Question

A mass \(m\) suspended from a spring stretches it by \(5\,\text{cm}\) when on the surface of the earth. The mass is then taken to a height of \(1600\,\text{km}\) above earth’s surface and again suspended from the same spring. At this altitude the extension of the spring is (Radius of earth \(= 6400\,\text{km}\))

• A. \(6.4\,\text{cm}\)
• B. \(1.6\,\text{cm}\)
• C. \(3.2\,\text{cm}\)
• D. \(0.8\,\text{cm}\)

Hint:

Spring extension depends on weight, and weight depends on the value of gravity at that location.

Answer 1

The Correct Option is C

Step 1: Relation between extension and gravity.
Extension of a spring is directly proportional to the weight acting on it:
\[ x \propto g \]
Step 2: Acceleration due to gravity at height \(h\).
At a height \(h\) above the earth’s surface,
\[ g_h = g\left(\frac{R}{R+h}\right)^2 \]
Step 3: Substitute given values.
\[ R = 6400\,\text{km}, \quad h = 1600\,\text{km} \] \[ g_h = g\left(\frac{6400}{6400+1600}\right)^2 = g\left(\frac{6400}{8000}\right)^2 = g\left(\frac{4}{5}\right)^2 = \frac{16}{25}g \]
Step 4: Calculate new extension.
Original extension \(x = 5\,\text{cm}\).
\[ x_h = x \times \frac{g_h}{g} = 5 \times \frac{16}{25} = 3.2\,\text{cm} \]
Step 5: Conclusion.
The extension of the spring at the given altitude is \(3.2\,\text{cm}\).