 m$ are placed on a fixed triangular wedge by means of massless, inextensible string and a frictionless pulley as shown. The wedge is inclined at $45^{\circ}$ to horizontal on both sides. The coefficient of friction between block $A$ and wedge is $2 / 3$ and that between block $B$ and wedge is $1 / 3$. If system of $A$ and $B$ is released from rest then acceleration of $A$ is (Figure can't copy) (a) zero (b) $1 \mathrm{~ms}^{-2}$ (c) $2 \mathrm{~ms}^{-2}$ (d) $3 \mathrm{~ms}^{-2}$

Block $A$ of mass $m$ and block $B$ of mass $2 m$ are placed on a fixed triangular wedge by means of massless, inextensible string and a frictionless pulley as shown. The wedge is inclined at $45^{\circ}$ to horizontal on both sides. The coefficient of friction between block $A$ and wedge is $2 / 3$ and that between block $B$ and wedge is $1 / 3$. If system of $A$ and $B$ is released from rest then acceleration of $A$ is
(Figure can't copy)
(a) zero
(b) $1 \mathrm{~ms}^{-2}$
(c) $2 \mathrm{~ms}^{-2}$
(d) $3 \mathrm{~ms}^{-2}$

Key Concepts

Tension in a String-Pulley System

In a system where objects are connected by a string over a pulley, the tension is the force transmitted along the string. For massless, inextensible strings and frictionless pulleys, this tension is uniform along the string. This uniformity permits the linkage of the motion equations for both masses in the system, allowing for the determination of their acceleration.

Friction on Inclined Planes

Friction is the resistive force that acts parallel to the contact surface and opposes motion. On inclined planes, the frictional force is determined by the coefficient of friction and the normal force, which is the component of the gravitational force perpendicular to the surface. This concept is key when calculating net forces and subsequent accelerations.

Coefficient of Friction

The coefficient of friction is a dimensionless value that quantifies the interaction between two surfaces. It determines the magnitude of the frictional force when multiplied by the normal force. Different coefficients for different surfaces directly affect the motion and acceleration of objects on inclined planes.

Free-Body Diagrams

Free-body diagrams are graphical representations that depict all the forces acting on an object. By isolating each block and illustrating forces such as gravity, friction, normal force, and tension, it becomes easier to set up the equations needed to analyze the dynamics of the system on inclined planes.

Newton's Second Law

This principle states that the net force acting on an object is equal to the product of its mass and its acceleration. In systems like blocks on inclined planes, applying this law helps to establish equations of motion along the plane, which are essential for determining the acceleration of connected masses.

Inclined Plane Dynamics

Dynamics on an inclined plane involves resolving the gravitational force into components parallel and perpendicular to the plane. This resolution is crucial for understanding how gravitational forces contribute to an object's motion along the incline and how they are counteracted by friction and normal forces.

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