Angular Displacement

Q. A coin, placed on a rotating turn-table slips, when it is placed at a distance of 9 cm from the centre. If the angular velocity of the turn-table is trippled, it will just slip, if its distance from the centre is

1. 1 cm
2. 9 cm
3. 3 cm
4. 27 cm

Q. A wheel starts from the rest and attains an angular velocity of $20\text{rad/s}$ after being uniformly accelerated for $10\text{s}$. The total angle in radians through which it has turned in $10\text{s}$ is

1. 20
2. 40
3. 80
4. 100

Q. A uniform cylinder of height $h$ and radius $r$ is placed with its circular face on a rough inclined plane and the inclination of the plane to the horizontal is gradually increased. If $\mu$ is the coefficient of friction, then under what condition the cylinder will slide before toppling.

1. $\mu <\frac{2r}{h}$
   
2. $\mu >\frac{2r}{h}$
   
3. $\mu <\frac{r}{h}$
   
4. $\mu >\frac{r}{h}$
   

Q. The minimum and maximum distances of a planet revolving around sun are $r$ and $R$. If the minimum speed of planet on its trajectory is $v_0$, its maximum speed will be

1. $\frac{v_{0}R}{r}$
2. $\frac{v_{0}r}{R}$
3. $\frac{v_0{R}^2}{r^2}$
4. $\frac{v_0{r}^2}{R^2}$

Q.

A wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest, it reaches 100 rev/sec in 4 seconds. Find the angular acceleration. Find the angle rotated during these four seconds.

Q. A wheel starting from rest, rotates with a uniform angular acceleration of $2{\text{rad/s}}^2$. The number of rotation it performs in the tenth second is

1. 3
2. 6
3. 9
4. 12

Q. A balloon is moving upwards with a speed of $20\text{m/s}$. When it is at a height of $14\text{m}$ from the ground in front of a plane mirror as shown in figure, a boy drops himself from the balloon. Find the time duration for which he will see the image of source $S$ placed symmetrically before the plane mirror during free fall. Take $g=10{\text{m/s}}^2$.

1. $4.5\text{s}$
2. $3.7\text{s}$
3. $1.7\text{s}$
4. $2.7\text{s}$

Q. A car wheel is rotated to uniform angular acceleration about its axis. Initially its angular velocity is zero. It rotates through an angle ${\theta }_1$ in the first $2$s. In the next $2$ s, it rotates through an additional angle ${\theta }_2$, the ratio of $\frac{{\theta }_2}{{\theta }_1}$ is

1. 1
2. 2
3. 3
4. 4

Q. A particle is rotating in a circle of radius $1\text{m}$ with constant speed $4\text{m/s}$. In time $1\text{s}$
$\begin{array}{cccc}& \text{Column - I}& & \text{Column - II}\\ & \text{(Quantity)}& & \text{(Value in SI units)}\\ \text{(a)}& \text{Displacement}& \text{(p)}& 8\sin 2\\ \text{(b)}& \text{Distance}& \text{(q)}& 4\\ \text{(c)}& \text{Average velocity}& \text{(r)}& 2\sin 2\\ \text{(d)}& \text{Average acceleration}& \text{(s)}& 4\sin 2\end{array}$

1. a - r; b - p; c - q; d - s
2. a - r; b - q; c - p; d - s
3. a - r; b - p; c - r; d - s
4. a - r; b - q; c - r; d - p

Q.

Find the angular velocity of a body rotating with an acceleration of 2 rev/s${}^2$ as it completes the 5th revolution after the start.

Q. An opaque sphere of radius $a$ is just immersed in a transparent liquid as shown in figure. A point source is placed on the vertical diameter of the sphere at a distance $\frac{a}{2}$ from the top of the sphere. A ray from the point source after refraction from the air liquid interface forms tangent to the sphere. The angle of refraction for that particular ray is ${30}^{\circ }$. The refractive index of the liquid is

1. $\frac{2}{\sqrt{3}}$
2. $\frac{3}{\sqrt{5}}$
3. $\frac{4}{\sqrt{5}}$
4. $\frac{4}{\sqrt{7}}$

Q.

