Blocks A and B of weight 200N and 100N respectively, rest on a 30 inclined plane and are
attached to the post which is held perpendicular to the plane by force P, parallel to the plane,
as shown in fig. Assume that all surfaces are smooth and that the cords are parallel to the
plane. Determine the value of P. Also find the Normal reaction of Blocks A and B.
Resolve forces parallel to the plane:
P=(m1+m2)g cos(θ)
where
P=force up the plane
m1,m2 = masses of the blocks
Normal reactions of blocks are
m1g sin(θ)
and
m2g sin(θ)
unless blocks are piled up.
(confirm with your figure)
To find the value of force P and the normal reactions of blocks A and B, we can use the principles of equilibrium.
First, let's resolve the weight of each block into components parallel and perpendicular to the inclined plane.
The weight of block A (200N) has components:
- Perpendicular to the plane: w₁ = 200N * cos(30°)
- Parallel to the plane: w₁' = 200N * sin(30°)
The weight of block B (100N) has components:
- Perpendicular to the plane: w₂ = 100N * cos(30°)
- Parallel to the plane: w₂' = 100N * sin(30°)
Now, let's consider the forces acting on each block:
For block A:
- Perpendicular forces: Normal reaction (N₁) and w₁
- Parallel forces: P and frictional force (F₁)
For block B:
- Perpendicular forces: Normal reaction (N₂) and w₂
- Parallel fores: F₂
Since the surfaces are smooth, there is no friction between the blocks and the inclined plane.
Applying the principle of equilibrium in the perpendicular direction, we have:
N₁ + N₂ = w₁ + w₂
Applying the principle of equilibrium in the parallel direction, we have:
P = w₁' + F₁ = w₂' + F₂
Since friction is absent, F₁ and F₂ are both zero.
Substituting the components of weight and solving the equations:
N₁ + N₂ = 200N * cos(30°) + 100N * cos(30°)
P = 200N * sin(30°) + 100N * sin(30°)
Now, let's calculate the values:
N₁ + N₂ = 200N * cos(30°) + 100N * cos(30°)
N₁ + N₂ = 173.2N
P = 200N * sin(30°) + 100N * sin(30°)
P = 150N + 75N
P = 225N
Therefore, the value of force P is 225N.
The normal reaction of blocks A and B is 173.2N.
To determine the value of P, we need to consider the forces acting on the blocks A and B along the inclined plane. These forces include the weight of the blocks, the frictional force, and the force P.
1. Resolve the weight of each block:
The weight of block A is 200N, and the weight of block B is 100N. Since the inclined plane is at an angle of 30 degrees, we need to resolve the weight into two components: one parallel to the inclined plane (W_parallel) and one perpendicular to the inclined plane (W_perpendicular).
W_parallel_A = Weight of A * sin(30)
W_parallel_A = 200N * sin(30)
W_parallel_A = 100N
W_perpendicular_A = Weight of A * cos(30)
W_perpendicular_A = 200N * cos(30)
W_perpendicular_A = 173.2N
W_parallel_B = Weight of B * sin(30)
W_parallel_B = 100N * sin(30)
W_parallel_B = 50N
W_perpendicular_B = Weight of B * cos(30)
W_perpendicular_B = 100N * cos(30)
W_perpendicular_B = 86.6N
2. Determine the frictional force:
Since all surfaces are assumed to be smooth, there is no friction acting on the blocks.
3. Analyze the forces acting on the blocks along the inclined plane:
For block A:
- The force P acting parallel to the inclined plane.
- The weight component W_parallel_A acting down the inclined plane.
- The normal reaction N_A acting perpendicular to the inclined plane.
For block B:
- The weight component W_parallel_B acting down the inclined plane.
- The normal reaction N_B acting perpendicular to the inclined plane.
Since the blocks are in equilibrium, the sum of the forces along the inclined plane should be zero:
ΣF_parallel = P - W_parallel_A - W_parallel_B = 0
P = W_parallel_A + W_parallel_B
P = 100N + 50N
P = 150N
Thus, the value of P is 150N.
4. Find the normal reactions:
Since the blocks are at rest on the inclined plane, the sum of the forces perpendicular to the inclined plane should be zero:
ΣF_perpendicular = W_perpendicular_A + W_perpendicular_B - N_A - N_B = 0
N_A + N_B = W_perpendicular_A + W_perpendicular_B
N_A + N_B = 173.2N + 86.6N
N_A + N_B = 259.8N
The normal reaction of block A and B combined is 259.8N.
Note: The normal reaction of individual blocks cannot be determined without more information about the specific contact points. Only the combined normal reaction can be calculated.