Displacement in 1D Motion

Q. The magnitude of the displacement of a particle moving in a circle of radius $a$ with constant angular speed $\omega$ varies with time $t$ as

1. $2a\sin \frac{\omega t}{2}$
2. $2a\cos \omega t$
3. $2a\cos \frac{\omega t}{2}$
4. $2a\sin \omega t$

Q. In the given figure, $a=15{\text{m/s}}^2$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius $R=2.5\text{m}$ at a given instant of time. The speed of particle at that instant will be

1. $4.5\text{m/s}$
2. $5\text{m/s}$
3. $5.7\text{m/s}$
4. $6.2\text{m/s}$

Q. The distance of a particle moving on a circular path of radius $12\text{m}$ measured from a fixed point on the circle and measured along the circle is given by $S=2t^3$ (in metres). The ratio of its tangential to centripetal acceleration at $t=2\text{s}$ is

1. $1:1$
2. $1:2$
3. $2:1$
4. $3:1$

Q. A particle is launched from a horizontal plane with speed $u$ and angle of projection $\theta$. The angular velocity of the particle as observed from point of projection at the time of landing will be:

1. $\frac{g}{2u\cos \theta }$
2. $\frac{g}{u\cos \theta }$
3. $\frac{3g}{2u\cos \theta }$
4. $\frac{2g}{u\cos \theta }$

Q. A point moves along a circle with a velocity $v=kt$, where $k=0.5\text{m}/{\text{s}}^2$. Find the acceleration of the point at the moment when it has covered the $n^{th}$ fraction of the circle after the beginning of motion, where $n=\frac{1}{10}$.

1. $0.5\text{m}/{\text{s}}^2$
2. $0.3\text{m}/{\text{s}}^2$
3. $0.8\text{m}/{\text{s}}^2$
4. $1.1\text{m}/{\text{s}}^2$

Q. A point $P$ moves in clockwise direction on a circular path as shown in figure. The movement of $P$ is such that it sweeps a distance $s=2t^3+3t^2+6$, where $s$ is in metres and $t$ is in seconds. The radius of the path is $36\text{m}$. The acceleration of ${}^{\prime }{P}^{\prime }$ when $t=1\text{s}$ is

1. $18{\text{m/s}}^2$
2. $4{\text{m/s}}^2$
3. $\sqrt{350}{\text{m/s}}^2$
4. $\sqrt{340}{\text{m/s}}^2$

Q. A particle moves on a circular path of radius, $R$. Find magnitude of its displacement during an interval in which it covers an angle $\theta$

1. $R\theta$
2. $Rsin\theta$
3. $2Rsin\left(\theta /2\right)$
4. $2Rcos\left(\theta /2\right)$

Q. A wheel completes $2000$ revolutions which is equivalent to a linear distance of $9.5km$ distance, then the diameter of the wheel is

1. $1.51m$
2. $2.56m$
3. $5.54m$
4. $7.58m$

Q. The phase difference between the velocity and displacement of a particle executing SHM is

1. $\pi /2$ radian
2. $2\pi$ radian
3. $\pi$ radian
4. zero

Q. A particle moves on a given straight line with a constant speed v. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that $\overrightarrow{OP}\times \overrightarrow{v}$ is independent of the position P.

Q. The angular position of a point on a rotating wheel varies with time $t$ as $\theta =2t^3-6t^2$, where $\theta$ is in radian and $t$ is in $s$. The angular velocity at the momment when torque on the wheel becomes zero, is

1. $-3\text{rad/s}$
2. $-6\text{rad/s}$
3. $4\text{rad/s}$
4. $12\text{rad/s}$

Q. A man $A$ at radius $2\text{m}$ and other man $B$ at radius $3\text{m}$ perform uniform circular motion on the floor of a merry-go-round as the ride turns. They are on the same radial line. At one instant, the acceleration of the man $A$ is $(2\hat{i}+4\hat{j}){\text{m/s}}^2$. At that instant and in unit-vector notation, what is the acceleration of man $B$?

1. $(6\hat{i}+3\hat{j}){\text{m/s}}^2$
2. $(6\hat{i}-3\hat{j}){\text{m/s}}^2$
3. $(3\hat{i}+6\hat{j}){\text{m/s}}^2$
4. $(3\hat{i}-6\hat{j}){\text{m/s}}^2$

Q. The intensity on the screen at a certain point in a double-slit interference pattern is 64.0% of the maximum value. (a) What minimum phase difference (in radians) between sources produces this result? (b) Express this phase difference as a path difference for 486.1-nm light.

Q. If $u$ is perpendicular to $AB$ and $v$ be the velocity of the second particle, find $\frac{u}{v}$ to the nearest integer.