Before You Begin

Question #79190

Difficulty: 5/10 | 🟢 Free

The Question

A rigid rod $AB$ of length $L$ and mass $M$ lies on a smooth horizontal $xy$ -plane. The rod has a non-uniform linear mass density $\lambda(x) = \frac{2M}{L^2}x$ , where $x$ is the distance from end $A$ . End $A$ is constrained to slide without friction along a straight guide rail aligned with the $x$ -axis. At $t=0$ , the rod is at rest, making an angle $\theta = 60^\circ$ with the positive $x$ -axis. A time-varying thrust $F(t) = \beta t$ (where $\beta$ is a positive constant) is applied at end $B$ in the $xy$ -plane. The thrust is always perpendicular to the rod and has a positive $y$ -component. Determine the magnitude of the normal force exerted by the guide rail on end $A$ at $t=0^+$ .

Options

A.
$\frac{1}{2} \beta t$
B.
$\frac{4}{7} \beta t$
C.
$\frac{1}{4} \beta t$
D.
$\frac{1}{6} \beta t$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Free Body Diagram Isolation|Rigid Body Constraints|Recoil and Thrust Forces

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the rigorous application of rigid body constraints and non-inertial dynamics in a non-uniform mass system.

A common lethal mistake is applying $\tau_A = I_A \alpha$ (yielding Option D).

Students incorrectly assume that because end A is constrained to the x-axis, it can act as a fixed pivot for torque.

However, end A accelerates in the x-direction ( $a_{Ax} \neq 0$ ), making it a non-inertial reference frame.

Taking torque about A requires a pseudo-force torque $-M(\vec{r}_{C/A} \times \vec{a}_A)$ .

By forcing the student to use the Center of Mass (which shifts to $2L/3$ due to the density gradient) and coupling the translational and rotational equations via the $y_c$ constraint, the problem filters out those who rely on rote application of fixed-axis formulas.

Option B traps students who ignore the density gradient and assume the COM is at $L/2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Free Body Diagram Isolation
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free