A 12.0-kg box resting on a horizontal, frictionless surface...

A 12.0-kg box resting on a horizontal, frictionless surface is attached to a 5.00-kg weight by a thin, light wire that passes over a frictionless pulley ($\textbf{Fig. E10.16}$). The pulley has the shape of a uniform solid disk of mass 2.00 kg and diameter 0.500 m. After the system is released, find (a) the tension in the wire on both sides of the pulley, (b) the acceleration of the box, and (c) the horizontal and vertical components of the force that the axle exerts on the pulley. Figure E10.16 (CAN'T COPY THE FIGURE)

Key Concepts

Free-Body Diagram Analysis

Drawing free-body diagrams helps in visualizing and resolving all forces acting on each component of a system. By isolating each body—whether a mass on a frictionless surface, a hanging weight, or the rotating pulley—and identifying forces such as tension, gravitational force, and reaction forces, one can systematically apply the appropriate laws of motion to derive equations needed to analyze the system.

Tension Variation in a Rope Over a Rotating Pulley

Due to the rotational inertia of the pulley, the tension in the rope is not necessarily the same on either side of the pulley. The difference in these tensions creates the net torque on the pulley. Understanding this concept is fundamental when analyzing systems where the pulley has significant mass or when rotational dynamics play a role in the overall system behavior.

Constraint Relationships Between Linear and Angular Quantities

In systems with connected translational and rotational motion, such as a pulley system, the linear acceleration of a mass is directly related to the angular acceleration of the pulley through the relation a = ?r. This constraint is key to formulating the equations of motion that link the behavior of the masses with the rotational dynamics of the pulley.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and its distribution relative to the axis of rotation. In problems involving pulleys or other rotating bodies, knowing the moment of inertia is essential for relating applied torques to angular accelerations.

Newton's Second Law for Rotation

This law expresses the rotational analogue to F = ma, stating that the net torque acting on a rotating object equals the object’s moment of inertia multiplied by its angular acceleration (? = I?). It is crucial for analyzing systems with rotating components, like the pulley, where different tensions in the rope produce torques that affect its rotation.

Newton's Second Law for Translation

This principle states that the net force acting on an object equals the mass of the object multiplied by its acceleration (F = ma). It is used to relate the forces acting on bodies in motion, such as the box and the hanging weight, allowing the determination of linear accelerations based on the applied forces.

Transcript

00:01 Here we have a, there's a box resting on a table and then another box hanging from a string over a pulley here.

00:09 So i've drawn free by diagrams of these.

00:12 Technically this one is kind of down here.

00:15 So here this is the box that is free to move horizontally.

00:19 This is the one that's free to move vertically and here's our pulley.

00:23 So we have a weight and a tension in the pulley.

00:27 That tension is also in this side of the rope that's going around the pulley.

00:33 Because the pulley has inertia, this tension is not necessarily the same as this tension.

00:40 But this tension is the same as this tension because that's in the same part of the rope.

00:45 And then that tension is applied to this.

00:48 And so positive, i'm going to say to the right for this block, down for this one, and counterclockwise for the pulley.

00:56 I have a friction force acting on here.

00:58 And then these are reaction forces at the center of the pulley.

01:02 We're told that this one block weighs 12 kilograms.

01:07 This one, the one hanging, weighs 5.

01:10 The pulley weighs 2, and the pulley has a radius of 0 .25 meters.

01:18 And i guess they tell us that it's a frictional surface, so friction force is zero, so we don't have to worry about that.

01:25 So looking at this body here, we have, if this is positive, we have minus t1 plus wt equals m2a.

01:34 And we're told that the belt around here or the rope is not slipping on this pulley.

01:41 So we know that the angular acceleration of the pulley equals the acceleration of this block divided by r.

01:48 And we also know that that acceleration of this block must be the vertical acceleration of this block must be the horizontal acceleration of that block...

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