System of particles practice · Mixed difficulty

A body of mass 1 kg thrown with a velocity of $10\;\mathrm{m}{\mathrm{s}^{ - 1}}$ at an angle of ${60^0}$ with the horizontal. Its momentum at the highest point is

Two bodies of mass 10 kg and 2 kg are moving with velocities $\left( {2\hat i - 7\hat j + 3\hat k} \right)\,m/s$ and $\left( { - 10\hat i + 35\hat j - 3\hat k} \right)\,m/s$ respectively. The velocity of their centre of mass is

A particle of mass m moving with a velocity $\left( {3\hat i + 2\hat j} \right)m{s^{ - 1}}$ collides with a stationary body mass M and finally moves with a velocity $\left( { - 2\hat i + \hat j} \right)\mathrm{m}{\mathrm{s}^{ - 1}}.\,If\,\frac{m}{M} = \frac{1}{{13}},$ then

A tank filled with water is connected to another empty tank lying to the right side of the first tank. Until the level of water in the tanks become same
a) Centre of mass of water shifts down at constant rate
b) Centre of mass of water shifts down and rate of change of shift in centre of mass decreases gradually and becomes zero
c) Centre of mass shifts towards right as well

A shaft rotating at 6000 rpm is transmitting a power of $2\pi$ kilowatt. Then magnitude of the driving torque is

The moment of inertia of a straight thin rod of mass M and length L about an axis perpendicular to its length and passing through one end is

A ring is rolling down an inclined plane which is inclined at an angle of $30^{\circ}$ to the horizontal. Then the speed of ring after travelling a distance of $10 \mathrm{~m}$ is :-

Two particles move towards each other from rest under a mutual force of attraction. At the instant, when the speed of A is V, the speed of B is V/2. The speed of centre of mass of the system is

The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2 \hat{i}+2 \hat{j}+\hat{k}$, then the value of $\tau$ will be

As observed in the laboratory system, a $6 \mathrm{~MeV}$ proton is incident on a stationary $12 \mathrm{~C}$ target. The velocity of center of mass of the system is (Take mass of proton to be $1 \mathrm{amu}$)

The centre of mass of system of particles does not depend on

A solid sphere rolls down two different inclined planes of same length, but of different inclinations. In both cases

Two points of a rod move with velocities 3v and v perpendicular to the rod and in the same direction separated by a distance 'r'. The angular velocity of the rod is

Three particle of masses 8 kg, 4 kg, 4kg are at (4,1) (-2,2) and (1,-3) respectively. If they are acted upon by forces $16j, -6i$ and $14i$ newton’s respectively, acceleration of the centre of mass of the system is

A flywheel of mass 50 kg and radius of gyration about its axis of rotation of 0.5m is acted upon by a constant torque of 12.5 Nm. Its angular velocity at t=5s is

The moment of inertia $(I)$ of a sphere of radius $R$ and mass $M$ is given by

Statement-1: If the centre of mass of a system is the origin then the total mass to the right of origin is same as total mass to left of origin.
Statement-2: $x_{\mathrm{cm}}=\frac{\Sigma_{m_i x_i}}{\Sigma_{m_i}}$

The angular speed of fly wheel moving with uniform angular acceleration changes from 1200 rpm to 3120 rpm in 16 seconds. The angular acceleration in $\mathrm{rad/}{\mathrm{s}^2}$ is:

A particle performing uniform circular motion has

Which of the following relation is correct? (Where symbol have their usual meanings)

The two persons of masses 55 kg and 65 kg respectively, are at at opposite ends of a boat. The length of the boat is 30 m and weight 100 kg. The 55 kg man walkup to the 65 kg man and its with him. If the boat is in still water the centre of mass of the system shifts by

A ring of radius 1 m and mass 20 kg is rotating about is diameter with angular velocity of 40 rad/sec. Its kinetic energy is

The speed of a motor decreases from 1200 rpm to 600 rpm in 20s. The total number of rotations it makes before coming to rest.

An automobile engine develops 50 kW when rotating at a speed of 900 rev/min. The torque it delivers (in N-m)

$1^{\circ}$ is equal to

A solid cylinder begins to rolls down on ${30^0}$ incline of length 9 m without slipping. The speed of centre of mass at the bottom of plane is

The position vectors of two 1 kg particles, (A) and (B), are given by $\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{\mathrm{i}}+\alpha_2 \text{t}\hat{\mathrm{j}}+\alpha_3 \text{t} \hat{\mathrm{k}}\right) \mathrm{m}$ and $\overrightarrow{\mathrm{r}}_{\mathrm{B}}=\left(\beta_1 \mathrm{t} \hat{\mathrm{i}}+\beta_2 \mathrm{t}^2 \hat{\mathrm{j}}+\beta_3 \mathrm{t} \hat{\mathrm{k}}\right) \mathrm{m},$ respectively; $\left(\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{~nm} / \mathrm{s}, \alpha_3=2 \mathrm{~m} / \mathrm{s}, \beta_1=2 \mathrm{~m} / \mathrm{s},\beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{p} \mathrm{m} / \mathrm{s}\right),$ where t is time, n and p are constants, At $\text{t}=1 \mathrm{~s},$ $\left|\overrightarrow{\mathrm{V}}_{\mathrm{A}}\right|=\left|\overrightarrow{\mathrm{V}}_{\mathrm{B}}\right|$ and velocities $\mathrm{\vec{V}_A}$ and $\mathrm{\vec{V}_B}$ of the particles are orthogonal to each other. At $\text{t}=1 \mathrm{~s},$ the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{\mathrm{L}} \mathrm{~kgm}^2 \mathrm{~s}^{-1}.$ The value of 'L' is

An automobile engine develops ${10^5}\,\mathrm{W}$ power when rotating at an angular speed of 900 rad/minute. The instantaneous torque delivered by the engine is

A disc of radius '$r$' is removed from the disc of radius '$R$' then
a) The minimum shift in centre of mass is zero
b) The maximum shift in centre of mass cannot be greater than $r^2 /(R+r)$
c) Centre of mass must lie where mass exists
d) The shift in centre of mass is $r^2 /(R+r)$

A solid sphere of radius ${R}$ is placed on a smooth horizontal surface. A horizontal force $F$ is applied at height $h$ from the lowest point. For the maximum acceleration of centre of mass, which is correct?