A particle moves in the x-y plane according to rule $\mathrm{x}=\mathrm{asin\omega t}$ and $\mathrm{y}=\mathrm{acos\omega t}$. The particle follows:

1. an elliptical path.

2. a circular path.

3. a parabolic path.

4. a straight line path inclined equally to the x and y-axis.

The magnitude of a vector $\overrightarrow{\mathrm{A}}$ is constant but it is changing its direction continuously. The angle between $\overrightarrow{\mathrm{A}}$ and $\frac{\mathrm{d}\overrightarrow{\mathrm{A}}}{\mathrm{dt}}$ is :

1. 180°

2. 120°

3. 90°

4. 0°

What is the value of linear velocity if $\overrightarrow{\omega }=3\hat{i}-4\hat{j}+\hat{k}$ and $\overrightarrow{r}=5\hat{i}-6\hat{j}+6\hat{k}$ 

1. $6\hat{i}+2\hat{j}-3\hat{k}$

2. $-18\hat{i}-13\hat{j}+2\hat{k}$

3. $4\hat{i}-13\hat{j}+6\hat{k}$

4. $6\hat{i}-2\hat{j}+8\hat{k}$

The angle turned by a body undergoing circular motion depends on the time as given by the equation, $\theta ={\theta }_0+{\theta }_{1}t+{\theta }_2{t}^2$. It can be deduced that the angular acceleration of the body is 

1. θ₁

2. θ₂

3. 2θ₁

4. 2θ₂

A particle moves in a circle of radius 5 cm with constant speed and time period $0.2\mathrm{\pi }$ sec. The acceleration of the particle is:

1.  25 m/s²

2.  36 m/s²

3.  5 m/s²

4.  15 m/s²

If the equation for the displacement of a particle moving on a circular path is given by $\left(\theta \right)=2t^3+0.5$, where θ is in radians and t in seconds, then the angular velocity of the particle after 2 sec from its start is:

1. 8 rad/sec

2. 12 rad/sec

3. 24 rad/sec

4. 36 rad/sec

A car moves on a circular path such that its speed is given by v = Kt, where K =constant and t is time. Also given: radius of the circular path is r. The net acceleration of the car at time t will be

1. $\sqrt{{\mathrm{K}}^2 + {\left(\frac{{\mathrm{K}}^2{\mathrm{t}}^2}{\mathrm{r}}\right)}^2}$

2. 2K

3. K

4. $\sqrt{{\mathrm{K}}^2 + {\mathrm{K}}^2{\mathrm{t}}^2}$

Two particles A and B are moving in a uniform circular motion in concentric circles of radii $r_A and r_B$ with speeds $v_A and v_B$ respectively. Their time periods of rotation are the same. The ratio of angular speed of A  to that of B will be:

1. 1: 1

2. $r_A : r_B$

3. $v_A : v_B$

4. $r_B : r_A$

A stone tied to the end of a 1 m long string is whirled in a horizontal circle at a constant speed. If the stone makes 22 revolutions in 44 seconds, what is the magnitude and direction of acceleration of the stone?

1. ${\pi }^2{\mathrm{ms}}^{-2}$ and direction along the tangent to the circle.

2. ${\pi }^2{\mathrm{ms}}^{-2}$ and direction along the radius towards the centre.

3. $\frac{{\pi }^2}{4}{\mathrm{ms}}^{-2}$ and direction along the radius towards the centre.

4. ${\pi }^2{\mathrm{ms}}^{-2}$ and direction along the radius away from the centre.

The position vector of a particle is $\overrightarrow{\mathrm{r}} = \left(\mathrm{a} \sin \mathrm{\omega t}\right)\hat{\mathrm{i}} - \left(\mathrm{a} \cos \mathrm{\omega t}\right)\hat{\mathrm{j}}$. The velocity of the particle is:

1. parallel to the position vector.

2. at 60° with position vector.

3. parallel to the acceleration vector.

4. perpendicular to the position vector.