Consider the rectangular cabinet of Problem 50 shown in Figure P 12.50 , but with a force 𝐅 appl

Consider the rectangular cabinet of Problem 50 shown in...

Consider the rectangular cabinet of Problem 50 shown in Figure $\mathrm{P} 12.50$ , but with a force $\overrightarrow{\mathbf{F}}$ applied horizontally at the upper edge. (a) What is the minimum force required to start to tip the cabinet? (b) What is the minimum coefficient of static friction required for the cabinet not to slide with the application of a force of this magnitude? (c) Find the magnitude and direction of the minimum force required to tip the cabinet if the point of application can be chosen anywhere on the cabinet.

Key Concepts

Static Equilibrium

This concept involves ensuring that a body is at rest under the action of various forces. It requires that both the net force and the net moment (or torque) on the body are zero. In the context of a cabinet being tipped or slid, the equilibrium equations help determine the conditions under which the cabinet will start to tip or begin to slide.

Torque and Moment Analysis

A fundamental aspect of analyzing tipping involves computing moments about a chosen pivot point. The moment is the product of a force and its lever arm (the perpendicular distance from the line of action to the pivot). When a force is applied to a body, its ability to cause rotation (or tipping) is determined by the torque it produces relative to the resisting moment due to gravity acting through the center of mass.

Center of Mass

The center of mass of a rigid body is the point at which the mass of the body can be considered to be concentrated for the purpose of analyzing translational motion and generating gravitational moments. Its position is critical when determining the tipping point since the gravitational force acting through it produces a stabilizing moment against which any applied moment must work to tip the body.

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It is proportional to the normal force and depends on the coefficient of static friction. In tipping problems, static friction must be overcome for sliding to occur, and understanding its magnitude is essential for establishing the minimum conditions under which a body will tip rather than slide.

Force Application Geometry

The point and direction at which an external force is applied can greatly affect the moments produced on a body. For the cabinet, applying a force at different points (such as the upper edge versus a more optimal position) alters the lever arm and the resulting torque, thereby affecting the minimum force needed to tip the cabinet. This concept emphasizes the importance of where and how forces are applied in structural stability analyses.

Transcript

00:01 We have to consider the cabinet that we were pulling in a previous problem.

00:06 What problem was that 50? so we have this box here, and we're worried about it tipping over.

00:14 So in that case, we're just going to have a force from the floor is just going to be exerted at point a.

00:21 So just as this thing lifts off the ground, we only have reaction force there.

00:26 And so we'll take moments about that.

00:28 So i'm not worried about what these forces are.

00:31 Which so i haven't drawn them to clutter up the figure.

00:36 We are told in this case that h2, we're applying the force at the very top here and in the horizontal direction.

00:44 So theta is zero and h2 is l in this case.

00:47 Everything else is the same.

00:51 And, well, no, f1, we don't know in this case.

00:58 We're trying to find it.

00:59 So we can take moments about this point.

01:02 And we get our moment equation here.

01:04 And again, this just winds up being 1.

01:07 And so f1 is just the weight of the cabinet times the width over 2 and divided by the height of the cabinet.

01:23 And that winds up being 0 .3 times the weight of the cabinet or 120 newtons.

01:30 Now, let's see here...

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