In a playground,there is a merry-go-round of mass $120 \ kg$ and radius $4 \ m$. The radius of gyration is $3 \ m$. $A$ child of mass $30 \ kg$ runs at a speed of $5 \ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y = \alpha x + \beta$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

Which of the following kingdoms includes multicellular producers?

For the reaction $A + 2B \rightleftharpoons 2C + D$,the initial concentration of $A$ is $a$ and the initial concentration of $B$ is $1.5$ times that of $A$. If the concentrations of $A$ and $D$ are equal at equilibrium,what will be the concentration of $B$ at equilibrium?

As shown in the figure,two particles each of mass $m$ tied at the ends of a light string of length $2 a$ are kept on a frictionless horizontal surface. When the mid-point $(P)$ of the string is pulled vertically upwards with a small but constant force $F$,the particles move towards each other on the surface. The magnitude of acceleration of each particle,when the separation between them becomes $2 x$,is:

$A$ straight line is equally inclined to all the three coordinate axes. Then, the angle made by the line with the $y$-axis is