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JEE Advance - Physics (2007 - Paper 2 Offline - No. 20)

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

Column I Column II
(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. (P) The object executes a simple harmonic motion.
(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. (Q) The object does not change its direction.
(C) The object is attached to one end of a massless 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. (R) The kinetic energy of the object keeps on decreasing
(D) The object is projected from the earth's surface vertically upwards with a speed $$2\sqrt {GMe/{\mathop{\rm Re}\nolimits} } $$, 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. (S) The object can change its direction only once.

$$[\mathrm{A} \rightarrow(\mathrm{P}); \mathrm{B} \rightarrow(\mathrm{Q}, \mathrm{R}); \mathrm{C} \rightarrow(\mathrm{P}); \mathrm{D} \rightarrow(\mathbf{Q}, \mathrm{R})]$$
$$[\mathrm{A} \rightarrow(\mathrm{Q, R}); \mathrm{B} \rightarrow(\mathrm{Q}, \mathrm{R}); \mathrm{C} \rightarrow(\mathrm{P}); \mathrm{D} \rightarrow(\mathbf{S}, \mathrm{R})]$$
$$[\mathrm{A} \rightarrow(\mathrm{P, S}); \mathrm{B} \rightarrow(\mathrm{Q}, \mathrm{R}); \mathrm{C} \rightarrow(\mathrm{P}); \mathrm{D} \rightarrow(\mathbf{R})]$$
$$[\mathrm{A} \rightarrow(\mathrm{P, R}); \mathrm{B} \rightarrow(\mathrm{Q}, \mathrm{R}); \mathrm{C} \rightarrow(\mathrm{P}); \mathrm{D} \rightarrow(\mathbf{S})]$$

Explanation

Reason:

IIT-JEE 2007 Paper 2 Offline Physics - Waves Question 11 English Explanation 1

In above figure shown that S.H.M

Therefore, $$f_s=Kx$$

Given, $$v = {c_1}\sqrt {{c_2} - {x^2}} $$ ..... (i)

For a simple harmonic motion,

$$v = \omega \sqrt {{A^2} - {x^2}} $$ ..... (ii)

On comparing eq. (i) & (ii) therefore it shows the S.H.M of particle.

A $$\to$$ P

(b) Given,

$$v=-kx$$

At $$x=0,v=0$$

$$x=2,v=-2k$$

$$x=-5,v=+5k$$

The above consideration show that particle is coming from minus x-axis.

IIT-JEE 2007 Paper 2 Offline Physics - Waves Question 11 English Explanation 2

Velocity is decreasing with time.

Graph:

IIT-JEE 2007 Paper 2 Offline Physics - Waves Question 11 English Explanation 3

Therefore, no change in direction, velocity keeps on decreasing. So that kinetic energy is decreasing with time.

B $$\to q_1r$$

(C) From the reference frame of the observer in the elevator a constant force acts on the object. Hence the motion of the object will remain simple harmonic.

Therefore, option 1 is correct.

Velocity of object

$$v = 2\sqrt {G{{{M_e}} \over {{R_e}}}} = \sqrt 2 \sqrt {2G{{{M_e}} \over {{R_e}}}} = \sqrt 2 {v_e}$$

Since, $$v > {v_e}$$ (where $$v_e$$ is escape velocity) object will move out of earth.

Hence, option (Q) and (R) are correct.

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