Question

A particle hanging from a spring stretches it by 1 cm at earth’s surface. Radius of earth is 6400 km. At a place 800 km above the earth’s surface, the same particle will stretch the spring by:
A. 0.79 cm
B. 1.2 cm
C. 4 cm
D. 17 cm

Answer 1

Hint: The particle hanging from the spring, the elongation in the spring is due to the weight of the particle. So, the spring force balances the weight of the particle. Therefore, use the formula for spring force and weight to equate the forces. Calculate the value of acceleration due to gravity 800 km above the surface of earth and use it to determine the elongation in the spring.

Formula used:
The spring force is given by the expression,
\[F = - kx\]
Here, k is spring constant and x is the elongation in spring.

Complete step by step answer:
We know that if a particle hanging from a spring of spring constant k stretches the spring by a distance x, then the spring force is given by the expression,
\[F = - kx\]

The negative sign indicates that the force opposes the elongation of the spring.

We see the particle hanging from the spring, the elongation in the spring is due to the weight of the particle. So, the spring force balances the weight of the particle. Therefore, we can write,
\[kx = mg\]
\[ \Rightarrow x = \dfrac{{mg}}{k}\] …… (1)

Here, g is the acceleration due to gravity at the surface of earth and m is the mass of the particle.

At the height h above the surface of earth, we have the acceleration due to gravity is expressed by the equation,
\[{g_h} = g\left( {1 - \dfrac{{2h}}{R}} \right)\]

Here, h is the height above the surface and R is the radius of the earth.

Therefore, if we substitute 800 km for h, we will get the acceleration due to gravity at height 800 km above the surface.
\[{g_h} = g\left( {1 - \dfrac{{2\left( {800} \right)}}{{6400}}} \right)\]
\[ \Rightarrow {g_h} = g\left( {1 - 0.25} \right)\]
\[ \Rightarrow {g_h} = \dfrac{3}{4}g\]

Therefore, we can express the elongation in the spring at height 800 km as follows,
\[{x_h} = \dfrac{{m{g_h}}}{k}\] …… (2)

Now, we have to divide equation (2) by equation (1).
\[\dfrac{{{x_h}}}{x} = \dfrac{{\dfrac{{m{g_h}}}{k}}}{{\dfrac{{mg}}{k}}}\]
\[ \Rightarrow \dfrac{{{x_h}}}{x} = \dfrac{{{g_h}}}{g}\]

Substitute \[\dfrac{3}{4}g\] for \[{g_h}\] in the above equation.
\[\dfrac{{{x_h}}}{x} = \dfrac{{\dfrac{3}{4}g}}{g}\]
\[ \Rightarrow {x_h} = \dfrac{3}{4}x\]
\[ \Rightarrow {x_h} = \left( {0.75} \right)\left( {1\,cm} \right)\]
\[ \Rightarrow {x_h} = 0.75\,cm \approx 0.79\,cm\].

So, the correct answer is “Option A”.

Note:
To solve this question, you don’t need to calculate the magnitude of acceleration due to gravity at 800 km above the earth’s surface. You can express it in terms of acceleration due to gravity at the surface of earth. Then to calculate the elongation in the spring, take the ratio of elongation of spring on the earth’s surface and elongation of spring at 800 km above the earth’s surface.