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Q. No.A block of mass \(m\) is attached to two fixed light springs of identical stiffness \(k,\) and the block rests on the \(x\text-y\) plane. The springs are along the \(x\) & \(y\) axes. The system is viewed from above. The block undergoes small oscillations along a line which makes \(45^\circ\) with the \(x\text-\)axis. The angular frequency of these oscillations is:
1. \(\sqrt{\Large\frac{2k}{m}}\)
2. \(\sqrt{\Large\frac{\sqrt2k}{m}}\)
3. \(\sqrt{\Large\frac{2\sqrt2k}{m}}\)
4. \(\sqrt{\Large\frac{k}{m}}\)
Subtopic: Combination of Springs |
Level 4: Below 35%
Hints
A particle is subjected to two SHMs, one along the \(x\text-\)axis and the other along the \(y\text-\)axis:
\(x=A\sin\omega t\\ y=A\sin(\omega t+\pi)\)
The resulting motion is:
\(x=A\sin\omega t\\ y=A\sin(\omega t+\pi)\)
The resulting motion is:
| 1. | Uniform circular motion. |
| 2. | Elliptic motion. |
| 3. | SHM along a straight line. |
| 4. | SHM along a circle. |
Subtopic: Simple Harmonic Motion |
51%
Level 3: 35%-60%
Hints
A spring 'lengthens' by \(l\) when a block of mass \(m\) is suspended from it. This block is suspended from the same spring and the system is allowed to oscillate vertically, in space where the gravity is \(\bigg({\large\frac19}\bigg)^{\text{th}}\) of its original value. The time period of small oscillations is:
(where \(g\) is the acceleration due to gravity on the surface of the Earth)
| 1. | \(2\pi{\sqrt{\large\frac{l}{g}}}\) | 2. | \(6\pi{\sqrt{\large\frac{l}{g}}}\) |
| 3. | \(2\pi{\sqrt{\large\frac{9l}{8g}}}\) | 4. | \(2\pi{\sqrt{\large\frac{3l}{g}}}\) |
Subtopic: Spring mass system |
Level 4: Below 35%
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Hints
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Two SHMs given by their displacements along the respective directions are superposed: \(x=A\sin\omega t~~~~(1^{\text{st}}~\text{SHM along }x\text{-axis})\\ y=A\sin\bigg(\omega t+{\large\frac{\pi}{2}}\bigg)~~~~(2^{\text{nd}}~\text{SHM along }y\text{-axis}).\)
The resultant motion is:
The resultant motion is:
| 1. | SHM along a straight line |
| 2. | SHM along a circular arc |
| 3. | uniform circular motion |
| 4. | motion along an elliptic path |
Subtopic: Types of Motion |
Level 3: 35%-60%
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Hints
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Two identical masses are connected by a spring of spring constant \(k,\) and the individual masses are observed to undergo SHM with their centre of mass remaining at rest. The amplitude of oscillation of one of the masses is \(A.\) The total energy of oscillation is:
1. \({\Large\frac{1}{2}}kA^2\)
2. \(kA^2\)
3. \(2kA^2\)
4. \(4kA^2\)
1. \({\Large\frac{1}{2}}kA^2\)
2. \(kA^2\)
3. \(2kA^2\)
4. \(4kA^2\)
Subtopic: Energy of SHM |
Level 4: Below 35%
Hints
A spring-mass system is undergoing small oscillations of amplitude \(A.\) When the block is at its mean position, it is given an impulse \(J\) in the direction of its motion, and its new amplitude is \(A'.\) Then, (given \(\alpha,\beta,\gamma\) are constants)
1. \(A'=A+\alpha J\)
2. \(A'^{\Large^2}=A^2+\alpha J^2\)
3. \(A'^{\Large^2}=A^2+\alpha J+\beta A\)
4. \(A'^{\Large^2}=A^2+\alpha J^2+\beta AJ\)
1. \(A'=A+\alpha J\)
2. \(A'^{\Large^2}=A^2+\alpha J^2\)
3. \(A'^{\Large^2}=A^2+\alpha J+\beta A\)
4. \(A'^{\Large^2}=A^2+\alpha J^2+\beta AJ\)
Subtopic: Spring mass system |
Level 3: 35%-60%
Hints
Two particles undergo SHM along the same straight line, moving with the same frequency and amplitude but with a phase difference of \(60^\circ\) between each other. If they have a maximum speed of \(v_0,\) the maximum relative velocity between them will be:
| 1. | \(v_0\) | 2. | \(2v_0\) |
| 3. | \({\dfrac{\sqrt3}{2}}v_0\) | 4. | \(\sqrt3v_0 \) |
Subtopic: Phasor Diagram |
Level 3: 35%-60%
Hints
A physical pendulum consists of a uniform rod \(AB\) of mass \(m\) and length \(L,\) suspended from one end \(A\) – so as to rotate freely under gravity. If it is displaced slightly from its mean position, it executes SHM. Let the maximum kinetic energy of the rod be \(E_0.\)
If the time period of a simple pendulum of the same length is \(T_0,\) then the time period of this pendulum is:
If the time period of a simple pendulum of the same length is \(T_0,\) then the time period of this pendulum is:
| 1. | \(\sqrt{\dfrac{2}{3}} T_0\) | 2. | \(\sqrt{\dfrac{1}{12}} T_0\) |
| 3. | \(\sqrt{\dfrac{3}{2}} T_0\) | 4. | \(T_0\) |
Subtopic: Angular SHM |
53%
Level 3: 35%-60%
Hints
Consider a particle undergoing uniform circular motion, with angular speed \(\omega.\) The projection of its motion on any straight line in its plane is:
| 1. | periodic but not SHM |
| 2. | SHM with angular frequency \(\omega\) |
| 3. | SHM but angular frequency \(2\omega\) |
| 4. | neither SHM nor periodic |
Subtopic: Phasor Diagram |
51%
Level 3: 35%-60%
Hints
A small block of mass \(m\) slides a distance \(L\) down a smooth incline and rebounds elastically back up. The period of the motion is: (using standard symbols where necessary)
| 1. | \(\sqrt{\dfrac{4L\sin\theta}{g}} \) | 2. | \(\sqrt{\dfrac{8L\sin\theta}{g}} \) |
| 3. | \(\sqrt{\dfrac{8L}{g\sin\theta}} \) | 4. | \(\sqrt{\dfrac{4L}{g\sin\theta}} \) |
Subtopic: Linear SHM |
Level 3: 35%-60%
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