NEET 2026 ALL Q #1924
COMPETENCY BASED
APPLY
4 Marks 2026 NTA-RE-NEET-2026 MCQ SINGLE
A particle of mass M moves along a horizontal x axis from $x=0$ to $x=L$. The coefficient of kinetic friction varies as a function of x as $\mu_{k}(x)=\mu_{0}-\alpha x,$ where $\mu_0$, $\alpha$ are constants of appropriate dimensions, so that $\mu_{k}(L)=0$. The total work done by the frictional force during the motion is $n\mu_0 MgL$, where g is the acceleration due to gravity. The value of n is:
(A) $\frac{1}{2}$
(B) 3
(C) 1
(D) $\frac{1}{3}$
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Correct Answer: A

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Detailed Solution

Step 1: Identify the expression for frictional force

The frictional force $f_k$ acting on the particle is given by $f_k = \mu_k(x) N$. Since the motion is on a horizontal surface, the normal force $N = Mg$. Thus, $f_k(x) = (\mu_0 - \alpha x) Mg$.

Step 2: Determine the constant alpha

We are given that $\mu_k(L) = 0$. Substituting this into the expression: $$ \mu_0 - \alpha L = 0 \implies \alpha = \frac{\mu_0}{L} $$

Step 3: Calculate the work done by friction

Work done by friction is the integral of force over the displacement. Since friction opposes motion, $W = -\int_{0}^{L} f_k(x) dx$: $$ W = -\int_{0}^{L} (\mu_0 - \alpha x) Mg \, dx $$ $$ W = -Mg \left[ \mu_0 x - \frac{\alpha x^2}{2} \right]_{0}^{L} $$

Step 4: Substitute alpha and solve

Substitute $\alpha = \frac{\mu_0}{L}$ into the integral result: $$ W = -Mg \left( \mu_0 L - \frac{\mu_0}{L} \cdot \frac{L^2}{2} \right) $$ $$ W = -Mg \left( \mu_0 L - \frac{\mu_0 L}{2} \right) = -\frac{1}{2} \mu_0 MgL $$ The magnitude of work done is $\frac{1}{2} \mu_0 MgL$. Comparing this with $n \mu_0 MgL$, we get $n = \frac{1}{2}$.

Final Answer: 1/2

AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the calculus-based definition of work done by a variable force to a specific physical scenario involving friction.
Knowledge Dimension: PROCEDURAL
Justification: The problem requires a sequence of steps: defining the variable force, determining the constant alpha, and performing definite integration.
Syllabus Audit: In the context of NEET, this is classified as COMPETENCY. This question tests the ability to handle variable forces, a key skill for high-scoring NEET aspirants, moving beyond simple formula substitution.

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