A mass is attached to a spring that hangs vertically. The extension produced in the spring is 6 cm on Earth. The acceleration due to gravity on the surface of the Moon is one-sixth of its value on the surface of the Earth. The extension of the spring on the Moon would be :
6 cm
1 cm
0 cm
36 cm
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC NDA-1 – 2023
– Weight (W) = mass (m) × acceleration due to gravity (g).
– On Earth: $W_{Earth} = mg_{Earth} = kx_{Earth}$.
– On the Moon: $W_{Moon} = mg_{Moon} = kx_{Moon}$.
Acceleration due to gravity on the Moon $g_{Moon} = \frac{1}{6} g_{Earth}$.
From Hooke’s Law on Earth: $mg_{Earth} = k \times 6$ cm. So, $\frac{mg_{Earth}}{k} = 6$ cm.
On the Moon: $mg_{Moon} = kx_{Moon}$.
Substitute $g_{Moon}$: $m \left(\frac{g_{Earth}}{6}\right) = kx_{Moon}$.
$\frac{1}{6} (mg_{Earth}) = kx_{Moon}$.
Substitute $mg_{Earth} = k \times 6$ cm: $\frac{1}{6} (k \times 6 \text{ cm}) = kx_{Moon}$.
$k \times 1 \text{ cm} = kx_{Moon}$.
$x_{Moon} = 1$ cm.
The extension is directly proportional to the weight, and thus directly proportional to gravity. Since gravity on the Moon is one-sixth of Earth’s, the extension will also be one-sixth.