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A particle is moving in a circular orbit with constant speed. Select wrong alternate

One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively \(\text{I}_{A}\) and \(\text{I}_{B}\) such that
1. \(\text{I}_{\text{A}}=\text{I}_{\text{B}}\)
2. \(\text{I}_{\text{A}}>\text{I}_{\text{B}}\)
3. \(\text{I}_{\text{A}}<\text{I}_{\text{B}}\)
4. \(\frac{\text{I}_{\text{A}}}{\text{I}_{\text{B}}}=\frac{d_A}{d_B}\)

A particle of mass \(1 ~\text{kg}\) is kept at (1m, 1m, 1m). \((1~\text{m},~1~\text{m},~1~\text{m}),\) The moment of inertia of this particle about \(z-\)axis would be
1. \(1~\text{kg}-\text{m}^2\)
2. \(2~\text{kg}-\text{m}^2\)
3. \(3~\text{kg}-\text{m}^2\)
4. None of these

One-quarter sector is cut from a uniform circular disc of radius \(R.\)  This sector has mass \(M.\)  It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc.  Its moment of inertia about the axis of rotation is:
       
1. \(\frac{1}{2} M R^2\)
2. \(\frac{1}{4} M R^2\)
3. \(\frac{1}{8} M R^2\)
4. \(\sqrt{2} M R^2\)

The radius of gyration of a uniform rod of length \(L\) about an axis passing through its centre of mass is
1. \(\frac{L}{2 \sqrt{3}}\)
2. \(\frac{L^2}{12}\)
3. \(\frac{L}{\sqrt{3}}\)
4. \(\frac{L}{\sqrt{2}}\)

A solid sphere, a hollow sphere, and a disc, all having the same mass and radius, are placed at the top of a smooth incline and released. Least time will be taken in reaching the bottom by  

A bullet of mass m and velocity \(\text v\) is fired into a block of mass \(M\) and sticks to it. The final velocity of the system equals
1. \(\frac{\text{M}}{\text{~m}+\text{M}} \cdot \text{v}\)
2. \(\frac{\text{~m}}{\text{~m}+\text{M}} \cdot \text{v}\)
3. \(\frac{\text{~m}+\text{M}}{m} \cdot \text{v}\)
4. None of these

A bullet of mass \(5 ~\text g\) is fired at a velocity of \(900~\text{ms}^{-1}\) from a rifle of mass \(2.5 ~\text{kg.}\)  What is the recoil velocity of the rifle
1. \(0.9~\text{ms}^{-1}\)
2. \(180~\text{ms}^{-1}\)
3. \(900~\text{ms}^{-1}\)
4. \(1.8~\text{ms}^{-1}\)
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Two bodies with moment of inertia $I_{2}and{I}_2$(such that $I_1>I_2$) have equal angular velocity. If their kinetic energy of rotation are $E_{1}and{E}_2$ then [Pb. PET 2000]
1. $E_1\ge E_2$
2. $E_1>E_2$
3. $E_1<E_2$
4. $E_1=E_2$

A particle P moving with speed v undergoes a head-on elastic collision with another particle Q of identical mass but at rest.  After the collision-
1. both P and Q move forward with speed $\frac{v}{2}$.
2. both P and Q move forward with speed $\frac{v}{\sqrt{2}}$.
3. P comes to rest and Q moves forward with speed v.
4. P and Q move in opposite directions with speed $\frac{v}{\sqrt{2}}$.

A man is sitting on a rotating table with his arms stretched outwards.  When he suddenly folds his arms inside, then
1. his angular velocity will decrease
2. his angular velocity remains constant
3. his moment of inertia decreases
4. angular momentum increases

One projectile moving with velocity v in space, gets burst into 2 parts of masses in the ratio 1:3.  The smaller part becomes stationary.  What is the velocity of the other part?
1. 4v                       
2. v
3. $\frac{4v}{3}$                     
4. $\frac{3v}{4}$

A wheel of radius R rolls on the ground with a uniform velocity v. The velocity of topmost point relative to the bottommost point is
1. v
2. 2v
3. v/2
4. zero
 

If the net external forces acting on the system of particles is zero, then which of the following may vary ?
1. Momentum of the system
2. Velocity of centre of mass
3. Position of centre of mass
4. None of the above

