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Applying NLM along 3 Axis

Two blocks, w...

Question

Two blocks, with masses $m_{1}$ and $m_{2}$ , are stacked as shown in Fig. and placed on a frictionless horizontal surface. There is a friction between two blocks. An external force applied to the top block at an angle $α$ below the horizontal.
a. If the two blocks move together, find their acceleration.
b. Calculate the maximum value of force so that blocks will move together.

A

a.

\frac{F c o s α}{(m_{1} + m_{2})}

; b.

\frac{μ_{s} (m_{1} + m_{2}) m_{1} g}{[2 m_{2} c o s α - μ_{s} (m_{1} + m_{2}) s i n α]}

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B

a.

\frac{F c o s α}{(m_{1} + m_{2})}

; b.

\frac{μ_{s} (m_{1} + m_{2}) m_{1} g}{[m_{2} c o s α - μ_{s} (m_{1} + m_{2}) s i n α]}

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C

a.

\frac{F c o s α}{(m_{1} + m_{2})}

; b.

\frac{μ_{s} (m_{1} + m_{2}) m_{1} g}{[3 m_{2} c o s α - μ_{s} (m_{1} + m_{2}) s i n α]}

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D

a.

\frac{F c o s α}{(m_{1} + m_{2})}

; b.

\frac{μ_{s} (m_{1} + m_{2}) m_{1} g}{[4 m_{2} c o s α - μ_{s} (m_{1} + m_{2}) s i n α]}

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Solution

The correct option is B a. $\frac{F c o s α}{(m_{1} + m_{2})}$ ; b. $\frac{μ_{s} (m_{1} + m_{2}) m_{1} g}{[m_{2} c o s α - μ_{s} (m_{1} + m_{2}) s i n α]}$

$Soln. (a) Both the blocks are moving together 50 we can consider$
$as a big block of mass (m_{1} + n_{2})$
$Force equation:$
$F cos α = (m_{1} + m_{2}) a$

(b) $N = (m_{1} g + F sin θ)$
$F_{f} = μ N$
$F cos θ - F_{f} = m_{1} a$
$F_{1} cos θ - μ (n_{1} g + F sin θ) = m_{1} - (1)$
$F cos θ = (m_{1} + m_{2}) a - 2$
$∴ F cos α - μ (m_{1} g + F sin α) = \frac{m_{1} F cos θ}{m_{1} + m_{2}}$
$∴ F = \frac{μ m_{1} (m_{1} + m_{2}) g}{m_{2} cos α - μ (m_{1} + m_{2}) sin α}$

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