A car is moving in a circular horizonta track of radius 10m with a constant speed of 10 m/s. A pendulum bob is suspended from the roof of the cat by a light rigid rod of length 1.00m. The angle made by the rod with track is

Options

A. zero
B. `30^@`
C. `45^@`
D. `60^@`

Explanation

To find the angle made by the pendulum bob with the vertical when the car is moving in a circular horizontal track, we can follow these steps:

### Step 1: Identify the Forces Acting on the Pendulum Bob
The forces acting on the pendulum bob are:
- The gravitational force (weight) acting downwards, \( mg \).
- The tension \( T \) in the rod acting along the rod.

### Step 2: Analyze the Motion
Since the car is moving in a circular path, the pendulum bob will experience a centripetal force directed towards the center of the circular path. This force can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the pendulum bob,
- \( v \) is the speed of the car (10 m/s),
- \( r \) is the radius of the circular path (10 m).

### Step 3: Set Up the Equations
For the pendulum bob, we can resolve the tension \( T \) into two components:
- The vertical component \( T \cos \theta \) balances the weight \( mg \).
- The horizontal component \( T \sin \theta \) provides the required centripetal force.

Thus, we have two equations:
1. \( T \cos \theta = mg \) (1)
2. \( T \sin \theta = \frac{mv^2}{r} \) (2)

### Step 4: Divide the Equations
To eliminate \( T \), we can divide equation (2) by equation (1):
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{\frac{mv^2}{r}}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{v^2}{rg} \]

### Step 5: Substitute Values
We know:
- \( v = 10 \, \text{m/s} \)
- \( r = 10 \, \text{m} \)
- \( g = 10 \, \text{m/s}^2 \)

Substituting these values into the equation gives:
\[ \tan \theta = \frac{10^2}{10 \times 10} = \frac{100}{100} = 1 \]

### Step 6: Find the Angle
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(1) = 45^\circ \]

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

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