Uniform Circular Motion

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

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

Q. A tunnel is dug along the diameter of Earth.There is a particle of mass $m$ at the centre of tunnel.The maximum velocity given to the particle, so that it just reaches the surface of earth is: $(R$ is radius of Earth and $M$ is the mass of the earth $)$

1. $\sqrt{\frac{GM}{R}}$
2. $\sqrt{\frac{GM}{2R}}$
3. $\sqrt{\frac{2GM}{R}}$
4. Zero

Q. When a ceiling fan is switched on, it makes 10 revolutions in the first 3 seconds. Assuming a uniform angular acceleration, how many rotations it will make in the next 3 seconds?

1. 10
2. 20
3. 40
4. 30

Q. Total kinetic energy of solid sphere of mass $m$ rolling with velocity $v$ without slipping is

1. $\frac{7}{10}m{v}^2$
2. $\frac{10}{7}m{v}^2$
3. $\frac{5}{6}m{v}^2$
4. $\frac{7}{5}m{v}^2$

Q. A particle is moving along a circular path with a constant speed of $10{\text{ms}}^{-1}$. What is the magnitude of the change in velocity of the particle, when it moves through an angle of ${60}^{\circ }$ around the centre of the circle?

1. $10\text{m/s}$
2. $\text{Zero}$
3. $10\sqrt{2}\text{m/s}$
4. $10\sqrt{3}\text{m/s}$

Q.

A person of $80kg$ mass is standing on the rim of a circular platform of mass $200kg$ rotating about its axis at $5$ revolutions per minute (rpm). The person now starts moving towards the center of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its center _____?

Q. A charged particle of specific charge $\alpha$ is released from origin at time $t=0$ with velocity $\overrightarrow{v}=v_0(\hat{i}+\hat{j})$ in magnetic field $\overrightarrow{B}=B_0\hat{i}$. The coordinate of the particle at time $t=\frac{\pi }{B_0\alpha }$ will be

1. $(\frac{v_0\pi }{B_0\alpha },0,\frac{-2v_0}{B_0\alpha })$
2. $(0,\frac{2v_0\pi }{B_0\alpha },\frac{2v_0}{B_0\alpha })$
3. $(\frac{3v_0\pi }{B_0\alpha },\frac{-2v_0}{B_0\alpha },0)$
4. $(\frac{2\sqrt{2}{v}_0\pi }{B_0\alpha },\frac{v_0\pi }{B_0\alpha },\frac{\sqrt{2}{v}_0}{B_0\alpha })$

Q.

When a body moves with a constant speed along a circle

1. No acceleration is produced in the body

2. No work is done on it

3. No force acts on the body

4. Its velocity remains constant

Q. The seconds hand of a watch has length $6cm$. Speed of the end point and magnitude of difference in velocities at positions $A$ and $B$, as shown in the figure will be

1. $2\pi \frac{mm}{s},2\sqrt{2}\pi \frac{mm}{s}$
2. $\pi \frac{mm}{s},2\sqrt{2}\pi \frac{mm}{s}$
3. $2\pi \frac{mm}{s},\sqrt{2}\pi \frac{mm}{s}$
4. $\pi \frac{mm}{s},\sqrt{2}\pi \frac{mm}{s}$

Q.

If a body is moving at constant speed in a circular path, its

1. velocity is constant and its acceleration is zero

2. velocity and acceleration are both changing directions only

3. velocity and acceleration are both increasing

4. velocity is constant and acceleration is changing direction

Q. A particle is projected vertically up with velocity v=4gR/3 from earth surface. The velocity of particle at height equal to half of maximum height reach by it

Q. A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with speed of 0.5 m/s. What is the height of the plane of circle from vertex of the funnel?

1. 0.25 cm
2. 2 cm
3. 4 cm
4. 2.5 cm

Q. If a particle moves in a circle describing equal angles in equal intervals of times, its velocity vector

1. Changes both in magnitude and direction
2. Remains constant
3. Changes in magnitude
4. Changes in direction

Q. A stone tied to a string is rotated with a uniform speed in a vertical plane. If mass of the stone is $m$, the length of the string is $r$ and the linear speed of the stone is $v$, when the stone is at its lowest point, then the tension in the string will be ($g=$ acceleration due to gravity)

1. $\frac{m{v}^2}{r}-mg$
2. $\frac{mv}{r}$
3. $\frac{m{v}^2}{r}+mg$
4. $mg$

Q. A charged particle of mass $m$ and charge $q$ is accelerated through a potential difference of $V$. It enters a region of uniform magnetic field $B$, which is directed perpendicular to the direction of motion of the particle. The particle will move on a circular path of radius given by -

1. $\sqrt{\frac{Vm}{2q{B}^2}}$
2. $\sqrt{\frac{Vm}{3q{B}^2}}$
3. $\sqrt{\frac{2Vm}{q}}\times \left(\frac{1}{B}\right)$
4. $\sqrt{\frac{Vm}{q}}\times \left(\frac{1}{B}\right)$

Q. A particle starting from rest rotates in a circle of radius $R=\sqrt{2}\text{m}$ with an angular acceleration $\alpha =\frac{\pi }{4}{\text{rad/sec}}^2$. The magnitude of average velocity of the particle over the time in rotating a quarter of a circle is

1. $1\text{m/s}$
2. $2\text{m/s}$
3. $4\text{m/s}$
4. $6\text{m/s}$

Q. An unbalanced force acts on a body. The body must

1. remain at rest.
2. move with uniform velocity.
3. be accelerated.
4. Can't predict

Q.

