THE FUNDAMENTAL EQUATION OF DYNAMICS

A small bar starts sliding down an inclined plane forming an angle $\alpha$ with the horizontal. The friction coefficient depends on the distance $x$ covered as $k=ax$, where $a$ is a constant. Find the distance covered by the bar till it stops, and its maximum velocity over this distance.

A body of mass $m$ rests on a horizontal plane with the friction coefficient $k$. At the moment $t=0$ a horizontal force is applied to it, which varies with time as $\overrightarrow{F}=\overrightarrow{a}t$, where $\overrightarrow{a}$ is a constant vector. Find the distance traversed by the body during the first $t$ seconds after the force action began.

A body of mass $m$ is thrown straight up with velocity $v_0$. Find the velocity $v^{\text{'}}$ with which the body comes down if the air drag equals $k{v}^2$, where $k$ is a constant and $v$ is the velocity of the body.

A particle of mass $m$ moves in a certain plane $P$ due to a force $F$ whose magnitude is constant and whose vector rotates in that plane with a constant angular velocity $\omega$. Assuming the particle to be stationary at the moment $t=0$, find,

$\left(a\right)$ its velocity as a function of time,

$\left(b\right)$ the distance covered by the particle between two successive stops, and the mean velocity over this time.

A small disc $A$ is placed on an inclined plane forming an angle $\alpha$ with the horizontal and is imparted an initial velocity $v_0$. Find out how the velocity of the disc depends on the angle $\phi$, if the friction coefficient $k=\tan \alpha$ and at the initial moment is ${\phi }_0=\frac{\mathrm{\pi }}{2}$ .

A chain of length $l$ is placed on a smooth spherical surface of the radius $R$ with one of its ends fixed at the top of the sphere. What will be the acceleration $w$ of each element of the chain when its upper end is released? It is assumed that the length of the chain $l<\frac{1}{2}\mathrm{\pi }R$.

A small body is placed on the top of a smooth sphere of radius $R$. Then the sphere is imparted a constant acceleration $\overrightarrow{w_0}$ in the horizontal direction and the body begins sliding down. Find, 

$\left(a\right)$ the velocity of the body relative to the sphere at the moment of break-off, 

$\left(b\right)$ the angle ${\text{θ}}_0$ between the vertical and the radius vector drawn from the centre of the sphere to the break-off point; calculate ${\text{θ}}_0$ for $w_0=g$.

A sleeve $A$ can slide freely along a smooth rod bent in the shape of a half-circle of radius $R$ (Figure.). The system is set in rotation with a constant angular velocity $\omega$ about a vertical axis $O{O}^{\text{'}}$. Find the angle $\text{θ}$ corresponding to the steady position of the sleeve.