Q. No.A solid sphere \((A)\) of mass \(5~\text{m}\) and a spherical shell \((B)\) of mass \(m\), both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of \(A\) and \(B,\) they start rolling without slipping with an acceleration of \(\alpha_A\) and \(\alpha_B,\) respectively. The ratio of \(\alpha_A\) and \(\alpha_B\) is:
1. \(5 : 21\)
2. \(6: 10\)
3. \(21:25\)
4. \(1 : 5\)
1. \(5 : 21\)
2. \(6: 10\)
3. \(21:25\)
4. \(1 : 5\)
Subtopic: Rotational Motion: Dynamics |
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A solid sphere of radius \(4~\text{cm}\) and mass \(5~\text{kg}\) is rotating (rotation axis is passing through the centre of the sphere) with an angular velcocity of \(1200~\text{rpm}\). It is brought to rest in \(10~\text{s}\) by applying a constant torque. The torque applied and the number of rotation it made before it come to rest are _____________and _____________________respectively.
1. \(0.128 \pi~ \text{Nm}, 100\)
2. \( 0.0128 \pi ~\text{Nm}, 50\)
3. \(0.128 \pi ~\text{Nm}, 50\)
4. \(0.0128 \pi ~\text{Nm}, 100\)
1. \(0.128 \pi~ \text{Nm}, 100\)
2. \( 0.0128 \pi ~\text{Nm}, 50\)
3. \(0.128 \pi ~\text{Nm}, 50\)
4. \(0.0128 \pi ~\text{Nm}, 100\)
Subtopic: Rotational Motion: Dynamics |
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The position of an object having mass \(0.1 \) kg as a function of time \(t\) is given as \(\vec{r}=\left(10 t^2 \hat{i}+5 t^3 \hat{j}\right)~ \text{m}.\) At \(t=1~\text{s}\), which of the following statements are correct?
Choose the correct answer from the options given below:
1. \(\mathrm{A,B} \) and \(\mathrm{C}\) only
2. \(\mathrm{B,C}\) and \(\mathrm{D}\) only
3. \(\mathrm{A,C}\) and \(\mathrm{D}\) only
4. \(\mathrm{A,B}\) and \(\mathrm{D}\) only
| \(\mathrm A.\) | The linear momentum \(\vec{p}=(2 \hat{i}+1.5 \hat{j}) ~\text{kg} \cdot \text{m/s}.\) |
| \(\mathrm B.\) | The force acting on the object \(\overrightarrow{F}=(2 \hat{i}+3 \hat{j}) ~\text{N} .\) |
| \(\mathrm C.\) | The angular momentum of the object about its origin \(\overrightarrow{L}=15 \hat{k} ~\text{J s}.\) |
| \(\mathrm D.\) | The torque acting on the object about its origin \(\vec{\tau}=20 \hat{k} ~\text{Nm} .\) |
1. \(\mathrm{A,B} \) and \(\mathrm{C}\) only
2. \(\mathrm{B,C}\) and \(\mathrm{D}\) only
3. \(\mathrm{A,C}\) and \(\mathrm{D}\) only
4. \(\mathrm{A,B}\) and \(\mathrm{D}\) only
Subtopic: Rotational Motion: Dynamics |
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A particle is rotating in a circular path and at any instant its motion can be described as \(\theta=\dfrac{5 t^4}{40}-\dfrac{t^3}{3}\). The angular acceleration of the particle after \(10\) seconds is: (in \(\text{rad/s}^{2}\))
1. \(150\)
2. \(120\)
3. \(130\)
4. \(170\)
1. \(150\)
2. \(120\)
3. \(130\)
4. \(170\)
Subtopic: Rotational Motion: Dynamics |
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An object of uniform density rolls up the curved path with initial velocity \(v_0\) as shown in the figure. If the maximum height attained by an object is \(\dfrac{7 v_{{0}}^2}{10{g}}\) (\(g=\)acceleration due to gravity), the object is a:
1. solid cylinder
2. ring
3. disc
4. solid sphere
1. solid cylinder
2. ring
3. disc
