A wedge $Y$ with mass of 10 kg and all frictionless surfaces and the inclined surface making $37^\circ$ with horizontal. A block $X$ with mass 2 kg is placed at the highest point of the wedge as shown in figure is at rest. At $t = 0$ wedge ($Y$) is pulled toward right with constant force ($f$) of 24 N. Taking the block $X$ at rest at $t = 0$, the time taken by it to slide down 8.8 m on the slope, while $Y$ is on the move, is ________s. (take $\tan (37^\circ) = 3/4$ and $g = 10 \text{ m/s}^2$)
Step 1: Understanding the Question:
A block $X$ is on a wedge $Y$. The wedge is pulled to the right by a force $f$. We need to find the relative acceleration of the block down the incline to calculate the time taken to travel a given distance. Step 2: Key Formula or Approach:
1. Find horizontal acceleration of the wedge system: $A = \frac{f}{M_{total}}$.
2. Use pseudo-force logic: Relative acceleration $a_{rel} = g \sin \theta - A \cos \theta$ (since pseudo-force acts opposite to wedge acceleration).
3. Kinematics: $s = \frac{1}{2} a_{rel} t^2$. Step 3: Detailed Explanation:
First, calculate the acceleration of the wedge system. Since all surfaces are frictionless and the wedge is pulled with $f = 24$ N:
\[ A = \frac{f}{M_X + M_Y} = \frac{24}{2 + 10} = 2 \text{ m/s}^2 \]
Now, consider the motion of block $X$ relative to the wedge.
The forces acting on the block along the incline in the wedge's frame are:
- Component of gravity: $mg \sin \theta$ (down the incline)
- Component of pseudo-force: $mA \cos \theta$ (up the incline, because $A$ is to the right)
Net relative acceleration $a_{rel} = g \sin 37^\circ - A \cos 37^\circ$.
Given $\tan 37^\circ = 3/4 \implies \sin 37^\circ = 3/5$ and $\cos 37^\circ = 4/5$.
\[ a_{rel} = 10 \left( \frac{3}{5} \right) - 2 \left( \frac{4}{5} \right) = 6 - 1.6 = 4.4 \text{ m/s}^2 \]
Using the kinematic equation for distance $s = 8.8$ m:
\[ s = u_{rel}t + \frac{1}{2} a_{rel} t^2 \]
\[ 8.8 = 0 + \frac{1}{2} (4.4) t^2 \implies 8.8 = 2.2 t^2 \]
\[ t^2 = \frac{8.8}{2.2} = 4 \implies t = 2 \text{ s} \] Step 4: Final Answer:
The time taken is 2 s.
