A body is projected up along a rough inclined plane from the bottom with some velocity. It travels up the incline and then returns. If the time of ascent is $t_{\mathit{a}}$ and time of descent is $t_{\mathit{d}}$, then

Starting from rest, a flat car is given a constant acceleration $a_0=2 \mathrm{m} {\mathrm{s}}^{-2}$. A cable is connected to a crate $A$ of mass $50 \mathrm{kg}$ as shown. Neglect the friction between floor and car wheels and mass of pulley. The coefficient of friction between crate & floor of the car is $\mathrm{\mu } = 0.3$. The tension in the cable is –

A block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined at an angle $\mathrm{\theta }$ to the vertical. The minimum value of the coefficient of friction between the block and the surface for which the block will remain in equilibrium is:

A plank is held at an angle $\mathrm{\alpha }$ to the horizontal (figure) on two fixed supports $A$ and $B$. The plank can slide against the supports (without friction) because of its weight $Mg$. With what acceleration and in which direction, a man of mass $m$ should move so that the plank does not move.

A $1.5 \mathrm{kg}$ box is initially at rest on a horizontal surface when at $t=0$ a horizontal force $\stackrel{\mathit{\rightharpoonup }}{\mathit{F}}=(1.8t)\hat{\mathrm{i}} \text{N}$ (with $t$ in seconds) is applied to the box. The acceleration of the box as a function of time $t$ is given by:
$(g=10 \mathrm{m} {\mathrm{s}}^{-2})$

$\begin{array}{c}\stackrel{\mathit{\rightharpoonup }}{a}=0\end{array}$ for $0\le t\mathit{ }\le 2.85$

$\begin{array}{c}\stackrel{\mathit{\rightharpoonup }}{\mathit{a}}=(1.2t - 2.4)\hat{\mathrm{i}} {\text{m s}}^{-2}\end{array}$ for $t\mathit{ }>2.85$

The coefficient of kinetic friction between the box and the surface is:

In the given figure the coefficient of friction between $4 \mathrm{kg}$ and $5 \mathrm{kg}$ blocks is $0.2$ and between $5 \mathrm{kg}$ block and ground is $0.1$. Choose the correct statements.

Given $m_{\mathit{A}}=30 \mathrm{kg}, m_{\mathit{B}} = 10 \mathrm{kg}, m_{\mathit{C}}=20 \mathrm{kg}$. Between $A$ and $B$ friction coefficient ${\mathrm{\mu }}_1 = 0.3$, between $B$ and $C$ friction coefficient ${\mathrm{\mu }}_2 = 0.2$ and between $C$ and ground ${\mathrm{\mu }}_3 = 0.1$. The least horizontal force $F$ to start the motion of any part of the system of three blocks resting upon one another as shown in the figure is $(g=10 \mathrm{m} {\mathrm{s}}^{-2})$