Q. No.From a circular ring of mass \({M}\) and radius \(R,\) an arc corresponding to a \(90^\circ\) sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is \(K\) times \(MR^2.\) The value of \(K\) will be:
1.
\(\frac{1}{4}\)
2.
\(\frac{1}{8}\)
3.
\(\frac{3}{4}\)
4.
\(\frac{7}{8}\)
A uniform rod of length \(200~ \text{cm}\) and mass \(500~ \text g\) is balanced on a wedge placed at \(40~ \text{cm}\) mark. A mass of \(2~\text{kg}\) is suspended from the rod at \(20~ \text{cm}\) and another unknown mass \(m\) is suspended from the rod at \(160~\text{cm}\) mark as shown in the figure. What would be the value of \(m\) such that the rod is in equilibrium?
(Take \(g=10~( \text {m/s}^2)\)
| 1. | \({\dfrac 1 6}~\text{kg}\) | 2. | \({\dfrac 1 {12}}~ \text{kg}\) |
| 3. | \({\dfrac 1 2}~ \text{kg}\) | 4. | \({\dfrac 1 3}~ \text{kg}\) |
Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration \(\vec{a}\). The centre of mass has an acceleration:
| 1. | zero | 2. | \(\vec{a}/2\) |
| 3. | \(\vec{a}\) | 4. | \(2\vec{a}\) |
A body falling vertically downwards under gravity breaks into two parts of unequal masses. The centre of mass of the two parts taken together shifts horizontally towards:
| 1. | heavier piece |
| 2. | lighter piece |
| 3. | does not shift horizontally |
| 4. | depends on the vertical velocity at the time of breaking |
Let \(\overrightarrow A\) be a unit vector along the axis of rotation of a purely rotating body and \(\overrightarrow B\) be a unit vector along the velocity of a particle P of the body away from the axis. The value of \(\overrightarrow A.\overrightarrow B\) is:
1. \(1\)
2. \(-1\)
3. \(0\)
4. None of these
Let \(\vec{F}\) be a force acting on a particle having position vector \(\vec{r}\). Let \(\vec{\tau}\) be the torque of this force about the origin, then:
| 1. | \(\vec{r} \cdot \vec{\tau}=0\) and \(\vec{F} \cdot \vec{\tau}=0\) |
| 2. | \(\vec{r} \cdot \vec{\tau}=0\) but \(\vec{F} \cdot \vec{\tau} \neq 0\) |
| 3. | \(\vec{r} \cdot \vec{\tau} \neq 0\) but \(\vec{F} \cdot \vec{\tau}=0\) |
| 4. | \(\vec{r} \cdot \vec{\tau} \neq 0\) and \(\vec{F} \cdot \vec{\tau} \neq 0\) |
A circular disc A of radius \(r\) is made from an iron plate of thickness \(t\) and another circular disc B of radius \(4r\) is made from an iron plate of thickness \(t/4.\) The relation between the moments of inertia \(I_A\) and \(I_B\) is:
| 1. | \(I_A>I_B\) |
| 2. | \(I_A=I_B\) |
| 3. | \(I_A<I_B\) |
| 4. | depends on the actual values of \(t\) and \(r\) |
From a circular disc of radius \(R\) and mass \(9M,\) a small disc of mass \(M\) and radius \(R/3\) is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is:
1. \(MR^{2}\)
2. \(4MR^{2}\)
3. \(\frac{4}{9}MR^{2}\)
4. \(\frac{40}{9}MR^{2}\)
Three masses are placed on the \(x\)-axis: \(300~\text{g}\) at origin, \(500~\text{g}\) at \(x= 40~\text{cm}\) and \(400~\text{g}\) at \(x= 70~\text{cm}.\) The distance of the centre of mass from the origin is:
1. \(45~\text{cm}\)
2. \(50~\text{cm}\)
3. \(30~\text{cm}\)
4. \(40~\text{cm}\)
A circular platform is mounted on a frictionless vertical axle. Its radius \(R = 2~\text{m}\) and its moment of inertia about the axle is \(200~\text{kg m}^2\). It is initially at rest. A \(50~\text{kg}\) man stands on the edge of the platform and begins to walk along the edge at the speed of \(1~\text{ms}^{-1}\) relative to the ground. The time taken by man to complete one revolution is:
1. \(\frac{3\pi}{2}\text{s}\)
2. \(2\pi~\text{s}\)
3. \(\frac{\pi}{2}\text{s}\)
4. \(\pi~\text{s}\)
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