For a conical pendulum of string length L, its angular speed is proportional to

Options

A. `sqrt(L)`
B. L
C. `(1)/(sqrt(L))`
D. `(1)/(L)`

Answer

To solve the problem of determining how the angular speed of a conical pendulum is related to its string length \( L \), we can follow these steps:

### Step 1: Understand the Setup
A conical pendulum consists of a mass (bob) attached to a string of length \( L \) that swings in a horizontal circle while the string traces out a cone. The angle between the string and the vertical is denoted as \( \theta \).

### Step 2: Identify Forces Acting on the Bob
The forces acting on the bob are:
- The gravitational force \( mg \) acting downwards.
- The tension \( T \) in the string acting along the string.

### Step 3: Resolve Forces into Components
We can resolve the tension \( T \) into two components:
1. The vertical component: \( T \cos \theta \) which balances the weight of the bob.
2. The horizontal component: \( T \sin \theta \) which provides the centripetal force necessary for circular motion.

### Step 4: Write the Equations
From the vertical balance of forces, we have:
\[ T \cos \theta = mg \quad \text{(1)} \]
From the horizontal motion, the centripetal force is given by:
\[ T \sin \theta = m \omega^2 r \quad \text{(2)} \]
where \( r \) is the horizontal radius of the circular path, which can be expressed as \( r = L \sin \theta \).

### Step 5: Substitute for \( r \)
Substituting \( r \) into equation (2):
\[ T \sin \theta = m \omega^2 (L \sin \theta) \]
This simplifies to:
\[ T = \frac{m \omega^2 L \sin \theta}{\sin \theta} = m \omega^2 L \quad \text{(3)} \]

### Step 6: Substitute \( T \) from Equation (1) into Equation (3)
From equation (1), we can express \( T \):
\[ T = \frac{mg}{\cos \theta} \]
Setting the two expressions for \( T \) equal gives:
\[ \frac{mg}{\cos \theta} = m \omega^2 L \]
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{g}{\cos \theta} = \omega^2 L \]

### Step 7: Solve for \( \omega \)
Rearranging gives:
\[ \omega^2 = \frac{g}{L \cos \theta} \]
Thus, we can express \( \omega \) as:
\[ \omega = \sqrt{\frac{g}{L \cos \theta}} \]

### Step 8: Determine Proportionality
From the equation \( \omega = \sqrt{\frac{g}{L \cos \theta}} \), we can see that:
- \( \omega \) is proportional to \( \frac{1}{\sqrt{L}} \) when \( \theta \) is constant.

### Conclusion
Thus, the angular speed \( \omega \) of a conical pendulum is inversely proportional to the square root of the length \( L \) of the string.

### Final Answer
The angular speed \( \omega \) is proportional to \( \frac{1}{\sqrt{L}} \).