A 50-story building is being planned. It is to be 180.0 m...

A 50-story building is being planned. It is to be 180.0 m high with a base 46.0 m by 76.0 m. Its total mass will be about $1.8 \times 10^7 kg$, and its weight therefore about $1.8 \times 10^8 N$. Suppose a 200-km/h wind exerts a force of over the 76.0-m-wide face (Fig. 9-80). Calculate the torque about the potential pivot point, the rear edge of the building (where $\overrightarrow{F}_E$ acts in Fig. 9-80), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the building's face, and that the building is not anchored in bedrock. [Hint: $\overrightarrow{F}_E$ in Fig. 9-80 represents the force that the Earth would exert on the building in the case where the building would just begin to tip.] FIGURE 9-80 Forces on a building subjected to wind $\left(\overrightarrow{\mathbf{F}}_{\mathrm{A}}\right)$, gravity $(m \overrightarrow{\mathbf{g}})$, and the force $\overrightarrow{\mathbf{F}}_{\mathrm{E}}$ on the building due to the Earth if the building were just about to tip. Problem 61 . (Figure can't copy)

A 50-story building is being planned. It is to be 180.0 m high with a base 46.0 m by 76.0 m. Its total mass will be about $1.8 \times 10^7 kg$, and its weight therefore about $1.8 \times 10^8 N$. Suppose a 200-km/h wind exerts a force of over the 76.0-m-wide face (Fig. 9-80). Calculate the torque about the potential pivot point, the rear edge of the building (where $\overrightarrow{F}_E$ acts in Fig. 9-80), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the building's face, and that the building is not anchored in bedrock. [Hint: $\overrightarrow{F}_E$ in Fig. 9-80 represents the force that the Earth would exert on the building in the case where the building would just begin to tip.]
FIGURE 9-80 Forces on a building subjected to wind $\left(\overrightarrow{\mathbf{F}}_{\mathrm{A}}\right)$, gravity $(m \overrightarrow{\mathbf{g}})$, and the force $\overrightarrow{\mathbf{F}}_{\mathrm{E}}$ on the building due to the Earth if the building were just about to tip. Problem 61 . (Figure can't copy)

Key Concepts

Toppling (Tipping) Analysis

Toppling analysis involves determining whether an applied external force, such as wind pressure, generates a torque that exceeds the stabilizing torque provided by the weight of a structure. It assesses the conditions under which a building might begin to rotate or tip about a pivot point, leading to potential instability or collapse.

Torque

Torque is the rotational equivalent of force and represents the tendency of a force to cause an object to rotate about a particular pivot point. It is calculated by multiplying the magnitude of the force by the distance (lever arm) from the pivot point to the line of action of the force, playing a crucial role in statics and stability analyses.

Lever Arm (Moment Arm)

The lever arm or moment arm is the perpendicular distance from the pivot point to the line through which the force acts. It determines the effectiveness of a force in generating torque, as a longer lever arm produces a larger rotational effect for the same magnitude of force.

Static Equilibrium

Static equilibrium refers to a state where all forces and torques acting on a body cancel each other out, resulting in no net translational or rotational motion. This principle is essential in analyzing structures to ensure that they remain stable and do not tip or collapse under applied loads.

Transcript

00:01 So chapter 9, problem 61.

00:08 So with this problem, let's just read it real quick.

00:12 A 50 -story building is being planned.

00:14 It is to be 180 meters high with a base 46 meters by 76 meters.

00:23 Its total mass will be about 1 .8 times 10 to the 7th kilograms and its weight therefore about 100 times more.

00:30 Which is 1 .8 times 10 to the 8 newtons.

00:36 Suppose the 200 kilometer per hour wind exerts the force of 950 newt per meter squared over the 76 meter wide face.

00:48 And what we want to do is calculate the torque about the potential pivot point, the rear edge of the building, the bottom left, where the force from the earth acts, and determine whether the building will topple or not.

01:00 And we can assume the total force of the wind acts the midpoint of the building's face and that the building is not anchored in bedrock.

01:07 We're also given a hint in the problem, but i'm not going to read that.

01:11 So let's just get right into it.

01:13 If we assume that the building has just begun to tip so that it is essentially vertical, but that all the force on the building due to contact with the earth is at the lower left hand corner, then we can draw a figure like this.

01:34 It's our building right here on this corner, so we can imagine here, and i'm just going to blow this up a little bit like this from the right right on the corner we're going to have a force up into the right to match our earth force that was diagonal originally so it's going to be f e x and this is f e y and this is the diagonal so if we take torques about that corner with counterclockwise torques is positive so it's going to be counterclockwise is going to be positive.

02:18 Clockwise will be negative.

02:22 Then what we can do is the newton's second mile, but in rotational form.

02:26 So some of the tors is equal to i -alpha.

02:33 But we can't say whether this is zero or not, because we don't know if it's well, it's definitely non -zero actually, because we know that the building is starting to tip over this way, right? the air is pushing it here.

02:44 So knowing that it's non -zero, we don't really to know what this is as long as we know what the sum of these torques.

02:53 All we really want to know is the sum of the torques.

02:56 We don't really want to know the specific moment of inertia or angular acceleration.

03:00 So let's just do that.

03:02 Some of the torques, they have a torque produced from the from the force of the air pushing leftwards.

03:12 And so since we have counterclockwise as positive, anything going in that direction, like the left direction is going to be positive.

03:18 So the torque is determined by, well, let's just write a general formula for torque first...

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