A 50-kg block is at rest on a 15 degree slope. A force of 250 N is acting on the block up the slope parallel to it. If the block does not slide up the slope, what is the minimum value of the coefficient of static friction between the block and the slope?
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To find the minimum value of the coefficient of static friction between the block and the slope, we need to consider the forces acting on the block.
1. Resolve the force acting parallel to the slope:
The force acting up the slope is 250 N.
The component of this force acting parallel to the slope is:
F_parallel = 250 N * sin(15°)
2. Calculate the gravitational force acting down the slope:
The mass of the block is 50 kg.
The gravitational force can be calculated as:
F_gravity = m * g
F_gravity = 50 kg * 9.8 m/s^2
3. Determine the frictional force:
Since the block is at rest, the force of static friction will be equal and opposite to the component of the force acting parallel to the slope.
F_friction = -F_parallel
4. Calculate the minimum coefficient of static friction:
The coefficient of static friction can be calculated using the formula:
μ_static = |F_friction| / |F_normal|
where F_friction is the frictional force and F_normal is the normal force.
5. Calculate the normal force:
Since the block is on an incline, the normal force no longer fully opposes gravity. It can be calculated as:
F_normal = F_gravity * cos(15°)
Now we can calculate the minimum value of the coefficient of static friction:
F_parallel = 250 N * sin(15°)
F_gravity = 50 kg * 9.8 m/s^2
F_friction = -F_parallel
F_normal = F_gravity * cos(15°)
μ_static = |F_friction| / |F_normal|
Plug in the values and calculate:
F_parallel ≈ 250 N * 0.259
F_gravity ≈ 50 kg * 9.8 m/s^2
F_friction ≈ -64.75 N (negative because it acts in the opposite direction)
F_normal ≈ F_gravity * cos(15°)
μ_static ≈ |F_friction| / |F_normal|
After evaluating the above calculations, you will get the minimum value of the coefficient of static friction between the block and the slope.
To find the minimum value of the coefficient of static friction between the block and the slope, we need to analyze the forces acting on the block.
The gravitational force (weight) acting on the block can be determined using the formula:
Weight = Mass x Acceleration due to gravity
Weight = 50 kg x 9.8 m/s^2
Weight = 490 N
Next, determine the force component acting parallel to the slope. This is done by finding the force's projection onto the slope. The force acting up the slope can be resolved into two components: one perpendicular to the slope and one parallel to the slope.
Force parallel to the slope = Force x sinθ
Force parallel to the slope = 250 N x sin(15°)
Force parallel to the slope = 250 N x 0.259
Force parallel to the slope = 64.75 N
Since the block does not slide up the slope, the force parallel to the slope must be balanced by the force of static friction pointing down the slope.
The force of static friction can be determined using the formula:
Force of static friction = Coefficient of static friction x Normal force
To find the normal force, we need to determine the force component acting perpendicular to the slope. This is given by:
Force perpendicular to the slope = Force x cosθ
Force perpendicular to the slope = 250 N x cos(15°)
Force perpendicular to the slope = 250 N x 0.966
Force perpendicular to the slope = 241.5 N
Since the block is at rest, the normal force (N) is equal to the weight of the block. Therefore, N = 490 N.
Now we can calculate the force of static friction:
Force of static friction = Coefficient of static friction x Normal force
64.75 N = Coefficient of static friction x 490 N
Solving for the coefficient of static friction:
Coefficient of static friction = 64.75 N / 490 N
Coefficient of static friction = 0.1327
Therefore, the minimum value of the coefficient of static friction between the block and the slope is approximately 0.1327.