A hollow sphere of radius R lies on a smooth horizontal surface. It is pulled by a horizontal force acting tangentially from the highest point. Find the distance travelled by the sphere during the time it makes one full rotation.

Q.

A capillary tube of radius 1 mm is kept vertical with the lower end in water . (a) Find the height of water raised in the capillary. (b) If the length of the capillary tube is half the answer of part (a), find the angle $\theta$ made by the water surface in the capillary with the wall.

Q.

A bird is flying up at angle $si{n}^{-1}\left(3/5\right)$ with the horizontal. A fish in a pond looks at that bird. When it is vertically above the fish, the angle (in degrees) at which the bird appears to fly (to the fish) is $[n_{water}=4/3]$

Q. Find the angular displacement of the hour hand of a normal clock in $4\text{h}$

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{2\pi }{3}$
4. $\frac{3\pi }{2}$

Q. If an object covers a distance of 5 cm moving in a circular path of radius 5 cm, find its angular displacement in radians.

1. 1
2. 0
3. 5
4. 25

Q. The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta \left(t\right)=3t^3-9t^2$ (Here $t$ is in seconds). The angular acceleration of the wheel becomes zero at

1. $t=1\text{s}$
2. $t=0.5\text{s}$
3. $t=0.25\text{s}$
4. $t=2\text{s}$

Q. A fly wheel is accelerated uniformly from rest and rotates through $5\text{rad}$ in the first second. The angle rotated by the fly wheel in the next second, will be :

1. $20\text{rad}$
2. $30\text{rad}$
3. $15\text{rad}$
4. $7.5\text{rad}$

Q. Neena goes around a circular track that has a diameter of $7m$. If she runs around the entire track for a distance of $50m$, what is her angular displacement?

1. $6.28radians$
2. $3.14radians$
3. $7.14radians$
4. $14.28radians$

Q. Why is angular displacement vector quantity only for small values? Why is the direction of angular displacement perpendicular to the plane of rotation of object?

Q. A boy running on a circular track of radius $10\text{m}$ completes $3$ revolution and comes back to the point from where he started. Calculate the angular displacement of the boy $\left(\text{in rad}\right)$.

1. $3\pi$
2. $6\pi$
3. $\frac{\pi }{3}$
4. $0$

Q.

Give an example of variable velocity when the magnitude is constant and the direction is changing.

Q. An athlete running with constant speed, completes one round of a circular track of radius $R$ in $40\text{seconds}$. What will be his angular displacement at the end of $20\text{seconds}$?

1. $\frac{\pi }{2}\text{rad}$
2. $\frac{\pi }{3}\text{rad}$
3. $\frac{\pi }{4}\text{rad}$
4. $\pi \text{rad}$

Q. A spotlight is fixed $4\text{m}$ from the vertical wall and is rotating at a rate ${\text{1 rads}}^{-1}$. The spot moves vertically on the wall. Find the speed ( in ${\text{ms}}^{-1}$) of the spot on the wall when the spotlight makes an angle of ${45}^{\circ }$ with the wall.

Q. Find out the dimensional formula for following quantity: 1) Angular displacement and 2) Angular velocity

Q. A particle is moving in a circle of radius $R$ with initial speed $v$. It starts retarding with constant retardation $\frac{v^2}{4\pi R}$. The number of the revolution it makes in time $\frac{8\pi R}{v}$ is

1. 3
2. 4
3. 5
4. 2

Q.

A point initially at rest moves along $X-axis$. Its acceleration varies with time as $a=(6t+5)m{s}^{-2}$. If it starts from origin, the distance covered in $2s$ is

1. $20m$

2. $18m$

3. $16m$

4. $25m$

Q.

Derive the second equation of motion.

Q. When a wheel completes $2000$ revolutions to cover $15\text{km}$ distance, then the radius of the wheel is?

1. $2.4\text{m}$
2. $2\text{m}$
3. $1.19\text{m}$
4. $0.8\text{m}$

Q. why infinitesimal angular displacement is a vector rather avg angular displacement or velocity a scalar ?!!!