This question has Statement-1 and statement-2. Of the four choices given after the Statements, choose the one that best describes the two statements. If two springs S1 and S2 of force constants k1 and k2, respectively, are stretched by the same force, it is found that more work is done on spring S1 than on spring S2. Statement-1 : If stretched by the same amount, work done on S1, will be more than that on S2. Statement -2 : k1

Engineering

Physics

Work Done by a Constant Force

Newtons Second Law of Motion

SHM of Spring Block System

Question

This question has Statement-1 and statement-2. Of the four choices given after the Statements, choose the one that best describes the two statements.

If two springs S₁ and S₂ of force constants k₁ and k₂, respectively, are stretched by the same force, it is found that more work is done on spring S₁ than on spring S₂.

Statement-1 : If stretched by the same amount, work done on S₁, will be more than that on S₂.

Statement -2 : k₁ < k₂

Statement-1 is true, Statement-2 is true and Statement-2 is the correct explanation of Statement-1.

Statement-1 is true, Statement-2 is true and Statement-2 is not the correct explanation of Statement-1

Statement-1 is true, Statement-2 is false

Statement-1 is false, Statement-2 is true.

Stretched by same force hence k₁x₁ = k₂x₂

More work is done on spring-1 hence

$\frac{1}{2} k_{1} x_{1}^{2} > \frac{1}{2} k_{2} x_{2}^{2}$

$\Rightarrow$ x₁ > x₂

$\Rightarrow$ k₁ < k₂

This question involves analyzing two statements about springs and work done. Let's break down the concepts step by step.

Step 1: Understand the Given Information

Two springs S₁ and S₂ have force constants k₁ and k₂. They are stretched by the same force F. It is given that more work is done on S₁ than on S₂.

Step 2: Work Done by a Force on a Spring

The work done W in stretching a spring is given by:

W = \frac{1}{2} k x^{2}

where k is the spring constant and x is the extension.

Also, from Hooke's law, the force F required to stretch a spring by x is:

F = k x

Step 3: Analyze the Given Condition (Same Force)

For the same force F, the extension x in each spring is:

x_{1} = \frac{F}{k_{1}}, x_{2} = \frac{F}{k_{2}}

Work done on S₁ is:

W_{1} = \frac{1}{2} k_{1} {(\frac{F}{k_{1}})}^{2} = \frac{F^{2}}{2 k_{1}}

Similarly, work on S₂ is:

W_{2} = \frac{F^{2}}{2 k_{2}}

Given that W₁ > W₂, we have:

\frac{F^{2}}{2 k_{1}} > \frac{F^{2}}{2 k_{2}} \Rightarrow \frac{1}{k_{1}} > \frac{1}{k_{2}} \Rightarrow k_{2} > k_{1} \Rightarrow k_{1} < k_{2}

So, Statement-2 is true: k₁ < k₂.

Step 4: Analyze Statement-1 (Same Extension)

Statement-1 claims that if stretched by the same amount, work done on S₁ is more than on S₂.

For the same extension x, work done is:

W = \frac{1}{2} k x^{2}

So, W ∝ k. Since k₁ < k₂, we have:

W_{1} = \frac{1}{2} k_{1} x^{2} < \frac{1}{2} k_{2} x^{2} = W_{2}

Thus, for the same extension, work done on S₁ is less than on S₂, not more. Therefore, Statement-1 is false.

Step 5: Final Answer

Statement-1 is false, and Statement-2 is true. So, the correct option is:

Statement-1 is false, Statement-2 is true.

Detailed Solution

Step 1: Understand the Given Information

Step 2: Work Done by a Force on a Spring

Step 3: Analyze the Given Condition (Same Force)

Step 4: Analyze Statement-1 (Same Extension)

Step 5: Final Answer

Related Topics and Formulae

Detailed Solution

Step 1: Understand the Given Information

Step 2: Work Done by a Force on a Spring

Step 3: Analyze the Given Condition (Same Force)

Step 4: Analyze Statement-1 (Same Extension)

Step 5: Final Answer

Related Topics and Formulae

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