A hemisphere ball of radius $R$ is set rotating about its axis of symmetry which is kept vertical. A small block kept in the bowl rotates with the bowl without slipping on its surface. If the surface of the bowl is smooth and the angle made by the radius through the block with the vertical is $\theta$, find the angular speed at which the bowl is rotating.

1. $\omega =\sqrt{\frac{g\cos \theta }{R}}$
2. $\omega =\sqrt{\frac{g\sec \theta }{R}}$
3. $\omega =\sqrt{\frac{2g\cos \theta }{R}}$
4. $\omega =\sqrt{2gR\sec \theta }$

Q. A positively charged particle of specific charge $\left(\frac{q}{m}\right)$ is projected from the origin of coordinate with initial velocity $(u\hat{i}-v\hat{j})$. Uniform electric and magnetic fields exist in the region along the $+y-$direction, of magnitude $E$ and $B$. The particle will definitely return to the origin once if

1. $\frac{uB}{\pi E}$ is an integer
2. $\frac{vB}{\pi E}$ is an integer
3. $\frac{(u^2+v^2{)}^{1/2}B}{\pi E}$ is an integer
4. $\frac{vB}{2\pi E}$ is an integer

Q. Three identical particles are joined together by a thread as shown in figure. All the three particles are moving in a horizontal plane in circular path with $O$ as centre. If the velocity of the outermost particle $v_0$, then the ratio of tensions in the three sections of the string is

1. $3:5:7$
2. $3:4:5$
3. $7:11:6$
4. $3:5:6$

Q. The position vector of a particle is $\overrightarrow{r}=a[\text{cos}\omega t\hat{i}+\text{sin}\omega t\hat{j}]$. The velocity of the particle is

1. parallel to position vector
2. directed towards origin
3. directed away from origin
4. perpendicular to position vector.

Q. A body whose speed is constant -

1. must be accelerated
2. may be accelerated
3. never be accelerated
4. None of these

Q. In the figure a ball of mass $m$ is tied with two strings of equal length as shown. If the rod is rotated with angular velocity $\omega$ and $T_1,T_2$ be tension in the strings as shown, then

1. $T_1>T_2$
2. $T_2>T_1$
3. $T_1=2T_2$
4. $T_1=T_2$

Q. When a particle moves in a uniform circular motion, it has

1. radial velocity and radial acceleration.
2. tangential velocity and radial acceleration.
3. tangential velocity and tangential acceleration.
4. radial velocity and tangential acceleration.

Q. Velocity and acceleration vectors of a charged particle moving in a magnetic field at some instant are $\overrightarrow{v}=3\hat{i}+4\hat{j}$ and $\overrightarrow{a}=2\hat{i}+x\hat{j}$ (Select the correct alternative(s).

1. $x=-1.5$
2. $x=3$
3. Magnetic field is along z - direction.
4. Kinetic energy of the particle is constant

Q.

An athlete completes one round of a circular track of diameter 14m with a uniform speed of 11m/s. Find the time taken.

Q. A simple pendulum with bob of mass $m$ and conducting wire of length $L$ swings under gravity through an angle $\theta$. The component of earth's magnetic field, in a direction perpendicular to swing is $B$. Then

1. The maximum velocity of pendulum is $2\sqrt{gL}\sin \left(\frac{\theta }{2}\right)$
2. The maximum velocity of pendulum is $\sqrt{gL}\sin \left(\frac{\theta }{2}\right)$
3. The maximum potential difference induced across the pendulum is $BL\sqrt{gL}\sin \left(\frac{\theta }{2}\right)$
4. The maximum potential difference induced across the pendulum is $2BL\sqrt{gL}\sin \left(\frac{\theta }{2}\right)$

Q. A body is suspended from a spring balance kept in a satellite. The reading of the balance is $W_1$ when the satellite goes in a an orbit of radius $R$ and is $W_2$ when it goes in an orbit or radius $2R$.

1. $W_1=W_2$
   
2. $W_1<W_2$
   
3. $W_1>W_2$
   
4. $W_1\ne W_2$
   

Q.

A particle moves with constant angular velocity in a circle. During the motion its

1. Energy is conserved

2. Momentum is conserved

3. Energy and momentum both are conserved

4. None of the above is conserved

Q.

The average acceleration vector for a particle having a uniform circular motion is

1. A constant vector of magnitude $\frac{v^2}{r}$

2. A vector of magnitude $\frac{v^2}{r}$directed normal to the plane of the given uniform circular motion

3. Equal to the instantaneous acceleration vector at the start of the motion

4. A null vector