a 2kg block is placed on a frictionless wedge that is inclined at an angle of 60 degrees from the horizontal. if you release the block, it would slide down the wedge. it turns out that if you push the wedge to the right with acceleration a, the block will actually remain stationary with respect to the wedge. in other words, the block will not have any veritcal acceleration, though it does have the same horizontal acceleration as the wedge.
What acceleration is required for this to occur?
To determine the required acceleration, we need to analyze the forces acting on the block.
1. First, let's consider the forces acting along the inclined plane:
- The weight of the block (mg) acts vertically downward.
- The normal force (N) acts perpendicular to the inclined plane.
- The force of friction (f) acts parallel to the inclined plane, opposing the motion.
2. Since the block is stationary with respect to the wedge, the net force acting on it along the inclined plane is zero. Therefore, the force of friction equals the component of the weight parallel to the inclined plane:
f = mg * sin(θ)
where θ is the angle of inclination (θ = 60 degrees) and m is the mass of the block (m = 2 kg).
3. Now, let's analyze the horizontal forces acting on the block:
- The force of friction (f) acts in the opposite direction to the desired acceleration (a).
- The horizontal component of the weight (mg * cos(θ)) acts in the same direction as the desired acceleration (a).
4. Since the block is stationary with respect to the wedge, the net horizontal force acting on it is zero. Therefore, the force of friction equals the horizontal component of the weight:
f = mg * cos(θ)
5. Equating the two equations for the force of friction, we get:
mg * sin(θ) = mg * cos(θ)
6. Simplifying the equation, we find:
tan(θ) = 1
7. Substituting the value of θ (θ = 60 degrees), we find:
tan(60) = 1
8. Solving for the required acceleration (a), we find that it must be equal to the acceleration due to gravity (g):
a = g
Therefore, for the block to remain stationary with respect to the wedge, the required acceleration is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2.