$A$ thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of

If $a$ and $b$ are positive integers such that $b > a$,then $\lim _{n \rightarrow \infty} \left[ \frac{1}{na} + \frac{1}{na+1} + \frac{1}{na+2} + \ldots + \frac{1}{nb} \right] = $

The magnetic needle of a vibration magnetometer makes $12$ oscillations per minute in the horizontal component of the earth's magnetic field. When an external short bar magnet is placed at some distance along the axis of the needle in the same line,it makes $15$ oscillations per minute. If the poles of the bar magnet are interchanged,the number of oscillations it makes per minute is

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is

$A$ concave mirror of focal length $f$ produces an image $n$ times the size of the object. If the image is real,then the distance of the object from the mirror is:

The system shown in the figure is in equilibrium and at rest. The spring and string are massless. Now,the string is cut. The acceleration of mass $2m$ and $m$ just after the string is cut will be: