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Question: In the given diagram, when a weight of \(100\;{\rm{g}}\) is hung from the spring, its length is \(9\...

In the given diagram, when a weight of 100  g100\;{\rm{g}} is hung from the spring, its length is 9  cm9\;{\rm{cm}}. When a weight of 150  g150\;{\rm{g}} is hung from it, its length is 11  cm11\;{\rm{cm}}. What is the length of the spring when there’s no weight hanging from it?

(A) 2  cm2\;{\rm{cm}}
(B) 4  cm4\;{\rm{cm}}
(C) 5  cm5\;{\rm{cm}}
(D) 7  cm7\;{\rm{cm}}

Explanation

Solution

This question uses the concept of spring force and the force pulling object toward Earth, so you should know the formulas. You need to compare both the formulas and need to substitute the values simultaneously. Later solve both the obtained equations in order to get the length of the spring without load.

Complete step by step answer:
Given data
The length of the spring, when the load of mass 100 g is attached is 9 cm.
The length of the spring, when the load of mass 150 g is attached is 11 cm.

Let the length of the spring when no mass is attached is L.
As we know that the expression for the spring force is given as,
F=−kxF = - kx
Here, k is the spring constant and x is the spring length.

Further, the expression for the force exerted by the load on the spring can be written as,
F=mgF = mg
Here, m is the mass of the load and g is the acceleration due to gravity.

Using both the expression, we can write it as,
mg=−kx......(1)mg = - kx......\left( 1 \right)

As it is given that, the length of the spring is 9 cm, when the load of mass 100 g. So we substitute the values in equation (1).
(100  g)×g=−k×(9  cm−L) (100g)  g=−k(9  cm−L)......(1)  \left( {100\;{\rm{g}}} \right) \times g = - k \times \left( {9\;{\rm{cm}} - L} \right)\\\ \left( {100g} \right)\;{\rm{g}} = - k\left( {9\;{\rm{cm}} - L} \right)......\left( 1 \right)\\\

Similarly, the length of the spring is 11 cm, when the load of mass 150 g. So we substitute the values in equation (1).
(150  g)×g=−k×(11  cm−L) (150g)  g=−k(11  cm−L)......(2)  \left( {150\;{\rm{g}}} \right) \times g = - k \times \left( {11\;{\rm{cm}} - L} \right)\\\ \left( {150g} \right)\;{\rm{g}} = - k\left( {11\;{\rm{cm}} - L} \right)......\left( 2 \right)\\\

Now by equation (1) and (2), we can easily calculate the value of L.

(100g)  g(150g)  g=−k(9  cm−L)  −k(11  cm−L)   ⟹  2(11  cm−L)=3(9  cm−L)   ⟹  22  cm−2L=27  cm−3L ∴L=5  cm \dfrac{{\left( {100g} \right)\;{\rm{g}}}}{{\left( {150g} \right)\;{\rm{g}}}} = \dfrac{{ - k\left( {9\;{\rm{cm}} - L} \right)\;}}{{ - k\left( {11\;{\rm{cm}} - L} \right)}}\\\ \implies 2\left( {11\;{\rm{cm}} - L} \right) = 3\left( {9\;{\rm{cm}} - L} \right)\\\ \implies 22\;{\rm{cm}} - 2L = 27\;{\rm{cm}} - 3L\\\ \therefore L = 5\;{\rm{cm}}

Therefore, the length of the spring when no load is attached is 5 cm.

So, the correct answer is “Option C.

Note:
You need to equate both the equations properly, otherwise you can get your answer wrong. Also, you should not substitute the values on parameters that are going to be cancelled in the solution later on. This can save time and calculation.