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Chapter 12: Problem 88

The leaning Tower of Pisa is \(59.1 \mathrm{~m}\) high and \(7.44 \mathrm{~m}\) in diameter. The top of the tower is displaced \(4.01 \mathrm{~m}\) from the vertical. Treat the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge of toppling? (b) What angle would the tower then make with the vertical?

Short Answer

Expert verified
No additional displacement needed; angle is about 3.60 degrees.

Step by step solution

01

Understand the Problem

We are asked to find the additional displacement that would cause the Tower of Pisa to topple, and the angle it would make with the vertical at that point. We will treat the tower as a vertical uniform cylindrical object.
02

Find the Center of Mass

For a uniform cylinder, the center of mass is at its geometric center. Thus, for the Tower of Pisa, which is 59.1 m high and 7.44 m in diameter, the center of mass is vertically at 29.55 m (half the height) from the base, along the center axis.
03

Determine the Verge of Toppling Condition

A cylinder will be on the verge of toppling when its center of mass is directly above the edge of its base. Since the cylinder's base has a diameter of 7.44 m, the maximum horizontal displacement before toppling should equal the radius of the base, which is \( \frac{7.44}{2} = 3.72 \) m.
04

Calculate Additional Displacement

Currently, the top of the tower is displaced 4.01 m from the vertical. To achieve the toppling condition, the center of mass must move horizontally an additional \(3.72 - 4.01 = -0.29\) m. This implies the tower is already past the verge of toppling (because 4.01 m > 3.72 m). To actually compute additional for true context, technically zero additional is required but otherwise calculated as a trait of system in its equilibrium safety margin negative.
05

Calculate the Angle with Vertical

If the top displacement is at 3.72 m (verge of toppling), we use \(\theta = \tan^{-1}(\frac{3.72}{59.1})\). Thus, \(\theta = \tan^{-1}(0.063) \approx 3.60\) degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of Mass
In physics, the center of mass is a crucial point that represents the average position of all the mass in an object. When dealing with simple shapes like cylinders, finding the center of mass is straightforward. It can often be located at the geometric center.
For the Tower of Pisa, modeled as a uniform cylindrical structure, its center of mass is situated halfway up the tower's height. Since the tower is 59.1 meters tall, the center of mass is at 29.55 meters from the ground. This centralized position helps simplify balance and stability analyses of objects like towers, especially when considering forces acting upon them.
Cylindrical Objects
Cylindrical objects, such as the Tower of Pisa, can be described as having a circular base and a specific height. These symmetrical objects are prevalent in engineering and physics due to their uniform shape and predictable behavior.
The circular base has properties like diameter and radius, which are critical when considering stability. For the tower, the diameter is 7.44 meters, giving it a base radius of half this value, which is 3.72 meters. Cylinders are commonly assumed to have uniform mass distribution, allowing their center of mass to align with their geometric center. This simplifies calculations, making it easier to predict how they will react to various forces and torques.
Equilibrium
Equilibrium in physics refers to a state where an object experiences no net force or torque. For the cylindrical Tower of Pisa, achieving equilibrium means its center of mass must be directly above its base. In this state, there are no unbalanced forces causing rotation or translation.
In studying equilibrium, engineers often check if structures rest stably, not teetering or toppling. Equilibrium considerations help ensure safety and durability. If the center of mass moves outside the cylinder base's effective area, equilibrium is lost, and the object might topple. Thus, maintaining equilibrium is crucial for architectural integrity and safety.
Toppling Condition
The condition for toppling of a cylindrical object, like the Tower of Pisa, is reached when the center of mass aligns with the edge of the base. When the object鈥檚 center of mass is above or beyond this edge, it risks toppling.
For the tower, we established that it already surpasses its toppling condition, as evidenced by the 4.01-meter displacement from vertical, beyond the 3.72-meter safety margin. Typically, toppling is imminent when the horizontal displacement equals the base radius. This insight underscores the importance of maintaining structural adjustments to restore stability.

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Most popular questions from this chapter

A uniform cubical crate is 0.750 m on each side and weighs 500 N. It rests on a floor with one edge against a very small, fixed obstruction. At what least height above the floor must a horizontal force of magnitude 350 N be applied to the crate to tip it?

A crate, in the form of a cube with edge lengths of \(1.2 \mathrm{~m}\). contains a piece of machinery; the center of mass of the crate and its contents is located \(0.30 \mathrm{~m}\) above the crate's geometrical center. The crate rests on a ramp that makes an angle \(\theta\) with the horizontal. As \(\theta\) is increased from zero, an angle will be reached at which the crate will either tip over or start to slide down the ramp. If the coefficient of static friction \(\mu,\) between ramp and crate is \(0.60,\) (a) does the crate tip or slide and (b) at what angle \(\theta\) does this occur? If \(\mu_{1}=0.70\), (c) does the crate tip or slide and (d) at what angle \(\theta\) does this occur? (Hint: At the onset of tipping, where is the normal force located?)

A uniform sphere of mass \(m=0.85 \mathrm{~kg}\) and radius \(r=4.2 \mathrm{~cm}\) is held in place by a massless rope attached to a frictionless wall a distance \(L=8.0 \mathrm{~cm}\) above the center of the sphere. Find (a) the tension in the rope and (b) the force on the sphere from the wall.

A uniform ladder whose length is \(5.0 \mathrm{~m}\) and whose weight is \(400 \mathrm{~N}\) leans against a frictionless vertical wall. The coefficient of static friction between the level ground and the foot of the ladder is 0.46 . What is the greatest distance the foot of the ladder can be placed from the base of the wall without the ladder immediately slipping?

A makeshift swing is constructed by making a loop in one end of a rope and tying the other end to a tree limb. A child is sitting in the loop with the rope hanging vertically when the child's father pulls on the child with a horizontal force and displaces the child to one side. Just before the child is released from rest, the rope makes an angle of \(15^{\circ}\) with the vertical and the tension in the rope is \(280 \mathrm{~N}\). (a) How much does the child weigh? (b) What is the magnitude of the (horizontal) force of the father on the child just before the child is released? (c) If the maximum horizontal force the father can exert on the child is \(93 \mathrm{~N},\) what is the maximum angle with the vertical the rope can make while the father is pulling horizontally?
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