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Q. No.Moment of inertia about an axis \(AB~\) for a rod of mass \(40~\text{kg}\) and length \(3~\text{m}\) is same as that of a solid sphere of mass of \(10~\text{kg}\) and radius \(R\) about an axis parallel to \(AB~\) axis with separation of \(3~\text{m}\) as shown in figure below. The value of \(R\) is given as \(\sqrt{\dfrac{\alpha}{2}}\). The value of \(\alpha\)
1. \(5\)
2. \(10\)
3. \(15\)
4. \(20\)
1. \(5\)
2. \(10\)
3. \(15\)
4. \(20\)
Subtopic: Moment of Inertia |
Level 4: Below 35%
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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 |
Level 4: Below 35%
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Two identical bodies \(A\) and \(B\) of equal masses have initial velocities \(\overrightarrow{v_1}=4 \hat{i} ~\text{m/s}\) and \(\overrightarrow{v_2}=4 \hat{j}~ \text{m/s}\) respectively. The body \(A\) has acceleration \(\overrightarrow{a_1}=6 \hat{i}+6 \hat{j}~\text{m/s}^2\) while the acceleration of the other body \(B\) is zero. In what type of path does the centre of mass of two bodies move?
1. circular
2. parabolic
3. straight line
4. elliptical
1. circular
2. parabolic
3. straight line
4. elliptical
Subtopic: Center of Mass |
Level 3: 35%-60%
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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 |
Level 4: Below 35%
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The position of centre of mass of three masses \(2~\text{kg}.\) \(3~\text{kg}\) and \(15~\text{kg}\) placed with respected to mid point \((p)\) of normal bisector, as shown in the figure is:
1. \( \left(\dfrac{\sqrt{3}}{4}, 1.25\right)~\)
2. \(\left(\dfrac{\sqrt{3}}{4}, 1.0\right) ~\)
3. \( (0,0) \)
4. \((1.25,0)\)
1. \( \left(\dfrac{\sqrt{3}}{4}, 1.25\right)~\)
2. \(\left(\dfrac{\sqrt{3}}{4}, 1.0\right) ~\)
3. \( (0,0) \)
4. \((1.25,0)\)
Subtopic: Center of Mass |
75%
Level 2: 60%+
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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 |
77%
Level 2: 60%+
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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 |
84%
Level 1: 80%+
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A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first \(2~\text{s}\) it rotates through an angle \(\theta_1\) and in the next \(2~\text{s}\) it rotates through an angle \(\theta_2\). The ratio \(\dfrac{\theta_2}{\theta_1}\) is:
1. \(6\)
2. \(3\)
3. \(4\)
4. \(\dfrac{1}{3}\)
1. \(6\)
2. \(3\)
3. \(4\)
4. \(\dfrac{1}{3}\)
Subtopic: Rotational Motion: Kinematics |
83%
Level 1: 80%+
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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 |
Level 4: Below 35%
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A solid sphere of mass \(M\) \(R\) \(M/8\) \(r\) \(t\) \(2R~\) \(I_1\) \(I_2\)
1. \(35\)
2. \(70\)
3. \(140\)
4. \(210\)
1. \(35\)
2. \(70\)
3. \(140\)
4. \(210\)
Subtopic: Moment of Inertia |
71%
Level 2: 60%+
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