A particle is moving in a circle of radius 1 cm and with a constant speed of `8 cm s^(-1)` . Centre of the circle lies on principle axis of a converging lens of focal length 50 cm and at a distance of 75 cm from the lens . Plane of the circle is perpendicular to principle axis. The correct statement is

Options

A. Speed of the image is `4 cm s^(-1)`
B. Speed of the image is `8 cm s^(-1)`
C. Speed of the image is `16 cm s^(-1)`
D. Speed of the image is `3.2 cm s^(-1)`

Answer

To solve the problem step by step, let's break down the information given and apply the relevant physics concepts.

### Step 1: Identify the parameters
- Radius of the circular path (r₀) = 1 cm
- Speed of the particle (v₀) = 8 cm/s
- Focal length of the lens (F) = 50 cm
- Object distance (U) = -75 cm (negative because it is in the direction opposite to the light travel)

### Step 2: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{v} = \frac{1}{50} + \frac{1}{-75} \]
Finding a common denominator (150):
\[ \frac{1}{v} = \frac{3}{150} - \frac{2}{150} = \frac{1}{150} \]
Thus,
\[ v = 150 \text{ cm} \]

### Step 3: Calculate magnification
Magnification (M) is given by:
\[ M = \frac{v}{u} = \frac{150}{-75} = -2 \]
This means the image is inverted and twice the size of the object.

### Step 4: Relate angular speed and radii
The angular speed (ω) is the same for both the object and the image. The relationship between the radius of the object (r₀) and the radius of the image (rᵢ) is given by the magnification:
\[ \frac{rᵢ}{r₀} = |M| = 2 \]
Thus,
\[ rᵢ = 2 \times r₀ = 2 \times 1 \text{ cm} = 2 \text{ cm} \]

### Step 5: Calculate the speed of the image
The speed of the image (vᵢ) can be calculated using the relationship:
\[ v = ω \cdot r \]
For the object:
\[ v₀ = ω \cdot r₀ \implies ω = \frac{v₀}{r₀} = \frac{8 \text{ cm/s}}{1 \text{ cm}} = 8 \text{ rad/s} \]
For the image:
\[ vᵢ = ω \cdot rᵢ = 8 \text{ rad/s} \cdot 2 \text{ cm} = 16 \text{ cm/s} \]

### Conclusion
The speed of the image is 16 cm/s. Therefore, the correct statement is that the speed of the image is 16 cm/s.