A vector $\vec { A }$ is rotated by a small angle $\Delta \theta$ radians ( $\Delta \theta \ll 1$ ) to get a new vector $\vec { B }$. In that case $| \vec { B } - \vec { A } |$ is : (1) $| \vec { A } | \left[ 1 - \frac { ( \Delta \theta ) ^ { 2 } } { 2 } \right]$ (2) 0 (3) $| \vec { A } | \Delta \theta$ (4) $| \vec { B } | \Delta \theta - | \vec { A } |$
From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of $48 \mathrm {~m} / \mathrm { s }$. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g = 32 \mathrm {~m} / \mathrm { s } ^ { 2 }$, is: (1) 112 (2) 88 (3) 128 (4) 100
A large number ( $n$ ) of identical beads, each of mass $m$ and radius $r$ are strung on a thin smooth rigid horizontal rod of length $L ( L \gg r )$ and are at rest at random positions. The rod is mounted between two rigid supports. If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic): (1) $\frac { m v ^ { 2 } } { L - n r }$ (2) $\frac { m v ^ { 2 } } { L - 2 n r }$ (3) $\frac { m v ^ { 2 } } { 2 ( L - n r ) }$ (4) Zero
A particle is moving in a circle of radius $r$ under the action of a force $F = \alpha r ^ { 2 }$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $= 0$ for $r = 0$ ): (1) $\frac { 5 } { 6 } \alpha r ^ { 3 }$ (2) $\alpha r ^ { 3 }$ (3) $\frac { 1 } { 2 } \alpha r ^ { 3 }$ (4) $\frac { 4 } { 3 } \alpha r ^ { 3 }$
A uniform thin rod AB of length $L$ has linear mass density $\mu ( x ) = a + \frac { b x } { L }$, where $x$ is measured from A. If the CM of the rod lies at a distance of $\left( \frac { 7 } { 12 } L \right)$ from A, then $a$ and $b$ are related as: (1) $2 a = b$ (2) $a = 2 b$ (3) $a = b$ (4) $3 a = 2 b$
A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is $12 \mathrm { rad } \mathrm { s } ^ { - 1 }$, the magnitude of its angular momentum about a point on the ground right under the center of the circle is: (1) $14.4 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$ (2) $11.52 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$ (3) $20.16 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$ (4) $8.64 \mathrm {~kg} \mathrm {~m} ^ { 2 } \mathrm {~s} ^ { - 1 }$
A pendulum with the time period of 1 s is losing energy due to damping. At a certain time, its energy is 45 J. If after completing 15 oscillations its energy has become 15 J, then its damping constant (in $\mathrm { s } ^ { - 1 }$) will be (1) $\frac { 1 } { 2 }$ (2) $\frac { 1 } { 15 } \ln 3$ (3) $\frac { 1 } { 30 } \ln 3$ (4) 2
A cylindrical block of wood (density $= 650 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$), of base area $30 \mathrm {~cm} ^ { 2 }$ and height 54 cm, floats in a liquid of density $900 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly): (1) 52 cm (2) 26 cm (3) 39 cm (4) 65 cm