Two blocks of masses m and 2m are held in equilibrium on a frictionless incline as in the figure. In terms of m and θ, find the following.
Incomplete.
To solve this problem, we need to consider the forces acting on each block individually and determine the conditions for equilibrium. Let's break down the problem step by step.
Step 1: Identify the forces acting on each block
For the block with mass m:
- The weight force acts vertically downward, given by mg, where g is the acceleration due to gravity.
- The normal force acts perpendicular to the inclined plane.
- The force of gravity component along the inclined plane can be calculated as mg*sin(θ), where θ is the angle of the incline.
For the block with mass 2m:
- The weight force acts vertically downward, given by 2mg.
- The normal force acts perpendicular to the inclined plane.
- The force of gravity component along the inclined plane can be calculated as 2mg*sin(θ).
Step 2: Analyze the conditions for equilibrium
In order for the blocks to be in equilibrium, the net force acting on each block must be zero, and the net torque acting on each block must be zero about any point.
Since the incline is frictionless, we only need to consider the forces acting parallel to the inclined plane for equilibrium.
For the block with mass m:
- The force of gravity component along the inclined plane, mg*sin(θ), must be balanced by a force acting in the opposite direction for equilibrium.
For the block with mass 2m:
- The force of gravity component along the inclined plane, 2mg*sin(θ), must be balanced by a force acting in the opposite direction for equilibrium.
Step 3: Solve for the forces to achieve equilibrium
For the block with mass m:
mg*sin(θ) = F
For the block with mass 2m:
2mg*sin(θ) = 2F
Simplifying the equations, we find:
F = mg*sin(θ)
2F = 2mg*sin(θ)
F = mg*sin(θ)
F = mg*sin(θ)
Since the equations are identical, the force acting on both blocks is the same, and the equilibrium condition is satisfied.
Therefore, the force required for equilibrium is F = mg*sin(θ), and it is the same for both blocks with masses m and 2m.