A 2kg block situated on a rough incline is connected to a spring of negligible mass having a spring constant of 100N/m. The pulley is frictionless. The block is released from rest when the spring is unstretched . The block moves 20cm down the incline before coming to rest. Using work energy theorem, find the coefficient of kinetic friction between the block and inclined plane
To find the coefficient of kinetic friction between the block and inclined plane, we can use the work-energy theorem.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the block is equal to the work done by the friction force and the work done by the gravitational force.
The work done by the friction force is given by the equation:
Work_friction = Force_friction * distance * cos(theta)
where:
- Force_friction is the friction force exerted on the block,
- distance is the distance traveled by the block down the incline, which is given as 20 cm (or 0.2 m),
- theta is the angle between the incline and the horizontal.
The work done by the gravitational force is given by the equation:
Work_gravity = m * g * h
where:
- m is the mass of the block, given as 2 kg,
- g is the acceleration due to gravity, approximately 9.8 m/s^2,
- h is the vertical height from the starting point to the ending point of the block's motion down the incline.
Since the block is initially at rest and comes to rest at the end, its change in kinetic energy is zero. Therefore, the work done by the friction force and the work done by the gravitational force must cancel each other out:
Work_friction + Work_gravity = 0
Substituting the equations for work friction and work gravity:
Force_friction * distance * cos(theta) + m * g * h = 0
We can rearrange the equation to solve for the friction force:
Force_friction = -(m * g * h) / (distance * cos(theta))
The friction force can be further expressed as:
Force_friction = coefficient_friction * normal_force
where:
- coefficient_friction is the coefficient of kinetic friction between the block and inclined plane,
- normal_force is the normal force exerted on the block by the inclined plane, which is equal to the weight of the block in this case.
Substituting this equation into the previous one:
coefficient_friction * normal_force = -(m * g * h) / (distance * cos(theta))
The normal force can be calculated as:
normal_force = m * g * cos(theta)
Substituting this equation back into the previous one:
coefficient_friction * (m * g * cos(theta)) = -(m * g * h) / (distance * cos(theta))
Simplifying the equation:
coefficient_friction = -(m * g * h) / (distance * (m * g * cos^2(theta)))
Now, we can plug in the given values: m = 2 kg, g = 9.8 m/s^2, h = 0.2 m, distance = 0.2 m, and theta is the angle between the incline and the horizontal (which is not given).