Two particles of the same mass m are moving in circular orbits because of force, given by
`F(r)=(-16)/(r)-r^(3)`
The first particle is at a distance r =1 , and the second, at r = 4. The best estimate for the ratio of kinetic energies of the first and the second particle is closet to :

Options

A. `3xx10^(-3)`
B. `6xx10^(2)`
C. `10^(-1)`
D. `6xx10^(-2)`

Answer

To solve the problem, we need to find the ratio of the kinetic energies of two particles moving in circular orbits under the influence of a given force. The force is defined as:

\[ F(r) = -\frac{16}{r} - r^3 \]

### Step 1: Understand the relationship between force and centripetal motion

For a particle moving in a circular orbit, the centripetal force required to keep it in that orbit is provided by the net force acting on it. The centripetal force \( F_c \) is given by:

\[ F_c = \frac{mv^2}{r} \]

where \( m \) is the mass of the particle, \( v \) is its tangential velocity, and \( r \) is the radius of the circular orbit.

### Step 2: Set up the equation for each particle

We can equate the given force to the centripetal force for each particle:

\[ F(r) = \frac{mv^2}{r} \]

Thus, we have:

\[ -\frac{16}{r} - r^3 = \frac{mv^2}{r} \]

Multiplying through by \( r \) to eliminate the denominator gives:

\[ -16 - r^4 = mv^2 \]

So, we can express \( mv^2 \) as:

\[ mv^2 = -16 - r^4 \]

### Step 3: Calculate \( mv^2 \) for both particles

1. **For the first particle at \( r = 1 \)**:

\[ mv^2 = -16 - (1)^4 = -16 - 1 = -17 \]

2. **For the second particle at \( r = 4 \)**:

\[ mv^2 = -16 - (4)^4 = -16 - 256 = -272 \]

### Step 4: Calculate the kinetic energy for both particles

The kinetic energy \( K \) of a particle is given by:

\[ K = \frac{1}{2} mv^2 \]

1. **Kinetic energy of the first particle**:

\[ K_1 = \frac{1}{2} mv^2 = \frac{1}{2} (-17) = -\frac{17}{2} \]

2. **Kinetic energy of the second particle**:

\[ K_2 = \frac{1}{2} mv^2 = \frac{1}{2} (-272) = -136 \]

### Step 5: Find the ratio of the kinetic energies

Now, we can find the ratio of the kinetic energies of the first particle to the second particle:

\[ \text{Ratio} = \frac{K_1}{K_2} = \frac{-\frac{17}{2}}{-136} = \frac{17}{2 \times 136} = \frac{17}{272} \]

### Step 6: Simplify the ratio

Calculating the ratio:

\[ \frac{17}{272} = \frac{1}{16} = 0.0625 \]

### Step 7: Convert to scientific notation

\[ 0.0625 = 6.25 \times 10^{-2} \]

### Conclusion

The best estimate for the ratio of kinetic energies of the first and the second particle is closest to:

\[ \boxed{6 \times 10^{-2}} \]