A long plank begins to move at t = 0 and accelerates along a straight track with a speed given by v = 2t^2 for 0 ≤ t ≤ 2. After 2 s, the plank continues to move at the constant speed acquired. A small block initially at rest on the plank begins to slip at t = 1 s and stops sliding at t = 3 s. Find the coefficient of static and kinetic friction between the block and the plank.


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Question Number 22456 by Tinkutara last updated on 18/Oct/17

$A long plank begins to move at t = 0$ $and accelerates along a straight track$ $with a speed given by v = 2 t^{2} for 0 ⩽ t$ $⩽ 2 . After 2 s, the plank continues to$ $move at the constant speed acquired .$ $A small block initially at rest on the$ $plank begins to slip at t = 1 s and stops$ $sliding at t = 3 s . Find the coefficient of$ $static and kinetic friction between the$ $block and the plank .$
	Answered by ajfour last updated on 18/Oct/17

	$A c c e l e r a t i o n o f p l a n k A = 4 t$ ${(a c c e l e r a t i o n)}_{m a x} o f b l o c k = \frac{μ_{s} m g}{m}$ $= μ_{s} g$ $g i v e n : b l o c k b e g i n s s l i d i n g a t t = 1 s$ $m e a n s v e l o c i t y a n d h e n c e a c c -$ $e l e r a t i o n o f p l a n k e x c e e d s t h a t$ $o f b l o c k . b l o c k l a g s b e h i n d w . r . t .$ $p l a n k .$ $\Rightarrow μ_{s} g = 4 (1 s) \Rightarrow μ_{s} = 0.4$ $A t t = 1 s r e l a t i v e v e l o c i t y o f b l o c k$ $i s z e r o .$ $A t t = 2 s i t i s$ $= \int_{1}^{2} a_{r} d t = \int_{1}^{2} (μ_{k} g - 4 t) d t$ $= μ_{k} g - 6$ $v_{r} = u_{r} + a_{r} △ t; A f t e r t = 2 s,$ $a_{r} = μ_{k} g - 0$ $\Rightarrow v_{r} = (- 6 + μ_{k} g) + μ_{k} g △ t$ $s i n c e a t t = 3 s v_{r} = 0, w i t h △ t = 1 s$ $w e h a v e 0 = - 6 + 2 μ_{k} g$ $o r μ_{k} = 0.3$
	Commented by Tinkutara last updated on 18/Oct/17

	$Thank you very much Sir!$
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