A wheel is rotating about an axis through its centre at \(720~\text{rpm}.\) It is acted upon by a constant torque opposing its motion for \(8\) seconds to bring it to rest finally. The value of torque in \((\text{N-m })\) is:
(given \(I=\frac{24}{\pi}~\text{kg.m}^2)\) 
1. \(48\)
2. \(72\)
3. \(96\)
4. \(120\)

A stationary bomb explodes into two parts of masses 3kg and 1kg. The total KE of the two parts after explosion is 2400 J. The KE of smaller part is 
1. 600 J
2. 1800 J
3. 1200 J
4. 2160 J

A wheel is rotating 900 rpm about its axis. When power is cut off it comes to rest in 1 min. The angular retardation in rad/s² is 
1. $\mathrm{\pi }/2$
2. $\mathrm{\pi }/4$
3. $\mathrm{\pi }/6$
4. $\mathrm{\pi }/8$

An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver ?
1. 350 N-m
2. 440 N-m
3. 531 N-m
4. 628 N-m

In an orbital motion, the angular momentum vector is 
1. Along the radius vector
2. Parallel to the linear momentum
3. In the orbital plane
4. Perpendicular to the orbital plane

A wheel is at rest. Its angular velocity increases uniformly and becomes 80 rad/s after 5 s. The total angular displacement is 
1. 800 rad
2. 400 rad
3. 200 rad
4. 100 rad

A force\(- F \hat k\) acts on \(O\), the origin of the coordinate system. The torque at the point \((1,-1)\) will be:
1. \(-F(\hat i + \hat j)\)
2. \(F(\hat i + \hat j)\)
3. \(-F(\hat i -\hat j)\)
4. \(F(\hat i - \hat j)\)

When a ball of mass = 5kg hits a bat with a velocity = 3 m/s, in positive direction and it moves back with a velocity = 4 m/s, find the impulse in SI units
1. 5 
2. 15
3. 25
4. 35

A wheel whose moment of inertia is 12 ${\mathrm{Kgm}}^2$ has an initial angular velocity of 40 rad/sec. A constant torque of 20 Nm acts on the wheel. The time in which the wheel is accelerated to 100 rad/sec is
1.  72 seconds
2.  16 seconds
3.  8 seconds
4.  36 seconds

A constant torque acting on a uniform circular wheel changes its angular momentum from \(A_0\) to \(4A_0\) in \(4~\text{s}\). The magnitude of this torque is:
1. \(\dfrac{3A_0}{4}\)
2. \(4A_0\)
3. \(A_0\)
4. \(12A_0\)

A solid cylinder rolls down an inclined plane that has friction sufficient to prevent sliding. The ratio of rotational energy to total kinetic energy is
1.  $\frac{1}{2}$
2.  $\frac{1}{3}$
3.  $\frac{2}{3}$
4.  $\frac{3}{4}$
 

A swimmer while jumping into water from a height easily forms a loop in the air, if

Assertion : A ladder is more apt to slip when you are high up on it than when you just begin to climb
Reason : At the high point on a ladder, the torque is large and on climbing up the torque is small

Moment of inertia of a uniform circular disc about a diameter is I. Its moment of inertia about an axis perpendicular to its plane and passing through a point on its rim will be
1.  5 I
2.  3 I
3.  6 I
4.  4 I

If rotational kinetic energy is 50 % of translational kinetic energy, then the body is 
1. Ring
2. Cylinder
3. Hollow sphere
4. Solid sphere

Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration a. The centre of mass has an acceleration
1. zero
2. \(\frac{a}{2}\)
3. a
4. 2a

The moment of inertia of pulley is I and its radius is R. The string doesn't slips on the pulley, as system is released, the acceleration of the system is
 \(\)
1. $\frac{g}{\left(3+{\displaystyle \frac{I}{m{R}^2}}\right)}$
2. $\frac{g}{\left(1+{\displaystyle \frac{I}{m{R}^2}}\right)}$
3. $\frac{3g}{\left(1+{\displaystyle \frac{I}{m{R}^2}}\right)}$
4. $\frac{3g}{\left(2+{\displaystyle \frac{I}{m{R}^2}}\right)}$

