To determine the contact force between the block and the inclined plane, we need to analyze the forces acting on the block. The scenario involves a block of mass 2 kg on a 45-degree incline, with friction and an applied force at an angle. Let's break this down step by step.
Identifying the Forces
First, we need to identify all the forces acting on the block:
- Weight (W): This acts vertically downward and is calculated as W = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²).
- Normal Force (N): This acts perpendicular to the surface of the incline.
- Frictional Force (f): This opposes the motion of the block and is given by f = μN, where μ is the coefficient of friction.
- Applied Force (F): A force of 10 N is applied at an angle of 30 degrees upward from the incline.
Calculating the Weight of the Block
First, we calculate the weight of the block:
W = mg = 2 kg × 9.81 m/s² = 19.62 N
Resolving Forces
Next, we resolve the weight into components parallel and perpendicular to the incline:
- The component of weight parallel to the incline: W_parallel = W sin(θ) = 19.62 N × sin(45°) = 19.62 N × 0.7071 ≈ 13.88 N
- The component of weight perpendicular to the incline: W_perpendicular = W cos(θ) = 19.62 N × cos(45°) = 19.62 N × 0.7071 ≈ 13.88 N
Analyzing the Applied Force
The applied force also has components that need to be resolved:
- Component of the applied force parallel to the incline: F_parallel = F cos(30°) = 10 N × cos(30°) ≈ 10 N × 0.866 = 8.66 N
- Component of the applied force perpendicular to the incline: F_perpendicular = F sin(30°) = 10 N × sin(30°) = 10 N × 0.5 = 5 N
Calculating the Normal Force
The normal force can be found by considering the forces acting perpendicular to the incline:
N = W_perpendicular - F_perpendicular = 13.88 N - 5 N = 8.88 N
Determining the Frictional Force
Now, we can calculate the frictional force:
f = μN = 0.6 × 8.88 N ≈ 5.33 N
Finding the Net Force Parallel to the Incline
Next, we analyze the net force acting parallel to the incline:
Net force = F_parallel - W_parallel - f = 8.66 N - 13.88 N - 5.33 N = -10.55 N
This negative value indicates that the block is not moving up the incline; instead, it is being pulled down due to gravity and friction.
Final Contact Force Calculation
The contact force between the block and the plane is essentially the normal force we calculated earlier:
Contact Force (N): 8.88 N
In summary, the contact force between the block and the inclined plane is approximately 8.88 N. This analysis shows how various forces interact on an inclined plane, highlighting the importance of resolving forces into their components for accurate calculations.
