∙∙A block with mass m1 is placed on an inclined plane with slope angle αand is connected to a se

step 1 All right, so not a free body diagram, but here is the situation that we're presented with. You

∙∙A block with mass m1 is placed on an inclined plane with...

$\bullet$$\bullet$ A block with mass $m_{1}$ is placed on an inclined plane with slope angle $\alpha$ and is connected to a second hanging block with mass $m_{2}$ by a cord passing over a small, frictionless pulley (Fig. 5.68 ). The coefficient of static friction is $\mu_{3}$ and the coef- ficient of kinetic friction is $\mu_{\mathrm{k}}$ (a) Find the mass $m_{2}$ for which block $m_{1}$ moves up the plane at constant speed once it is set in motion. (b) Find the mass $m_{2}$ for which block $m_{1}$ moves down the plane at constant speed once it is set in motion. (c) For what range of values of $m_{2}$ will the blocks remain at rest if they are released from rest?

$\bullet$$\bullet$ A block with mass $m_{1}$ is placed on an inclined plane with
slope angle $\alpha$ and is connected to a second hanging block with
mass $m_{2}$ by a cord passing over a small, frictionless pulley
(Fig. 5.68 ). The coefficient of static friction is $\mu_{3}$ and the coef-
ficient of kinetic friction is $\mu_{\mathrm{k}}$ (a) Find the mass $m_{2}$ for which
block $m_{1}$ moves up the plane at constant speed once it is set in
motion. (b) Find the mass $m_{2}$ for which block $m_{1}$ moves down
the plane at constant speed once it is set in motion. (c) For
what range of values of $m_{2}$ will the blocks remain at rest if they
are released from rest?

Key Concepts

System Equilibrium

In problems where objects are moving at constant speed or are at rest, the system is in equilibrium, meaning that the sum of forces and, if applicable, torques is zero. This concept is essential for setting up the equations of motion that account for gravitational, frictional, and tension forces to determine conditions under which the system remains static or moves uniformly.

Inclined Plane Analysis

This concept involves decomposing an object’s weight into components parallel and perpendicular to the inclined surface. The parallel component influences the motion along the plane, while the perpendicular component affects the normal force, which is directly related to the frictional force acting on the object.

Tension in a Pulley System

This concept involves analyzing the forces transmitted through a cord over a pulley. In idealized systems with massless cords and frictionless pulleys, the tension is the same on both sides. This helps in relating the forces acting on different masses and establishing the conditions for equilibrium or constant velocity motion.

Newton's Second Law

This principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In problems where objects move at constant speed, the net force is zero, which leads to equilibrium equations that balance forces in different directions.

Static and Kinetic Friction

Friction is the resistive force that opposes motion. Static friction acts when there is no relative motion between surfaces and must be overcome to start moving an object, while kinetic friction acts when the object is already in motion. Understanding both types is crucial when analyzing the conditions for when an object on an incline starts to move or continues moving at constant speed.

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