Final answer:
The maximum compression of the spring is 3.27 cm.
Step-by-step explanation:
To find the maximum compression of the spring, we need to consider the forces acting on the block as it slides down the ramp.
First, let's calculate the normal force acting on the block. The normal force is equal to the weight of the block, which is given by the equation N = mg, where m is the mass of the block and g is the acceleration due to gravity. In this case, the mass is 4 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the normal force is N = 4 kg * 9.8 m/s^2 = 39.2 N.
Next, let's calculate the gravitational force component parallel to the ramp. The component of the weight of the block parallel to the ramp is given by the equation F_parallel = mg*sin(theta), where theta is the angle of the ramp. In this case, the angle of the ramp is 30 degrees, so F_parallel = 4 kg * 9.8 m/s^2 * sin(30 degrees) = 19.6 N.
The net force acting on the block is the difference between the gravitational force component parallel to the ramp (19.6 N) and the force of friction (F_friction). The force of friction is given by the equation F_friction = mu * N, where mu is the coefficient of friction. In this case, the coefficient of friction is 0.450, so F_friction = 0.450 * 39.2 N = 17.64 N.
Finally, the net force is F_net = F_parallel - F_friction = 19.6 N - 17.64 N = 1.96 N.
By Hooke's Law, we know that the force exerted by the spring is equal to the spring constant times the displacement. In this case, the force exerted by the spring is 1.96 N and the spring constant is 60 N/m. Therefore, the displacement of the spring is given by the equation F = k * x, where F is the force, k is the spring constant, and x is the displacement. Rearranging this equation, we can solve for x: x = F / k = 1.96 N / 60 N/m = 0.0327 m = 3.27 cm.