An electric bulb suspended from the roof of a train by a thread shifts through' an angle of 19.8° when the train goes round a curved path 100 m in radius. Find the speed of the train

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a bulb suspended from the roof of a train, which shifts at an angle of 19.8° when the train moves in a circular path of radius 100 m. We need to find the speed of the train.

### Step 2: Draw the Free Body Diagram (FBD)
- Draw the bulb hanging from the roof of the train.
- The tension (T) in the thread acts at an angle of 19.8° with the vertical.
- The weight (mg) of the bulb acts downward.

### Step 3: Resolve the Tension into Components
- The vertical component of tension: \( T \cos(19.8°) \)
- The horizontal component of tension (providing centripetal force): \( T \sin(19.8°) \)

### Step 4: Set Up the Equations
1. The vertical forces balance:
\[ T \cos(19.8°) = mg \quad \text{(Equation 1)} \]
2. The horizontal forces provide the centripetal force:
\[ T \sin(19.8°) = \frac{mv^2}{r} \quad \text{(Equation 2)} \]

### Step 5: Divide the Two Equations
To eliminate T, divide Equation 2 by Equation 1:
\[ \frac{T \sin(19.8°)}{T \cos(19.8°)} = \frac{\frac{mv^2}{r}}{mg} \]
This simplifies to:
\[ \tan(19.8°) = \frac{v^2}{rg} \]

### Step 6: Solve for the Speed (v)
Rearranging gives:
\[ v^2 = rg \tan(19.8°) \]
Taking the square root:
\[ v = \sqrt{rg \tan(19.8°)} \]

### Step 7: Substitute the Values
Given:
- \( r = 100 \, \text{m} \)
- \( g = 9.8 \, \text{m/s}^2 \)
- \( \theta = 19.8° \)

Now substitute these values into the equation:
\[ v = \sqrt{100 \times 9.8 \times \tan(19.8°)} \]

### Step 8: Calculate the Value
First, calculate \( \tan(19.8°) \):
\[ \tan(19.8°) \approx 0.361 \]
Now substitute:
\[ v = \sqrt{100 \times 9.8 \times 0.361} \]
Calculating this gives:
\[ v = \sqrt{353.378} \approx 18.7 \, \text{m/s} \]

### Final Answer
The speed of the train is approximately \( 18.7 \, \text{m/s} \).

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