Laws of Motion ...

If a particle of mass $$m$$ resting at the top of a fixed smooth hemisphere of radius $$R$$ is displaced, then the angle with vertical at which it looses contact with the surface is:

If the beads are released from rest, such that the bead $$A$$ is at the highest position of ring at time $$t = 0$$, the magnitude of acceleration of the bead $$B$$ in the vertical direction at the same instant is

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 particle is released on a vertical smooth semicircular track from point X so that OX makes angle $$\theta$$ form the vertical (see figure). The normal reaction of the track on the particle vanishes at point Y where OY makes angle $$\phi$$ with the horizontal. Then:

A uniform rod of length $$L$$ rests against a smooth wall as shown in figure. Find the friction coefficient between the ground and the lower end if the minimum angle that the rod can make with the horizontal is $$\theta$$.

A particle of mass 'm' strikes a smooth stationary wedge of mass M with a velocity $$v_0$$, at an angle $$\theta$$ with horizontal if the collision is perfectly inelastic, the impulse on the wedge is:

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.

Two masses $$A$$ and $$B$$ of $$10Kg$$ and $$5Kg$$ respectively are connected with a string passing over a frictionless pulley fixed at the corner of a table as in figure. The coefficient of friction of $$A$$ with the table is $$0.2$$. The minimum mass of $$C$$ that may be placed on $$A$$ to prevent it from moving is equal to:

A $$"V"$$ shaped rigid body has two identical uniform arms. What must be the angle between the two arms so that when the body is hung from one end, the other arm is horizontal?

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