Three uniform books, each having a weight W and length a, are stacked as shown. Determine the ma

Three uniform books, each having a weight W and length a,...

Key Concepts

Overhang Analysis

Overhang analysis is used to determine how far elements can be extended beyond a support without causing instability. In stacking problems, careful analysis of the progressive displacement and the resulting shifts in the center of mass are required to calculate the maximum allowable extension. This analysis often involves combining the effects of individual displacements and assessing their cumulative impact on the system’s stability.

Static Stability

Static stability refers to the condition where a body remains at rest or in equilibrium under applied forces without initiating movement. For objects in stacked configurations, static stability is maintained when the net force and net torque about any pivot point are zero, ensuring the structure does not collapse or tip over.

Center of Mass

The center of mass is the weighted average position of all the mass in a system and is crucial in determining stability. In statics, ensuring that the overall center of mass lies within the base of support is essential to prevent the system from toppling. This concept is used to determine the critical point where the moments about a pivot balance out, thereby ensuring equilibrium.

Torque Equilibrium

Torque, the product of force and its lever arm, determines the rotational effect of forces acting on a body. For a system to remain in equilibrium, the sum of clockwise moments must equal the sum of counterclockwise moments. Ensuring torque equilibrium about the edge of the support is vital in problems involving overhangs, as it dictates the maximum displacement possible without causing the system to rotate and topple.

Transcript

00:01 This is an interesting problem and that requires a little bit of thought, or should you should do a put a little bit of thought into how you're going to solve this before starting into things.

00:14 And why that is is because what we're asked to say the uniform, this is so do we have uniform books here, so the weight uniformly distributed.

00:23 They all have the same length, a, and we want to determine the maximum distance d.

00:30 Okay so here's d that the top book can extend out from the bottom so that the stack does not topple over so what we need to think about is you know is it so if say if we just push the first book off then obviously it's gonna get to so it's such that if we keep this one fixed once that the center of master the top book goes past this edge here then it's gonna topple but the other case is that both could topple over.

01:05 Obviously, if this one extends out and this one extends out, we'd expect that we could extend it further.

01:13 So we're going to look at the case of when both books topple over.

01:17 So now we need to be worried about when this point gets to here, and then the total moment falls over.

01:28 Or it gets kind of so that these books topple over.

01:33 And so what we do is we have just forces in the y direction.

01:40 So f1 is here.

01:44 So that's where that's kind of when they start the top, when the second book starts to topple over, then there's no distributed load here.

01:56 And there's just basically a point load where they're in contact.

02:00 So we just have a point load in.

02:02 The y direction and i'm going to call this my origin here...