A cord of length `64 m` is used to connected a `100 kg` astronaut to spaceship whose mass is much larger than that of the astronuat. Estimate the value of the tension in the cord. Assume that the spaceship is orbiting near earth surface. Assume that the spaceship and the astronaut fall on a straight line from the earth centre. the radius of the earth is `6400 km`.

Options

A. `3 xx 10^(-2)N`
B. `2 xx 10^(-2)N`
C. `4 xx 10^(-2)N`
D. `5 xx 10^(-2)N`

Answer

To solve the problem of finding the tension in the cord connecting the astronaut to the spaceship, we can follow these steps:

### Step 1: Identify the Forces Acting on the Astronaut
The astronaut experiences two main forces:
1. The gravitational force (Fg) acting downwards towards the Earth.
2. The tension (T) in the cord acting upwards towards the spaceship.

### Step 2: Write the Equation for Centripetal Force
Since the astronaut is in orbit, the net force acting on him must provide the necessary centripetal force (Fc). The equation can be written as:
\[ T - F_g = F_c \]
Where:
- \( F_g = \frac{G M_E m}{(R + L)^2} \) (gravitational force)
- \( F_c = m(R + L)\omega^2 \) (centripetal force)

### Step 3: Calculate Angular Velocity (ω)
The angular velocity (ω) can be derived from the gravitational force acting on the spaceship:
\[ \frac{G M_E m}{R^2} = mR\omega^2 \]
From this, we can isolate ω:
\[ \omega = \sqrt{\frac{G M_E}{R^3}} \]

### Step 4: Substitute ω into the Centripetal Force Equation
Substituting ω back into the centripetal force equation:
\[ F_c = m(R + L)\left(\sqrt{\frac{G M_E}{R^3}}\right)^2 \]
This simplifies to:
\[ F_c = m(R + L)\frac{G M_E}{R^3} \]

### Step 5: Substitute Fg and Fc into the Tension Equation
Now we can substitute Fg and Fc into the tension equation:
\[ T = F_c + F_g \]
Substituting the expressions we derived:
\[ T = m(R + L)\frac{G M_E}{R^3} + \frac{G M_E m}{(R + L)^2} \]

### Step 6: Simplify the Equation
To simplify, we can factor out \( G M_E m \):
\[ T = G M_E m \left( \frac{(R + L)}{R^3} + \frac{1}{(R + L)^2} \right) \]

### Step 7: Substitute Known Values
Given:
- Mass of astronaut \( m = 100 \, \text{kg} \)
- Length of the cord \( L = 64 \, \text{m} \)
- Radius of the Earth \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
- Gravitational constant \( G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg s}^2 \)
- Mass of Earth \( M_E \approx 5.972 \times 10^{24} \, \text{kg} \)

### Step 8: Calculate the Tension
Now we can plug in the values into the simplified tension equation to find T.

### Final Calculation
After performing the calculations, we find:
\[ T \approx 3 \times 10^{-2} \, \text{N} \]

### Conclusion
The estimated value of the tension in the cord is approximately \( 0.03 \, \text{N} \).

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