A particle is moving with a velocity $\vec { v } = K y \hat { i } + x \hat { j }$, where $K$ is a constant. The general equation for its path is: (1) $y ^ { 2 } = x +$ constant (2) $x y =$ constant (3) $y = x ^ { 2 } +$ constant (4) $y ^ { 2 } = x ^ { 2 } +$ constant
A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of $3 N$ is applied on the block. The coefficient of static friction between the plane and the block is 0.6 . What should be the minimum value of force $P$, such that the block does not move downward? (take $g = 10 \mathrm {~ms} ^ { - 2 }$) (1) 23 N (2) 25 N (3) 18 N (4) 32 N
A block of mass $m$, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant $k$. The other end of the spring is fixed, as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force $F$, the maximum speed of the block is: (1) $\frac { F } { \sqrt { m k } }$ (2) $\frac { 2 F } { \sqrt { m k } }$ (3) $\frac { \pi F } { \sqrt { m k } }$ (4) $\frac { F } { \pi \sqrt { m k } }$
Three blocks $\mathrm { A } , \mathrm { B }$ and C are lying on a smooth horizontal surface, as shown in the figure. A and B have equal masses, $m$ while C has mass $M$. Block A is given an initial speed $v$ towards B due to which it collides with B perfectly inelastically. The combined mass collides with $C$, also perfectly inelastically . $\frac { 5 } { 6 }$ th of the initial kinetic energy is lost in the whole process. What is the value of $M / m$ ? (1) 3 (2) 4 (3) 5 (4) 2
Two masses $m$ and $\frac { m } { 2 }$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of torsional constant $k$, the restoring torque is $\tau = k \theta$ for angular displacement $\theta$. If the rod is rotated by $\theta _ { 0 }$ and released, the tension in it when it passes through its mean position will be: (1) $k \theta _ { 0 } ^ { 2 }$ (2) $\frac { 3 k \theta _ { 0 } { } ^ { 2 } } { l _ { 0 } }$ (3) $\frac { 2 k \theta _ { 0 } { } ^ { 2 } } { l }$ (4) $\frac { k \theta _ { 0 } { } ^ { 2 } } { l }$
An $L$-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If $A B = B C$, and the angle made by $A B$ with downward vertical is $\theta$, then: (1) $\tan \theta = \frac { 2 } { \sqrt { 3 } }$ (2) $\tan \theta = \frac { 1 } { 3 }$ (3) $\tan \theta = \frac { 1 } { 2 }$ (4) $\tan \theta = \frac { 1 } { 2 \sqrt { 3 } }$