To solve this problem, we need to analyze the motion of both the block and the wedge while considering the forces acting on them. Since the surfaces are frictionless, we can apply the principles of conservation of momentum and energy to find the maximum velocity of the wedge, M, as the block, m, slides down.
Understanding the System
We have a block of mass m on a wedge of mass M. The wedge is initially at rest, and the block starts sliding down due to gravity. As the block descends, it exerts a force on the wedge, causing the wedge to move in the opposite direction. The spring attached to the wedge will also play a role in this dynamic.
Forces and Accelerations
Let’s denote the acceleration of the block relative to the wedge as a and the acceleration of the wedge as A. The gravitational force acting on the block is mg, where g is the acceleration due to gravity. The block will accelerate down the incline of the wedge, while the wedge will accelerate horizontally due to the reaction force from the block.
Applying Newton's Laws
Using Newton's second law, we can set up the equations of motion for both the block and the wedge. The force acting on the block can be expressed as:
- For the block: F = m * a
- For the wedge: F = M * A
Since the system is isolated and there are no external horizontal forces acting on it, we can apply the conservation of momentum. The momentum before the block starts moving is zero, so the momentum after must also be zero:
m * (a) + M * (-A) = 0
This leads to the relationship:
m * a = M * A
Energy Considerations
As the block slides down the wedge, it loses potential energy, which is converted into kinetic energy of both the block and the wedge. The potential energy lost by the block as it descends a height h can be expressed as:
PE = mgh
The kinetic energy gained by the block and the wedge can be expressed as:
KE = (1/2) * m * (v_m)^2 + (1/2) * M * (v_M)^2
Here, v_m is the velocity of the block and v_M is the velocity of the wedge. Since the block is sliding down the wedge, we can relate their velocities through the accelerations:
v_m = a * t and
v_M = A * t
Finding the Maximum Velocity of the Wedge
Substituting the expressions for v_m and v_M into the kinetic energy equation and setting it equal to the potential energy gives us:
mgh = (1/2) * m * (a * t)^2 + (1/2) * M * (A * t)^2
Now, using the relationship between a and A (m * a = M * A), we can express a in terms of A:
a = (M/m) * A
Substituting this back into the energy equation allows us to solve for A, leading to:
A = √(2mgh/(m + M))
Final Result
Thus, the maximum velocity of the wedge M, as the block m starts coming down, is given by:
√(2mgh/(m + M))
Therefore, the correct answer is option B: √(2mgh/(m + M)). This result illustrates how the motion of one object can influence another in a frictionless environment, showcasing the principles of conservation of momentum and energy in action.
