IIT-JEE 2007 · Paper 2 · Physics Q20

PhysicsMatrix match+6Question 20 of 66

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I

A
The object moves on the $x$ -axis under a conservative force in such a way that its "speed" and "position" satisfy $v=c_1\sqrt{c_2-x^2}$ , where $c_1$ and $c_2$ are positive constants.
B
The object moves on the $x$ -axis in such a way that its velocity and its displacement from the origin satisfy $v=-kx$ , where $k$ is a positive constant.
C
The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration $a$ . The motion of the object is observed from the elevator during the period it maintains this acceleration.
D
The object is projected from the earth's surface vertically upwards with a speed $2\sqrt{GM_e/R_e}$ , where, $M_e$ is the mass of the earth and $R_e$ is the radius of the earth. Neglect forces from objects other than the earth.

Column II

P
The object executes a simple harmonic motion.
Q
The object does not change its direction.
R
The kinetic energy of the object keeps on decreasing.
S
The object can change its direction only once.

Answer and worked solution

Official answer: A → P; B → Q, R; C → P; D → Q, R

A: $v^2=c_1^2(c_2-x^2)$ ; differentiating, $a=v\frac{dv}{dx}=-c_1^2x$ : SHM. → P.
B: $v=-kx$ gives $x=x_0e^{-kt}$ : the object approaches the origin without ever turning back, and its speed $k|x|$ keeps decreasing. → Q, R.
C: In the elevator frame a pseudo-force $ma$ acts downward. The equilibrium position shifts down by $\frac{ma}k$ , and the object, released from the old position, oscillates about the new one: SHM. → P.
D: The speed $2\sqrt{GM_e/R_e}$ exceeds the escape speed $\sqrt{2GM_e/R_e}$ , so the object never comes back: it keeps moving outward while slowing down. → Q, R.

The official key also accepts D → R alone.

Answer: A → P; B → Q, R; C → P; D → Q, R

Topics:Simple harmonic motion Escape velocity Pseudo forces and non-inertial frames Motion in a straight line

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IIT-JEE 2007 · Paper 2, Physics question 20