A particle moving in a straight line covers half the distance with speed $6 \mathrm {~m} / \mathrm { s }$. The other half is covered in two equal time intervals with speeds $9 \mathrm {~m} / \mathrm { s }$ and $15 \mathrm {~m} / \mathrm { s }$ respectively. The average speed of the particle during the motion is : (1) $10 \mathrm {~m} / \mathrm { s }$ (2) $8 \mathrm {~m} / \mathrm { s }$ (3) $9.2 \mathrm {~m} / \mathrm { s }$ (4) $8.8 \mathrm {~m} / \mathrm { s }$
A light unstretchable string passing over a smooth light pulley connects two blocks of masses $m _ { 1 }$ and $m _ { 2 }$. If the acceleration of the system is $\frac { g } { 8 }$, then the ratio of the masses $\frac { m _ { 2 } } { m _ { 1 } }$ is : (1) $8 : 1$ (2) $5 : 3$ (3) $4 : 3$ (4) $9 : 7$
A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt { x }$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = \mathrm { d }$, will be : (1) $\frac { m } { 2 \alpha ^ { 2 } d }$ (2) $\frac { \mathrm { md } } { 2 \alpha ^ { 2 } }$ (3) $2 m \alpha ^ { 2 } d$ (4) $\frac { m \alpha ^ { 2 } d } { 2 }$
A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is : (1) $\mathrm { W } \cos \theta$ (2) $\frac { W } { 2 }$ (3) W (4) $W \sin \theta$
An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac { \mathrm { GM } _ { \mathrm { e } } \mathrm { m } } { 21 \mathrm { R } _ { \mathrm { e } } }$. The value of $x$ is (take $\mathrm { R } _ { \mathrm { e } } = 6370 \mathrm {~km}$) : (1) 10 (2) 12 (3) 9 (4) 11
If $\vec { a }$ and $\vec { b }$ makes an angle $\cos ^ { - 1 } \left( \frac { 5 } { 9 } \right)$ with each other, then $| \vec { a } + \vec { b } | = \sqrt { 2 } | \vec { a } - \vec { b } |$ for $| \vec { a } | = n | \vec { b } |$ The integer value of n is $\_\_\_\_$
A string is wrapped around the rim of a wheel of moment of inertia $0.40 \mathrm { kgm } ^ { 2 }$ and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is $x \mathrm { rad } / \mathrm { s }$, where $x$ is $\_\_\_\_$
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 \mathrm {~m} , 2 \mathrm {~ms} ^ { - 1 }$ and $16 \mathrm {~ms} ^ { - 2 }$ at a certain instant. The amplitude of the motion is $\sqrt { x } , \mathrm {~m}$ where $x$ is $\_\_\_\_$