The Leaning Tower of Pisa is 59.1 m high and 7.44 m in diameter. The top of the tower is displaced 4.01 m from the vertical. Treat the tower as a uniform, circulary cylinder. What additional displacement (in m), measured at the top, would bring the tower to the verge of toppling ?

Answer

To solve the problem of determining the additional displacement that would bring the Leaning Tower of Pisa to the verge of toppling, we can follow these steps:

### Step 1: Understand the Problem
We are given the height of the Leaning Tower of Pisa (h = 59.1 m) and its diameter (d = 7.44 m). The top of the tower is already displaced 4.01 m from the vertical. We need to find out how much more displacement (let's call it \( x_{\text{additional}} \)) is needed to reach the critical point of toppling.

### Step 2: Calculate the Critical Angle
The tower will topple when the center of mass shifts beyond the base of the tower. We can find the critical angle \( \theta \) at which this occurs using the tangent function. The formula is:

\[ \tan(\theta) = \frac{d}{h} \]

Where:
- \( d = 7.44 \, \text{m} \) (diameter)
- \( h = 59.1 \, \text{m} \) (height)

Calculating \( \tan(\theta) \):

\[ \tan(\theta) = \frac{7.44}{59.1} \]

### Step 3: Calculate the Angle \( \theta \)
Now, we can find \( \theta \) by taking the arctangent:

\[ \theta = \tan^{-1}\left(\frac{7.44}{59.1}\right) \]

Calculating this gives:

\[ \theta \approx 7.175^\circ \]

### Step 4: Calculate the Maximum Displacement \( x_{\text{max}} \)
The maximum displacement \( x_{\text{max}} \) at the top of the tower can be calculated using the sine function:

\[ \sin(\theta) = \frac{x_{\text{max}}}{h} \]

Rearranging gives:

\[ x_{\text{max}} = h \cdot \sin(\theta) \]

Substituting the values:

\[ x_{\text{max}} = 59.1 \cdot \sin(7.175^\circ) \]

Calculating \( \sin(7.175^\circ) \):

\[ x_{\text{max}} \approx 59.1 \cdot 0.125 \]

Thus,

\[ x_{\text{max}} \approx 7.38 \, \text{m} \]

### Step 5: Calculate the Additional Displacement
Now, we can find the additional displacement needed to reach the verge of toppling:

\[ x_{\text{additional}} = x_{\text{max}} - \text{initial displacement} \]

Where the initial displacement is 4.01 m:

\[ x_{\text{additional}} = 7.38 - 4.01 \]

Calculating this gives:

\[ x_{\text{additional}} \approx 3.37 \, \text{m} \]

### Final Answer
The additional displacement required to bring the Leaning Tower of Pisa to the verge of toppling is approximately **3.37 meters**.

---