A 2-kg small block is dropped from rest. The spring has a constant k = 500N/m. If the entire track is frictionless except for the 1m between points B and C, where the coefficient of kinetic friction is 0.15, find he maximum compression of the spring.
To find the maximum compression of the spring, we need to determine the point at which the block comes to a stop due to friction between points B and C on the track.
First, let's calculate the gravitational force on the block. The gravitational force can be found using the formula:
F_gravity = mass * acceleration due to gravity
F_gravity = 2 kg * 9.8 m/s^2 = 19.6 N
Next, let's calculate the frictional force acting on the block between points B and C. The frictional force can be found using the formula:
F_friction = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the block, which is F_gravity. Therefore,
F_friction = 0.15 * F_gravity = 0.15 * 19.6 N = 2.94 N
Now, let's calculate the net force acting on the block between points B and C. The net force is the difference between the gravitational force and the frictional force:
Net force = F_gravity - F_friction = 19.6 N - 2.94 N = 16.66 N
According to Newton's second law of motion, the net force is equal to the product of mass and acceleration:
Net force = mass * acceleration
Since the block is coming to a stop, the acceleration is zero. Therefore,
16.66 N = 2 kg * 0
This equation implies that the net force is zero.
Now, let's consider the force exerted by the spring, which obeys Hooke's law:
F_spring = -k * compression
Since there is no net force acting on the block at the maximum compression, the force exerted by the spring should be equal in magnitude but opposite in direction to the frictional force:
F_spring = F_friction = 2.94 N
Using Hooke's law, we can solve for compression:
compression = -F_friction / k
compression = -(2.94 N) / (500 N/m) = -0.00588 m = -5.88 mm
Therefore, the maximum compression of the spring is 5.88 mm.