A block (m = 38.0kg) sits on an inclined plane of 38 degrees. If there is no friction,k what magnitude force is needed to keep the block from sliding? If there is friction, with coefficients 0.580 and 0.630, what is the maximum magnitude force that can be applied before the block begins sliding?
To solve this problem, we will need to analyze the forces acting on the block and use Newton's laws of motion.
1. If there is no friction:
When there is no friction, the force required to keep the block from sliding is the force component along the incline that opposes the gravitational force pulling the block downward.
Step 1: Determine the force component along the incline:
The force component along the incline can be calculated using the formula:
F = m * g * sin(θ)
Where:
F is the force component along the incline,
m is the mass of the block (38.0 kg),
g is the acceleration due to gravity (9.8 m/s²),
θ is the angle of the incline (38 degrees).
Step 2: Calculate the force required:
F = 38.0 kg * 9.8 m/s² * sin(38°)
F ≈ 228.47 N
Therefore, the magnitude force needed to keep the block from sliding, when there is no friction, is approximately 228.47 Newtons.
2. If there is friction:
If there is friction, we need to consider the maximum force that can be applied before the block begins sliding. We will use the concept of static friction.
Step 1: Calculate the maximum static friction:
The maximum static friction can be calculated using the formula:
Fstatic_max = μs * N
Where:
Fstatic_max is the maximum static friction force,
μs is the coefficient of static friction (0.580),
N is the normal force acting on the block.
Step 2: Determine the normal force:
The normal force can be calculated using the formula:
N = m * g * cos(θ)
Where:
N is the normal force,
m is the mass of the block (38.0 kg),
g is the acceleration due to gravity (9.8 m/s²),
θ is the angle of the incline (38 degrees).
Step 3: Calculate the maximum force before sliding:
Fstatic_max = 0.580 * (38.0 kg * 9.8 m/s² * cos(38°))
Step 4: Solve for the maximum force:
Fstatic_max ≈ 231.83 N
Therefore, the maximum magnitude force that can be applied before the block begins sliding, when friction coefficients are 0.580 and 0.630, is approximately 231.83 Newtons.