When a body moves with some friction on a surface,

When forces $F_1, F_2, F_3$ are acting on a particle of mass $m$ such that $F_2$ and $F_3$ are mutually perpendicular,the particle remains stationary. If the force $F_1$ is now removed,the acceleration of the particle is:

$A$ system consists of three masses $m_1, m_2$ and $m_3$ connected by a string passing over a pulley $P$. The mass $m_1$ hangs freely and $m_2$ and $m_3$ are on a rough horizontal table (the coefficient of friction $= \mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is (Assume $m_1 = m_2 = m_3 = m$)

$A$ particle of mass $m=1 \ kg$ moves in the $xy$-plane. The force on it at time $t$ is $F(t)=[2 \sin (\alpha t) \hat{i}+3 \cos (\alpha t) \hat{j}] \ N$, where $\alpha=1 \ s^{-1}$. At time $t=0$, the particle is at rest at the origin. Calculate the magnitude of its position vector $r$ (in $m$) and velocity vector $v$ (in $m/s$) at time $t=\frac{\pi}{2} \ s$.

Two pulley arrangements shown in the figure are identical. The mass of the rope is negligible. In figure $(a)$,the mass $m$ is lifted by attaching a mass $2m$ to the other end of the rope. In figure $(b)$,$m$ is lifted up by pulling the other end of the rope with a constant downward force $F = 2mg$. The accelerations of $m$ in the two cases are respectively:

Two masses $A$ and $B$ of $10\,kg$ and $5\,kg$ respectively are connected by a string passing over a frictionless pulley fixed at the corner of a table. The coefficient of friction between mass $A$ and the table is $0.2$. The minimum mass of $C$ that must be placed on $A$ to prevent it from moving is equal to $...........\,kg$.