A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by the string of the pendulum with the vertical if this angle does not change during the turn. Take `g=10 m/s^2`.

Explanation

To solve the problem of finding the angle made by the string of a pendulum with the vertical when a car is taking a turn, we can follow these steps:

### Step 1: Convert the speed from km/h to m/s
The speed of the car is given as 36 km/h. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ 36 \text{ km/h} = \frac{36}{3.6} \text{ m/s} = 10 \text{ m/s} \]

### Step 2: Calculate the centripetal acceleration
The centripetal acceleration \( a_c \) can be calculated using the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the velocity (10 m/s) and \( r \) is the radius of the turn (10 m). Therefore,
\[ a_c = \frac{(10)^2}{10} = \frac{100}{10} = 10 \text{ m/s}^2 \]

### Step 3: Analyze the forces acting on the pendulum
When the car is turning, two forces act on the pendulum:
1. The gravitational force \( mg \) acting downward.
2. The centrifugal force \( ma_c \) acting horizontally outward (from the perspective of the car).

### Step 4: Set up the relationship using trigonometry
Let \( \theta \) be the angle made by the string with the vertical. The forces can be resolved into components:
- The vertical component is \( mg \).
- The horizontal component is \( ma_c \).

From the geometry of the situation, we can use the tangent function:
\[ \tan(\theta) = \frac{a_c}{g} \]
Substituting the values \( a_c = 10 \text{ m/s}^2 \) and \( g = 10 \text{ m/s}^2 \):
\[ \tan(\theta) = \frac{10}{10} = 1 \]

### Step 5: Solve for the angle \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(1) \]
This gives us:
\[ \theta = 45^\circ \]

### Final Answer
The angle made by the string of the pendulum with the vertical is \( 45^\circ \).

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