A wheel has an angular acceleration of \(3.0\) rad/s² and an initial angular speed of \(2.00\) rad/s. In a time of \(2\) s, it has rotated through an angle (in radian) of:
1. \(6\)
2. \(10\)
3. \(12\)
4. \(4\)

For a rigid body rotating about a fixed axis, which of the following quantities is the same at an instant for all the particles of the body?

For a body, with angular velocity \( \vec{\omega }=\hat{i}-2\hat{j}+3\hat{k}\)  and radius vector \( \vec{r }=\hat{i}+\hat{j}++\hat{k},\)  its velocity will be:
1. \(-5\hat{i}+2\hat{j}+3\hat{k}\)
2. \(-5\hat{i}+2\hat{j}-3\hat{k}\)
3. \(-5\hat{i}-2\hat{j}+3\hat{k}\)
4. \(-5\hat{i}-2\hat{j}-3\hat{k}\)

If a body is moving in a circular path with decreasing speed, then: (symbols have their usual meanings):

1.  $\overrightarrow{r}.\overrightarrow{\omega }$ $=$ $$ $0$

2.  $\overrightarrow{\tau }.\overrightarrow{v}$ $=$ $0$

3.  $\overrightarrow{a}.\overrightarrow{v}$ $<$ $0$

4.  All of these

The angular speed of the wheel of a vehicle is increased from \(360~\text{rpm}\) to \(1200~\text{rpm}\) in \(14\) seconds. Its angular acceleration will be:
1. \(2\pi ~\text{rad/s}^2\)
2. \(28\pi ~\text{rad/s}^2\)
3. \(120\pi ~\text{rad/s}^2\)
4. \(1 ~\text{rad/s}^2\)

A rigid body rotates about a fixed axis with a variable angular velocity equal to \(\alpha\) \(-\) \(\beta t\), at the time \(t\), where \(\alpha , \beta\) are constants. The angle through which it rotates before it stops is:

Particles \(A\) and \(B\) are separated by \(10\) m, as shown in the figure. If \(A\) is at rest and \(B\) started moving with a speed of \(20\) m/s then the angular velocity of \(B\) with respect to \(A\) at that instant is:

Two gear wheels that are meshed together have radii of \(0.50\) cm and \(0.15\) cm. The number of revolutions made by the smaller one when the larger one goes through \(3\) revolutions is:
1. \(5\) revolutions 
2. \(20\) revolutions 
3. \(1\) revolution
4. \(10\) revolutions