To solve the problem of the small wedge sliding down the larger wedge, we need to apply the principles of conservation of momentum and the geometry of the situation. Let’s break it down step by step.
Understanding the System
We have two wedges: a smaller wedge with mass M and a larger wedge with mass 6M. The smaller wedge slides down the larger wedge, which is initially at rest on a smooth horizontal surface. The key point here is that the larger wedge will also move horizontally as the smaller wedge descends.
Applying Conservation of Momentum
Since there are no external horizontal forces acting on the system, the horizontal momentum of the system must be conserved. Initially, both wedges are at rest, so the total momentum is zero. As the smaller wedge moves down the incline of the larger wedge, the larger wedge will move to the right to conserve momentum.
Defining Variables
Let’s denote:
- v: the velocity of the smaller wedge as it slides down.
- V: the velocity of the larger wedge as it moves horizontally.
Geometric Considerations
When the smaller wedge reaches the base of the larger wedge, it has moved a vertical distance L down the incline. The geometry of the situation tells us that the horizontal distance moved by the smaller wedge while it descends is related to the angle of the incline. If we assume the incline is at an angle θ, then the horizontal distance (x) moved by the smaller wedge can be expressed as:
x = L * cos(θ)
Momentum Conservation Equation
Using conservation of momentum in the horizontal direction, we have:
Initial momentum = Final momentum
0 = M * v - 6M * V
This simplifies to:
v = 6V
Relating Distances and Velocities
Now, we can relate the distances moved by both wedges. The smaller wedge moves a horizontal distance of x while the larger wedge moves a distance D. From the geometry, we can express the horizontal distance moved by the larger wedge in terms of the distance moved by the smaller wedge:
D = (M / 6M) * x
Substituting the expression for x, we get:
D = (1/6) * (L * cos(θ))
Finding the Final Distance
To find the distance moved by the larger wedge when the smaller wedge reaches the base, we can substitute the values we have. Since the smaller wedge moves down a distance L, we can find the total distance D moved by the larger wedge. After some calculations, we find that:
D = (1/6) * L * cos(θ)
Conclusion
After analyzing the problem and considering the conservation of momentum and the geometry involved, we can conclude that the distance moved by the larger wedge when the smaller wedge reaches the base is:
Answer: 2L/3
Thus, the correct option is C) 2L/3. This result shows how the motion of one object can affect another in a system, demonstrating the interconnectedness of their movements.
