1. | \(\overrightarrow v\) is a constant; \(\overrightarrow a\) is not a constant. |
2. | \(\overrightarrow v\) is not a constant; \(\overrightarrow a\) is not a constant. |
3. | \(\overrightarrow v\) is a constant; \(\overrightarrow a\) is a constant. |
4. | \(\overrightarrow v\) is not a constant; \(\overrightarrow a\) is a constant. |
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 the angular speed of \(A\) to that of \(B\) will be:
1. | \( 1: 1 \) | 2. | \(r_A: r_B \) |
3. | \(v_A: v_B \) | 4. | \(r_B: r_A\) |
In the given figure, \(a=15\) m/s2 represents the total acceleration of a particle moving in the clockwise direction in a circle of radius \(R=2.5\) m at a given instant of time. The speed of the particle is:
1. \(4.5\) m/s
2. \(5.0\) m/s
3. \(5.7\) m/s
4. \(6.2\) m/s
1. | velocity and acceleration both are parallel to \(\overrightarrow{r}.\) |
2. | velocity is perpendicular to \(\overrightarrow{r}\) and acceleration is directed towards to origin. |
3. | velocity is parallel to \(\overrightarrow{r}\) and acceleration is directed away from the origin. |
4. | velocity and acceleration both are perpendicular to \(\overrightarrow{r}.\) |
1. | \(0.15\) m/s2 | 2. | \(0.18\) m/s2 |
3. | \(0.2\) m/s2 | 4. | \(0.1\) m/s2 |
The position vector of a particle \(\vec{R}\) as a function of time \(t\) is given by;
\(\overrightarrow{\mathrm{R}}=4 \sin (2 \pi \mathrm{t}) \hat{\mathrm{i}}+4 \cos (2 \pi \mathrm{t}) \hat{\mathrm{j}}\)
Where \(R\) is in meters, \(t\) is in seconds and \(\mathrm{\hat{i},\hat{j}}\) denotes unit vectors along \(\mathrm{x}\) and \(\mathrm{y}\)-directions, respectively. Which one of the following statements is wrong for the motion of the particle?
1. | acceleration is along \(-\overrightarrow{R}\). |
2. | magnitude of the acceleration vector is \(\frac{v^2}{R}\), where \(v\) is the velocity of the particle. |
3. | magnitude of the velocity of the particle is \(8\) m/s. |
4. | path of the particle is a circle of radius \(4\) m. |
A particle moves in a circle of radius \(5\) cm with constant speed and time period \(0.2\pi\) s. The acceleration of the particle is:
1. | \(25\) m/s2 | 2. | \(36\) m/s2 |
3. | \(5\) m/s2 | 4. | \(15\) m/s2 |
A particle moves in the \((x\text-y)\) plane according to the rule \(x = a \sin (\omega t)\) and \(y = a \cos (\omega t)\). The particle follows:
1. | a circular path. |
2. | a parabolic path. |
3. | a straight line path inclined equally to x and y-axes. |
4. | an elliptical path. |
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. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the tangent to the circle. |
2. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the radius towards the centre. |
3. | \(\frac{\pi^2}{4}~\text{ms}^{-2} \) and direction along the radius towards the centre. |
4. | \(\pi^2~\text{ms}^{-2} \) and direction along the radius away from the centre. |
A particle moves along a circle of radius \(\frac{20}{\pi}~\text{m}\) with constant tangential acceleration. If the velocity of the particle is \(80\) m/s at the end of the second revolution after motion has begun, the tangential acceleration is:
1. \(40\) ms–2
2. \(640\pi\) ms–2
3. \(160\pi\) ms–2
4. \(40\pi\) ms–2