Mechanics - Physics Previous Year Questions

Consider a mass-pulley system as shown in the figure. There is a wedge of mass $M$ and equal wedge angles $\theta$ lying on a rigid horizontal table. The coefficient of friction between the wedge and the table is $\mu$. There are two blocks of mass $m_1$ and $m_2$ lying on the incline of the wedge. The coefficients of friction between the blocks and wedge are $\mu_1$ and $\mu_2$ as shown in the figure. Consider $m_1>m_2$ and the coefficients of friction ( $\mu, \mu_1$ and $\mu_2$ ) to be less than $\tan \theta$. Gravity is acting downwards with acceleration due to gravity $g$. What should be the value of $\frac{m_1}{m_2}$ so that the system is in equilibrium?

Consider two waves, which are given by $y_1(x, t)=A \sin (k x-\omega t)$ and $y_2(x, t)=\sqrt{3} A \cos (k x-\omega t)$, where $k$ is the wave number and $\omega$ is the angular frequency. The amplitude of the resultant waveform obtained by the superposition of the two waves is $A_s$ and its phase difference with $y_1$ is $\phi_s$. What are $A_s$ and $\phi_s$ ?

The potential energy of a point particle of mass $m$ undergoing rectilinear motion along the $x$-axis is given by

$$V(x)=A x+B x^2$$

What is the maximum speed attained by the particle if it starts from rest at $x=\frac{A}{B}$ ?