JEE Advanced · Physics · 15. Oscillations

Paragraph :
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is $x(t) v s p(t)$ curve in this plane. The arrow on the
curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question :
The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_1$ and $E_2$ are the total mechanical energies respectively. Then,

A $E_1=\sqrt{2} E_2$
B $E_1=2 E_2$
C $E_1=2 E_2$
D $E_1=16 E_2$

Verified Solution

Answer & Solution

Correct Answer

(C) $E_1=2 E_2$

Step-by-step Solution

Detailed explanation

$E=\frac{1}{2} m \omega^2 A^2$
or
$
\begin{aligned}
E & \propto A^2 \\
\frac{E_2}{E_1} & =\left(\frac{A_2}{A_1}\right)^2 \\
& =\left(\frac{a}{2 a}\right)^2
\end{aligned}
$
or $\quad E_1=4 E_2$
$\therefore$ Correct option is (c).
Analysis of Question
(i) Question is simple.
(ii) Since, oscillator is same, $\omega_1=\omega_2$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

A charged particle is introduced at the origin

x = 0, y = 0, z = 0

with a given initial velocity

\vec{v}

. A uniform electric field

\vec{E}

and a uniform magnetic field

\vec{B}

exist everywhere. The velocity

\vec{v}

, electric field

\vec{E}

and a uniform magnetic field

\vec{B}

are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(i) $\vec{E} = E_{0} \hat{z}$	(P) $\vec{B} = - B_{0} \hat{x}$
2. Electron with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{y}$	(ii) $\vec{E} = - E_{0} \hat{y}$	(Q) $\vec{B} = B_{0} \hat{x}$
3.Proton with $\vec{v}$ =0	(iii) $\vec{E} = {- E}_{0} \hat{x}$	(R) $\vec{B} = - B_{0} \hat{y}$
4. Proton with $\vec{v} = 2 \frac{E_{0}}{B_{0}} \hat{x}$	(iv) $\vec{E} = E_{0} \hat{x}$	(S) $\vec{B} = - B_{0} \hat{z}$

In which case will the particle describe a helical path with axis along positive z direction?

JEE Advanced 2017 Medium

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is

0.5 mm

. The circular scale has

100

divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	$0$ division	$4$ divisions
Attempt- $1$ : With wire	$4$ divisions	$20$ divisions
Attempt- $2$ : With wire	$4$ divisions	$16$ divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

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From JEE Advanced

List I describes four systems, each with two particles

A

and

B

in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in

m s^{- 1}

) at time

t = \frac{π}{3} s

	List-I		List-II
(I)	$A$ and $B$ are moving on a horizontal circle of radius $1 m$ with uniform angular speed $ω = 1 rad s^{- 1}$ . The initial angular positions of $A$ and $B$ at time $t = 0$ are $θ = 0$ and $θ = \frac{π}{2}$ respectively.	(P)	$\frac{\sqrt{3} + 1}{2}$
(II)	Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t = 0$ and $t = 0.1 s$ respectively, with the same speed $v = \frac{5 π}{\sqrt{2}} m s^{- 1}$ and at $45 °$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide, $(g = 10 m s^{- 2})$ .	(Q)	$\frac{(\sqrt{3} - 1)}{\sqrt{2}}$
(III)	Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A} = x_{0} \sin \frac{t}{t_{0}}$ and $x_{B} = x_{0} \sin (\frac{t}{t_{0}} + \frac{π}{2})$ respectively, starting from $t = 0$ . Take $x_{0} = 1 m, t_{0} = 1 s$ .	(R)	$\sqrt{10}$
(IV)	Particle $A$ is rotating in a horizontal circular path of radius $1 m$ on the $x y$ plane, with constant angular speed $ω = 1 rad s^{- 1}$ . Particle $B$ is moving up at a constant speed $3 m s^{- 1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S)	$\sqrt{2}$
		(T)	$\sqrt{25 π^{2} + 1}$

Which one of the following options is correct?

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Column I	Column II
A. \(B _2\)	p. Paramagnetic
B. \(N_2\)	q. Undergoes oxidation
C. \(O _2^{-}\)	r. Undergoes reduction
D. \(O _2\)	s. Bond order
	t. Mixing of 's' and 'p' orbitals