Before You Begin

Question #42336

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m = 2 \text{ kg}$ is constrained to move along the vertical $y$ -axis. The effective potential energy of the system (including gravity) is given by $U(y) = 3y^4 - 28y^3 + 84y^2 - 96y$ (in Joules, where $y$ is in meters). The particle is initially at rest at its global stable equilibrium position. An impulsive force is applied to the particle, giving it just enough energy to cross the potential barrier and enter the meta-stable equilibrium well. Determine the minimum impulse $J_{\text{min}}$ required to achieve this transition, and the angular frequency $\omega'$ of small oscillations if the particle were to settle at this new meta-stable equilibrium.

Options

A.
$8\sqrt{2} \text{ N s}, \quad 3\sqrt{2} \text{ rad/s}$
B.
$6\sqrt{3} \text{ N s}, \quad 3\sqrt{2} \text{ rad/s}$
C.
$8\sqrt{2} \text{ N s}, \quad 6 \text{ rad/s}$
D.
$6\sqrt{3} \text{ N s}, \quad 6 \text{ rad/s}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Collision Impulses|Vertical Oscillations|Equilibrium Stability|Approximation (Small Angle)

Why This Question Is Tricky

Common Mistakes

This question tests the integration of impulsive mechanics with potential well analysis (bifurcation of equilibrium).

A common trap is calculating the required energy by taking the difference between the two minima ( $\Delta U = U(1) - U(4) = 27 \text{ J}$ ) rather than the energy required to cross the intermediate potential barrier ( $U(2) - U(4) = 32 \text{ J}$ ).

Additionally, students often incorrectly assume the effective spring constant (curvature of the potential) remains unchanged after transitioning to a new equilibrium state, leading them to use the initial angular frequency ( $\omega = 6 \text{ rad/s}$ ) instead of recalculating $U''(y)$ at the meta-stable point.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Collision Impulses
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free