A small block of mass m slides to the left on a...

A small block of mass m slides to the left on a frictionless, horizontal surface with speed v. Then, the block slides up a rough ramp that has an incline angle of ?. The coefficient of kinetic friction between the ramp and box is ?. The acceleration due to gravity is g. Final position bv m ?x Initial position h v m Rough surface ? Frictionless surface What is the height h of the block when its speed up the ramp equals bv, where 0 < b < 1? h = v^2(1 - b^2) / (2g + mg sin(2))

Transcript

0:00 Hi.

00:01 Here in this given problem there is a smooth horizontal surface which then converts into a rough inclined surface.

00:18 A block of mass m starts moving over the horizontal surface towards left with an initial velocity v.

00:28 Mass of the block that is m and then when it starts climbing over this inclined plane which is inclined at an angle theta there will be two components of its weight now weight mg acting vertically downward one of the component m g g cos theta normally to this incline plane because this angle will also be theta.

01:06 It will create the same reaction, normal reaction at the block created by this inclined plane.

01:15 And another component down the inclined plane, m .g, sine theta.

01:26 And when the block will be moving up a force of friction, kinetic friction f k that will also act in downward direction so finally it is said to achieve a velocity b v at this point where the corresponding height that is which is missing we have to find this corresponding height of the point where the final velocity becomes b, v.

02:02 Okay, initial velocity vi, that is v, when the block starts climbing over the inclined surface and final velocity f, vf, that is bv, total downward force acting on the block mg sine theta plus force of friction fk, that should be kept equal to m into a, and acceleration which is actually retardation in the direction in the motion of the block in upward direction.

02:38 Now this fk that will be given by mu into normal reaction means mu into m g cost theta that is equal to m a it implies retardation in the motion of the block upward that will be given by canceling m from both the sides and taking this g as a common out...

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