$A$ body of mass $2 \ kg$ is acted upon by two forces each of magnitude $1 \ N$ and inclined at $60^{\circ}$ with each other. The acceleration of the body in $m/s^2$ is $[\cos 60^{\circ}=0.5]$

Two situations are shown in figures $(a)$ and $(b)$. In each case,$m_1 = 3 \ kg$ and $m_2 = 4 \ kg$. If $a_1$ and $a_2$ are the respective accelerations of the blocks in these situations,then the values of $a_1$ and $a_2$ are respectively [ $g = 10 \ ms^{-2}$ ]

$A$ force $F = Kt$ (where $t$ is the time in seconds and $K = 2 \, N/s$) is applied on a $2 \, kg$ block at $t = 0$ as shown in the figure. The displacement of the $8 \, kg$ block until the time when the $2 \, kg$ block starts slipping on the $8 \, kg$ block will be (coefficient of friction between the $2 \, kg$ block and the $8 \, kg$ block is $0.2$ and between the $8 \, kg$ block and the surface is zero,$g = 10 \, m/s^2$).

$A$ solid ball is suspended from the ceiling of a motor car through a light string. $A$ transverse pulse travels at the speed $60 \text{ cm/s}$ on the string, when the car is at rest. When the car accelerates on a horizontal road, the speed of the pulse is $66 \text{ cm/s}$. The acceleration of the car is nearly $\left(g=10 \text{ m/s}^2\right)$ (in $\text{ m/s}^2$)

$A$ block of mass $m$ is on an inclined plane of angle $\theta$. The coefficient of friction between the block and the plane is $\mu$ and $\tan \theta > \mu$. The block is held stationary by applying a force $P$ parallel to the plane. The direction of force pointing up the plane is taken to be positive. As $P$ is varied from $P_1 = mg(\sin \theta - \mu \cos \theta)$ to $P_2 = mg(\sin \theta + \mu \cos \theta)$,the frictional force $f$ versus $P$ graph will look like:

$A$ helicopter of mass $2000 \,kg$ rises with a vertical acceleration of $15 \,m s^{-2}$. The total mass of the crew and passengers is $500 \,kg$. Calculate the magnitude and direction of the following (take $g = 10 \,m s^{-2}$):
$(a)$ Force on the floor of the helicopter by the crew and passengers.
$(b)$ Action of the rotor of the helicopter on the surrounding air.
$(c)$ Force on the helicopter due to the surrounding air.