With what minimum acceleration can a fireman slide down a rope if the breaking strength of the rope is $\frac{2}{3}$ of his weight?

Two masses $M$ and $m$ are connected by a weightless string. They are pulled by a force $F$ on a frictionless horizontal surface as shown in the figure. The tension in the string will be

For the given figure,find the value of $m$ (in $kg$) for which the blocks have an acceleration of $\frac{8}{13}g$. The block on the incline has a mass of $100 \, g = 0.1 \, kg$ and the hanging block on the left has a mass of $50 \, g = 0.05 \, kg$. The surface is frictionless $(\mu = 0)$.

The figure shows a uniform rod of length $30 \,cm$ having a mass of $3.0 \,kg$. The rod is pulled by constant forces of $20 \,N$ and $32 \,N$ as shown. Find the force exerted by the $20 \,cm$ part of the rod on the $10 \,cm$ part (all surfaces are smooth).

Two blocks,each having mass $M$,rest on frictionless surfaces as shown in the figure. If the pulleys are light and frictionless,and $M$ on the incline is allowed to move down,then the tension in the string will be

If a ship of mass $4 \times 10^{7} \,kg$ initially at rest is pulled by a force of $5 \times 10^{4} \,N$ through a distance of $4 \,m$, then the speed of the ship will be (resistance due to water is negligible). (in $\,ms^{-1}$)