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Q. No.A circular hole of radius \(\dfrac{{R}}{4}\) is made in a thin uniform disc having a mass \({M}\) and a radius \({R},\) as shown in the figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point \(O\) and perpendicular to the plane of the disc is:
1.
\(\dfrac{219~{MR}^2}{256}\)
2.
\(\dfrac{197{MR}^2}{256}\)
3.
\(\dfrac{19{MR}^2}{512}\)
4.
\(\dfrac{237{MR}^2}{512}\)
Subtopic: Moment of Inertia |
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A thin circular disk is in the \(xy\) plane as shown in the figure. The ratio of its moment of inertia about \(z\) and \(z'\) axes will be:
1. \(1 : 4\)
2. \(1 : 5\)
3. \(1 : 3\)
4. \(1 : 2\)
1. \(1 : 4\)
2. \(1 : 5\)
3. \(1 : 3\)
4. \(1 : 2\)
Subtopic: Moment of Inertia |
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Suppose that the angular velocity of the rotation of the earth is increased. Then as a consequence:
| 1. | Weight of the objects everywhere on the earth will decrease |
| 2. | Weight of the objects everywhere on the earth will increase |
| 3. | Except at the poles weight of the object on the earth will decrease |
| 4. | There will be no change in weight anywhere on the earth. |
Subtopic: Rotational Motion: Dynamics |
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A force of \(40 ~\text N\) acts on a point \(B\) at the end of an \(\mathrm L\text-\)shaped object as shown in the figure. The angle \(\theta\) that will produce the maximum moment of the force about the point \(A\) is given by:
1. \(\tan \theta = 4 \)
2. \(\tan \theta = \dfrac{1}{4} \)
3. \(\tan \theta = \dfrac{1}{2} \)
4. \(\tan\theta = 2 \)
1. \(\tan \theta = 4 \)
2. \(\tan \theta = \dfrac{1}{4} \)
3. \(\tan \theta = \dfrac{1}{2} \)
4. \(\tan\theta = 2 \)
Subtopic: Torque |
Level 3: 35%-60%
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The moment of inertia of an equilateral triangular lamina \(ABC,\) about the axis passing through its centre \({O}\) and perpendicular to its plane is \({I}_o\) as shown in the figure. A cavity \(DEF\) is cut out from the lamina, where \(D, E, F\) are the mind points of the sides. The moment of inertia of the remaining part of the lamina about the same axis is:
1. \(\frac{31}{32}I_0\)
2. \(\frac{3}{4}I_0\)
3. \(\frac{7}{8}{I}_0\)
4. \(\frac{15}{16}{I}_0\)
1. \(\frac{31}{32}I_0\)
2. \(\frac{3}{4}I_0\)
3. \(\frac{7}{8}{I}_0\)
4. \(\frac{15}{16}{I}_0\)
Subtopic: Moment of Inertia |
Level 3: 35%-60%
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A uniform disc of radius \({R}\) and mass \({M}\) is free to rotate only about its axis. A string is wrapped over its rim and a body of mass \({m}\) is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is:
1. \(\frac{2{Mg}}{2{m + M}}\)`
2. \(\frac{2{mg}}{2{m + M}}\)
3. \(\frac{2{Mg}}{2{M + m}}\)
4. \(\frac{2{mg}}{2{M + m}}\)
1. \(\frac{2{Mg}}{2{m + M}}\)`
2. \(\frac{2{mg}}{2{m + M}}\)
3. \(\frac{2{Mg}}{2{M + m}}\)
4. \(\frac{2{mg}}{2{M + m}}\)
Subtopic: Torque |
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A uniform rod \(AB \) is suspended from a point \(X, \) at a variable distance \(x \) from \(A , \) as shown in the figure. To make the rod horizontal, a mass \(m\) is suspended from its end \(A.\) A set of \((m,x) \) values is recorded. The appropriate variables that are given a straight line, when plotted are:
| 1. | \(m,x^2 \) | 2. | \(m,\dfrac{1}{x^2}\) |
| 3. | \(m,\dfrac{1}{x} \) | 4. | \(m,x \) |
Subtopic: Torque |
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A physical balance operates on the principle of moments. When a weight of \(5~\text{mg}\) is placed on the left pan, the beam becomes horizontal. The two empty pans are of equal mass. Which of the following statements is correct?
| 1. | The left arm is shorter than the right arm. |
| 2. | Both arms are of equal length. |
| 3. | Every object measured with this balance appears lighter than its actual weight. |
| 4. | The left arm is longer than the right arm. |
Subtopic: Torque |
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A uniform solid cylindrical roller of mass '\(m\)' is being pulled on a horizontal surface with force \(F\) parallel to the surface and applied at its centre. If the acceleration of the cylinder is '\(a\)' and it is rolling without slipping then the value of '\(F\)' is:
1. \({ma}\)
2. \(2{ma}\)
3. \(\frac{5}{3}{ma}\)
4. \(\frac{3}{2}{ma}\)
1. \({ma}\)
2. \(2{ma}\)
3. \(\frac{5}{3}{ma}\)
4. \(\frac{3}{2}{ma}\)
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A uniform thin rod \({AB}\) of length \({L}\) has linear mass density \({\mu(x)=}a+\frac{bx}{L},\) where \({x}\) is measured from \({A}.\) If the centre of mass of the rod lies at a distance of \(\left(7\over 12\right)L\) from \({A},\) then \({a}\) and \({b}\) are related as:
1. \({a=2b}\)
2. \({2a=b}\)
3. \({a=b}\)
4. \({3a=2b}\)
1. \({a=2b}\)
2. \({2a=b}\)
3. \({a=b}\)
4. \({3a=2b}\)
Subtopic: Center of Mass |
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Level 2: 60%+
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