Name: Inertial Frames of Reference
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Imagine if you are on a bus or a train, can you say you are in an inertial frame of reference? If yes, how can you predict it? Let us explore more in this vi...

Question

In the arrangement shown in figure the masses $m$ of the bar and $M$ of the wedge, as well as the wedge angle $α$ , are known. The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge $M$ .

Accepted Answer

Draw the free-body diagram of the system. According to the diagram, it is obvious that the wedge $M$ will move towards the right, and the block will move down along the wedge. And the distance travelled by the block on the wedge, and the distance travelled by the wedge on the floor must be equal because the length of the thread is constant. Hence,

$α$ .

Since direction of displacement is not changing, differentiating above equation with time we will get ${\vec{v}}_{mM} = {\vec{v}}_{M}$ , again after differentiating it we will get,
$w_{mM} = w_{M}$ .

Let $w_{mM} = w_{M} = w$

Accordingly, the diagram of the kinematical dependence is shown in the figure.

So from triangle law of vector addition, ${\vec{w}}_{m} = {\vec{w}}_{mM} + {\vec{w}}_{M}$ ,
$w_{m} = \sqrt{w_{M}^{2} + w_{mM}^{2} - 2 w_{mM} w_{M} \cos α} = w \sqrt{2 (1 - \cos α)}$
From $M$ , (for the wedge in horizontal direction),
$T - T \cos α + N \sin α = M w . . . (1)$
For the bar $m$ , write the equation using Newton's second law along the plane and normal to plane,
$m g \sin α - T = m w_{m (x)} = m [w_{mM (x)} + w_{M (x)}]$

$\Rightarrow m g \sin α - T = m [w_{mM} + w_{M} \cos (π - α)]$

$\Rightarrow m g \sin α - T = m w (1 - \cos α) . . . (2)$

$m g \cos α - N = m w_{m (y)} = m [w_{mM (y)} + w_{M (y)}] \Rightarrow m g \cos α - N = m [0 + w \sin α] . . . (3)$

Using the above equations $(1)$ , $(2)$ and $(3)$ ,

$w = \frac{m g \sin α}{M + 2 m (1 - \cos α)}$ .

In the arrangement shown in figure the masses $m$ of the bar and $M$ of the wedge, as well as the wedge angle $α$ , are known. The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge $M$ .

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

Learn and Prepare for any exam you want !

In the arrangement shown in figure the masses m of the bar and M of the wedge, as well as the wedge angle α, are known. The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge M.

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

Learn and Prepare for any exam you want !

In the arrangement shown in figure the masses $m$ of the bar and $M$ of the wedge, as well as the wedge angle $α$ , are known. The masses of the pulley and the thread are negligible. The friction is absent. Find the acceleration of the wedge $M$ .