Final answer:
Using conservation of energy and work-energy theorem, we find that the coefficient of kinetic friction for the block moving over the rough surface is 0.55.
Step-by-step explanation:
To solve the problem, we can apply the conservation of energy principle and the work-energy theorem. Initially, the block has gravitational potential energy which gets converted into kinetic energy as it slides down the ramp. When it reaches the rough surface, kinetic friction does work and removes some of this energy until it comes to rest compressing the spring.
First, we calculate the initial potential energy of the block which is equal to gravitational potential energy: PE = mgh where m is the mass of the block, g is the acceleration due to gravity (9.8 m/s²), and h is the height.
Next, we calculate the energy stored in the spring at the point of maximum compression using elastic potential energy = (1/2)kx², where k is the spring constant and x is the compression.
Using energy conservation, the initial potential energy equals the final elastic potential energy, plus the work done by friction (PE_initial = PE_spring + Work_friction). The work done by friction is Work_friction = frictional force * distance, where frictional force is μN, μ is the coefficient of kinetic friction and N is the normal force which equals mg for horizontal motion.
Solving these equations, the coefficient of kinetic friction is:
μ = μ = (mgh - (1/2)kx²) / (mgd)
Where:
- m = 10 kg (mass of the block)
- g = 9.8 m/s² (acceleration due to gravity)
- h = 3 m (height of the release point)
- k = 2250 N/m (spring constant)
- x = 0.3 m (compression of the spring)
- d = 6 m (distance traveled on the rough surface)
Substituting these values we get:
μ = (10 * 9.8 * 3 - 0.5 * 2250 * 0.3²) / (10 * 9.8 * 6)
After calculating, μ = 0.55. This is the coefficient of kinetic friction.