A shell is fired from a cannon with velocity v m/sec at an angle θ with the horizontal direction. At the highest point in its path it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed (in m/sec) of the other piece immediately after the explosion is 
(1) 3v cos θ
(2) 2v cos θ
(3) $\frac{3}{2}v\cos \theta$
(4) $\frac{\sqrt{3}}{2}v\cos \theta$

A rope is wound around a hollow cylinder of mass 3 kg and radius 40cm. What is the angular acceleration of the cylinder,if the rope is pulled with a force of 30 N?
1. 25$m/s^2$
2. 0.25 $rad/s^2$
3. 25 $rad/s^2$
4. 5 $m/s^2$

A rod of weight w is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is
1. $\frac{\mathrm{wx}}{\mathrm{d}}$
 
2. $\frac{\mathrm{wd}}{\mathrm{x}}$
 
3. w(d-x)/x
 
4. $\frac{w(d-x)}{d}$

A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis.A massless string is wound round the cylinder with one end attached to it and other hanging freely.Tension in the string required to produce an angular acceleration of 2 rev/ s² is
1. 25N
2. 50N
3. 78.5N
 
4. 157N

The moment of inertia of a disc of mass M and radius R about a tangent to its rim in its plane is
(1) $\frac{2}{3}M{R}^2$
(2) $\frac{3}{2}M{R}^2$
(3) $\frac{4}{5}M{R}^2$
(4) $\frac{5}{4}M{R}^2$

A disc is rotating with angular speed $\omega$. If a child sits on it, what is conserved?
(1) Linear momentum
(2) Angular momentum
(3) Kinetic energy
(4) Potential energy

If the angle between the vector \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{B }\) is \(\theta\), the value of the product \(\left(\overset{\rightarrow}{B} \times \overset{\rightarrow}{A}\right) . \overset{\rightarrow}{A}\) is equal to
(1) zero
(2) \(B A^{2} sin \theta cos \theta\)
(3) \(B A^{2} cos \theta\)
(4) \(B A^{2} sin \theta\)

Two bodies have their moments of inertia I and 2I, respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momentum will be in the ratio
(1) 1 : 2
(2) $\sqrt{2}<mspace\; width="1em"></mspace>:<mspace\; width="1em"></mspace>1$
(3) $1<mspace\; width="1em"></mspace>:<mspace\; width="1em"></mspace>\sqrt{2}$
(4) 2 : 1

A wheel has angular acceleration of 3.0 rad $s^{-2}$ and an initial angular speed of 2.00 rad $s^{-1}$. In a time 2 s it has rotated through an angle (in radian) of
(1) 10 
(2) 12
(3) 4
(4) 6

Two bodies of mass 1 kg and 3 kg have position vectors $\hat{i}+2\hat{j}+\hat{k}<mspace\; width="1em"></mspace>and<mspace\; width="1em"></mspace>-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The center of mass of this system has a position vector
(1) $-2\hat{i}-\hat{j}+\hat{k}$
(2) $2\hat{i}-\hat{j}-\hat{k}$
(3) $-\hat{i}+\hat{j}+\hat{k}$
(4) $-2\hat{i}+2\hat{k}$

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through

(1) D
(2) A
(3) B
(4) C

A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane and passing through its center with a constant angular velocity $\omega$. If objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity
(1) $\frac{\omega M}{M+2m}$
(2) $\frac{\omega (M+2m)}{M}$
(3) $\frac{\omega M}{M+m}$
(4) $\frac{\omega (M-2m)}{M+2m}$

A planet moving along an elliptical orbit is closest to the sun at a distance $r_1$ and farthest away at a distance of $r_2$. If $v_1<mspace\; width="1em"></mspace>and<mspace\; width="1em"></mspace>v_2$ are the linear velocities at these points respectively. Then the ratio $\frac{v_1}{v_2}$ is
(1) $\frac{r_1}{r_2}$
(2) ${\left(\frac{r_1}{r_2}\right)}^2$
(3) $\frac{r_2}{r_1}$
(4) ${\left(\frac{r_2}{r_1}\right)}^2$

Three masses are placed on the x-axis: 300 g at origin, 500 g at x = 40 cm and 400 g  at x = 70 cm.The distance of the center of mass from the origin is
(1) 50 cm
(2) 30 cm
(3) 40 cm
(4) 45 cm