A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is :


[The coefficient of static friction, $$\mu$$_s' is 0.4]

The function of time representing a simple harmonic motion with a period of $${\pi \over \omega }$$ is :

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $$\gamma$$ is the ratio of specific heats) :

A proton and an $$\alpha$$-particle, having kinetic energies K_p and K_$$\alpha$$ respectively, enter into a magnetic field at right angles.

The ratio of the radii of trajectory of proton to that of $$\alpha$$-particle is 2 : 1. The ratio of K_p : K_$$\alpha$$ is :