Work Energy and...

A small sphere is given vertical velocity of magnitude $$v_{0} = 5m/s$$ and it swings in a vertical plane about the end of massless string. The angle $$\theta$$ with the vertical at which string will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere, is $$[g = 10 m/s^{2}]$$.

Work done in time $$t$$ on a body of mass $$m$$ which is accelerated from rest to a speed $$v$$ in time as a function of time $$t$$ is given by

A mass tied to a string moves in a vertical circle and at the point $$P$$ its speed is $$5m/s$$. At the point $$P$$  the string breaks. The mass will reaches height above $$P$$ of nearly $$\left( g=10m/{ s }^{ 2 } \right) $$

A stone of mass $$1\ kg$$ tied to a light inextensible string of length $$L = \dfrac{10}{3}\ m$$, whirling in a circular path in a vertical plane. The ratio of maximum tension in the string to the minimum tension in the string is $$4$$. If g is taken to be $$10\ m/s^2$$ the speed of the stone at the highest point of the circle is

From $$x=0$$ to $$x=6$$, the force experienced by an object varies according to the function $$F\left(x\right)=\sqrt{6x-{x}^{2}}$$, as shown above. What is the work done by this force as the object moves from $$x=0$$ to $$x=3$$?
Assume all numbers are given in standard units.

An athlete in the Olympic games covers a distance of $$100$$m in $$10$$s. His kinetic energy can be estimated to be in the range. (Assume weight = 60kg)

Two blocks are connected by a massless string that passes over a frictionless peg as shown in fig. One end of the string is attached to a mass $$m_1 = 3kg$$, i.e. a distance $$R = 1.20 m$$ from the peg. The other end of the string is connected to a block of mass $$m_2 = 6 kg$$ resting on a table. When the 3 kg block be released at $$\displaystyle \theta = \frac{\pi}{k}$$, the 6 kg block just lift off the table? Find the value of k.

A car of mass $$m$$ starts moving so that its velocity varies according to the law $$v=\beta \sqrt { s }$$, where $$\beta$$ is a constant, and $$s$$ is the distance covered. The total work performed by all the forces which are acting on the car during the first $$t$$ seconds after the beginning of motion is:

By applying a force $$\vec{F} = (3xy - 5z)\hat{j} + 4z\hat{k}$$ a particle is moved along the path $$y=x^{2}$$ from point $$(0,0,0)$$ to point $$(2,4,0)$$.  The work done by the $$F$$ on the particle is 

A small block of mass m is pushed on a smooth track from position A with a velocity $$2\sqrt5$$ times the minimum velocity required to reach point D. The block will leave the contact with track at the point where normal force between them becomes zero.
At what angle $$\theta$$ with horizontal does the block gets separated from the track?