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Q. No.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 |
From NCERT
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A light rod \(AB\) is hinged at \(A\) so that it is free to rotate about \(A.\) It is initially horizontal with a small block of mass \(m\) attached at \(B,\) and a spring (constant - \(k\)) holding it vertically up at its mid-point. The time period of vertical oscillations of the system is:
| 1. | \(2 \pi \sqrt{\dfrac{m}{k}} \) | 2. | \(\pi \sqrt{\dfrac{m}{k}} \) |
| 3. | \(4\pi \sqrt{\dfrac{m}{k}}\) | 4. | \(\dfrac{\pi}{2} \sqrt{\dfrac{m}{k}}\) |
Subtopic: Spring mass system |
From NCERT
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The energy of the block is \(E\), and the plane is smooth, the wall at the end \(B\) is smooth. Collisions with walls are elastic. The distance \(AB=l\), the spring is ideal and the spring constant is \(k\). The time period of the motion is:
| 1. | \(2\pi\sqrt{\dfrac{m}{k}}\) |
| 2. | \(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{2m}{E}}\) |
| 3. | \(2\pi\sqrt{\dfrac{m}{k}}+2l\sqrt{\dfrac{2m}{E}}\) |
| 4. | \(\pi\sqrt{\dfrac{m}{k}}+l\sqrt{\dfrac{m}{2E}}\) |
Subtopic: Spring mass system |
From NCERT
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Two identical blocks are connected by an ideal spring and the system is allowed to oscillate, when undergoing horizontal displacements in opposite directions, with the centre-of-mass at rest. \(O\) is the mid-point of the spring, \(A\) is left end point, \(B\) is the right end-point. The motion of \(A\) is described by: \(x_A = A_0 \sin \omega t\) (displacement is taken to be positive rightward).
Call the mid-point of \(O\) and \(B\) as \(C\) and its \(x\text-\)coordinate as \(x_C.\) Then, the motion of the point \(C\) of the spring is described by:
1. \(x_{C}=A_{0} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
2. \(x_{C}=\dfrac{A_{0}}{2} \sin \omega t\)
3. \(x_{C}=\dfrac{A_{0}}{2} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
4. \(x_{C}=\dfrac{A_{0}}{2} \sin (\omega t+\pi)\)
Call the mid-point of \(O\) and \(B\) as \(C\) and its \(x\text-\)coordinate as \(x_C.\) Then, the motion of the point \(C\) of the spring is described by:
1. \(x_{C}=A_{0} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
2. \(x_{C}=\dfrac{A_{0}}{2} \sin \omega t\)
3. \(x_{C}=\dfrac{A_{0}}{2} \sin \left(\omega t+\dfrac{\pi}{2}\right)\)
4. \(x_{C}=\dfrac{A_{0}}{2} \sin (\omega t+\pi)\)
Subtopic: Spring mass system |
From NCERT
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A pendulum-bob \(A,\) after being released as shown, strikes a spring-block system when the bob \(A\) reaches its lowest position; the mass of the bob \(A\) being equal to that of the block (i.e., \(m\)) and the stiffness of the spring being \(k.\) The collision between the block and the bob \((A)\) is elastic. The time period of one complete oscillation is:
| 1. | \(2\pi\sqrt{\dfrac{l}{g}}+2\pi\sqrt{\dfrac{m}{k}}\) | 2. | \(\pi\sqrt{\dfrac{l}{g}}+\pi\sqrt{\dfrac{m}{k}}\) |
| 3. | \(\sqrt{\dfrac{g}{l}}+\sqrt{\dfrac{k}{m}}\) | 4. | \(\dfrac{1}{2\pi}\sqrt{\dfrac{g}{l}}+\dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}}\) |
Subtopic: Spring mass system |
From NCERT
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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 |
From NCERT
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