Question

The block $m_1$ is loaded on the block $m_2$. If there is no relative sliding between the blocks and the inclined plane is smooth, find the friction force between the blocks.

• A. $m_{1}gsin\theta$
• B. ($m_1+m_2$)g sin$\theta$
• C. 0
• D. $\mu m_{1}gcos\theta$

Answer 1

The correct option is C

0

So its given that both $m_1$ & $m_2$ have same acceleration.

That means they are moving together as a single system

Lets draw free body diagram of this system

No friction in the free body diagram as incline is smooth

$\Rightarrow$ ($m_1$ + $m_2$) g sin $\theta$ = ($m_1$ + $m_2$)a

a = g sin $\theta$

now draw free body diagram of $m_1$ alone

either friction can be above or below

so $m_{1}gsin\theta \pm fr=m_{1}a=m_{1}gsin\theta$

$\Rightarrow$ fr has to be 0

Alternate solution:

Take friction in any one direction assume $m_1$ has friction in upward direction then $m_2$ will have it in downward direction.

Both blocks are moving with same acceleration as no slipping

$\Rightarrow$ $m_1$ g sin $\theta$ - fr = $m_1$ a . . . . . . (1)

$m_2$ g sin $\theta$ + fr = $m_2$ a . . . . . . (2)

adding we get

($m_1$ + $m_2$) g sin $\theta$ = ($m_1$ + $m_2$)a

$\Rightarrow$ a = g sin $\theta$

Substituting a in equation (1) we get
fr = 0
so friction has to be 0