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Question: In the figure \(\left( A \right)\) and \(\left( B \right)\) \(AC,\,DG,\,\) and \(GF\) are fixed incl...

In the figure (A)\left( A \right) and (B)\left( B \right) AC,DG,AC,\,DG,\, and GFGF are fixed inclined planes, BC=EF=xBC = EF = x and AB=DE=yAB = DE = y. A small block of mass MM is released from the point AA . It slides down ACAC and reaches CC with a speed vc{v_c} . The same block is released from rest from the point DD . It slides down DGFDGF and reaches the point FF with speed vF{v_F} . The coefficient of kinetic frictions between the block and both the surfaces ACAC and DGFDGF are μ\mu . Calculate vc{v_c} and vF{v_F} .

Explanation

Solution

Hint Analyse the diagram given in the question, and derive the equation of the motion of the block from that. Use the potential energy and the kinetic energy in it. The simplification of the above equation provides the value of the velocity of the block.

Useful formula
(1) The formula of the potential energy is given by
P=mghP = mgh
Where PP is the potential energy, mm is the mass of the block, gg is the acceleration due to gravity and hh is the height.
(2) The formula of the kinetic energy is given by
K=12mv2K = \dfrac{1}{2}m{v^2}
KK is the kinetic energy, vv is the velocity of motion.

Complete step by step solution
It is given that the
AC,DG,AC,\,DG,\, and GFGF are fixed inclined planes
BC=EF=xBC = EF = x
AB=DE=yAB = DE = y
The speed at which the ACAC reaches CC is vc{v_c} .
The speed at which the block slides down ACAC and reaches CC is vc{v_c}
The coefficient of the kinetic friction is considered as μ\mu
It is known that the potential energy of the block is converted into the kinetic energy while moving.
PK=f2s1+f2s2P - K = {f_2}{s_1} + {f_2}{s_2}
Substituting the formula in it,
mgy12mvF2=μmgcosβS1μmgcosαS2mgy - \dfrac{1}{2}mv_F^2 = \mu mg\cos \beta {S_1} - \mu mg\cos \alpha {S_2}
By simplifying the above equation, we get

vF=2g(yμx){v_F} = \sqrt {2g\left( {y - \mu x} \right)}

Note The friction affects the motion of the body when it rolls or slides down the surface. This is because the surface with the diffraction is rough and this reduces the velocity . This sliding motion is due to the acceleration due to gravity and not due to the external force applied on it.