4. solid sphere
Subtopic: Rotational Motion: Dynamics |
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A solid cylinder having radius \(R\) and length \(L\) is slipping on a rough horizontal plane. At time \(t=0\) the cylinder has a translational velocity \(v_0=49~\text{m/s}\), perpendicular to axis and a rotational velocity \(\dfrac{v_0}{4R}\) about the centre. The time taken by the cylinder to start rolling is: (in seconds)
(coefficient of kinetic friction \(\mu_K=0.25 \text { and } g=9.8 ~\text{m/s}^2\))
1. \(15\)
2. \(5\)
3. \(10\)
4. \(7.5\)
(coefficient of kinetic friction \(\mu_K=0.25 \text { and } g=9.8 ~\text{m/s}^2\))
1. \(15\)
2. \(5\)
3. \(10\)
4. \(7.5\)
Subtopic: Rotational Motion: Dynamics |
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A uniform bar of length \(12~\text{cm}\) and mass \(20m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2m\) are moving in opposite directions with same speed of \(v\) and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency \(\omega\). The ratio of \(v\) and \(\omega\) is:
1. \(33\)
2. \(2\sqrt{88}\)
3. \(66\)
4. \(32\)
1. \(33\)
2. \(2\sqrt{88}\)
3. \(66\)
4. \(32\)
Subtopic: Rotational Motion: Dynamics |
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Two masses \(m\) and \(2m\) are connected by a light string going over a pulley (disc) of mass \(30m\) with radius \(r =0.1~\text{m}.\) The pulley is mounted in a vertical plane and it is free to rotate about its axis. The \(2m\) mass is released from rest and its speed when it has descended through a height of \(3.6~\text{m}\) is: (in m/s)
(Assume string does not slip and \(g = 10~ \text{m/s}^2)\)
1. \(2\)
2. \(3\)
3. \(6\)
4. \(9\)
(Assume string does not slip and \(g = 10~ \text{m/s}^2)\)
1. \(2\)
2. \(3\)
3. \(6\)
4. \(9\)
Subtopic: Rotational Motion: Dynamics |
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Two small balls with masses \(m\) and \(2m\) are attached to both ends of a rigid rod of length \(d\) and negligible mass. If angular momentum of this system is \(L\) about an axis \((A)\) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about \(A\) is:
1. \( \dfrac{{3L}}{2{md}^2}\)
2. \(\dfrac{2 {L}}{{md}^2}\)
3. \(\dfrac{{4L}}{{3md}^2}\)
4. \(\dfrac{2 {L}}{5 {md}^2}\)
1. \( \dfrac{{3L}}{2{md}^2}\)
2. \(\dfrac{2 {L}}{{md}^2}\)
3. \(\dfrac{{4L}}{{3md}^2}\)
4. \(\dfrac{2 {L}}{5 {md}^2}\)
Subtopic: Rotational Motion: Dynamics |
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A thin uniform rod \((X)\) of mass \( M\) and length \(L\) is pivoted at a height \(\left(\dfrac{{L}}{3}\right)\)as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is: \((g=\) gravitational acceleration\()\)
1. \(\sqrt{\dfrac{3}{2} \dfrac{{g}}{{L}}}\)
2. \(\dfrac{3}{\sqrt{2}} \sqrt{\dfrac{{g}}{{L}}}\)
3. \(\dfrac{1}{\sqrt{2}} \sqrt{\dfrac{{g}}{{L}}}\)
4. \(\sqrt{\dfrac{3 {g}}{{L}}}\)
1. \(\sqrt{\dfrac{3}{2} \dfrac{{g}}{{L}}}\)
2. \(\dfrac{3}{\sqrt{2}} \sqrt{\dfrac{{g}}{{L}}}\)
3. \(\dfrac{1}{\sqrt{2}} \sqrt{\dfrac{{g}}{{L}}}\)
4. \(\sqrt{\dfrac{3 {g}}{{L}}}\)
Subtopic: Rotational Motion: Dynamics |
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Level 3: 35%-60%
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