A 4.0-kg block slides down an inclined plane that makes an angle of 25° with the horizontal. Starting from rest, the block slides a distance of 2.5 m in 5.4 s. Find the coefficient of kinetic friction between the block and plane.
d=1/2 a t^2
solve for a, then
net force down plane=mass*acceleration
4*9.8*cosTheta-4*9.8*sinTheta*mu=m*a
solve for mu.
To find the coefficient of kinetic friction between the block and the plane, we need to analyze the forces acting on the block.
First, let's determine the gravitational force (weight) acting on the block. The gravitational force can be calculated using the formula:
weight = mass × acceleration due to gravity
Given that the mass of the block is 4.0 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the weight is:
weight = 4.0 kg × 9.8 m/s^2
weight = 39.2 N
Next, let's find the component of the weight that acts parallel to the incline. This component is given by:
weight_parallel = weight × sin(angle of incline)
The angle of the incline is given as 25°, so we can calculate the weight parallel to the incline:
weight_parallel = 39.2 N × sin(25°)
weight_parallel = 16.6 N
Now, we need to determine the acceleration of the block. We can use the equation of motion:
distance = initial velocity × time + (1/2) × acceleration × time^2
Given that the distance is 2.5 m, the initial velocity is 0 m/s (starting from rest), and the time is 5.4 s, we can rearrange the equation to solve for acceleration:
acceleration = (2 × (distance - initial velocity × time)) / time^2
acceleration = (2 × (2.5 m - 0 m/s × 5.4 s)) / (5.4 s)^2
acceleration = 0.78 m/s^2
Now, let's calculate the total force parallel to the incline acting on the block. This force can be found using Newton's second law:
force_parallel = mass × acceleration
force_parallel = 4.0 kg × 0.78 m/s^2
force_parallel = 3.12 N
Finally, we can determine the force of kinetic friction acting on the block. This force opposes the motion and is given by:
force_friction = coefficient of kinetic friction × force_normal
The force normal is the component of the weight perpendicular to the incline and can be calculated as:
force_normal = weight × cos(angle of incline)
force_normal = 39.2 N × cos(25°)
force_normal = 35.0 N
Substituting the values we have calculated, we can solve for the coefficient of kinetic friction:
force_friction = coefficient of kinetic friction × force_normal
3.12 N = coefficient of kinetic friction × 35.0 N
Solving for the coefficient of kinetic friction:
coefficient of kinetic friction = 3.12 N / 35.0 N
coefficient of kinetic friction ≈ 0.089 (rounded to three decimal places)
Therefore, the coefficient of kinetic friction between the block and the plane is approximately 0.089.