Two particles of masses in the ratio 1: 2 you're moving in circles of radii in the ratio 2:3 with time periods in the ratio 3: 4. The ratio of their centripetal forces is

Options

A. 9:4
B. 1:4
C. 9:16
D. 16:27

Answer

To find the ratio of the centripetal forces of two particles with given mass, radius, and time period ratios, we can follow these steps:

### Step 1: Understand the formula for centripetal force
The centripetal force \( F_c \) acting on an object moving in a circle is given by the formula:
\[ F_c = \frac{m v^2}{r} \]
where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius of the circular path.

### Step 2: Express velocity in terms of radius and time period
The velocity \( v \) of an object moving in a circle can be expressed as:
\[ v = \frac{2\pi r}{T} \]
where \( T \) is the time period of one complete revolution.

### Step 3: Substitute the expression for velocity into the centripetal force formula
Substituting the expression for \( v \) into the centripetal force formula gives:
\[ F_c = \frac{m \left(\frac{2\pi r}{T}\right)^2}{r} = \frac{m (4\pi^2 r^2)}{T^2 r} = \frac{4\pi^2 m r}{T^2} \]

### Step 4: Set up the ratio of centripetal forces for the two particles
Let the masses of the two particles be \( m_1 \) and \( m_2 \), the radii be \( r_1 \) and \( r_2 \), and the time periods be \( T_1 \) and \( T_2 \). The ratio of the centripetal forces \( F_1 \) and \( F_2 \) is:
\[ \frac{F_1}{F_2} = \frac{4\pi^2 m_1 r_1}{T_1^2} \div \frac{4\pi^2 m_2 r_2}{T_2^2} = \frac{m_1 r_1 T_2^2}{m_2 r_2 T_1^2} \]

### Step 5: Substitute the given ratios into the equation
We are given:
- Masses in the ratio \( \frac{m_1}{m_2} = \frac{1}{2} \)
- Radii in the ratio \( \frac{r_1}{r_2} = \frac{2}{3} \)
- Time periods in the ratio \( \frac{T_1}{T_2} = \frac{3}{4} \)

Now, substituting these ratios into the centripetal force ratio:
\[ \frac{F_1}{F_2} = \frac{1/2 \cdot (2/3) \cdot (4/3)^2}{1} = \frac{1/2 \cdot (2/3) \cdot (16/9)}{1} \]

### Step 6: Simplify the expression
Calculating the above expression:
\[ \frac{F_1}{F_2} = \frac{1 \cdot 2 \cdot 16}{2 \cdot 3 \cdot 9} = \frac{32}{54} = \frac{16}{27} \]

### Final Answer
Thus, the ratio of the centripetal forces \( F_1 : F_2 \) is:
\[ \frac{F_1}{F_2} = \frac{16}{27} \]