A bloc of 10 kg is put at the top of an inclined plan of 45 degrees (to the left), attached to a spring which has a spring constant of 250 N/m. The coefficient of kinetic friction between the bloc and the surface of the inclined plan is of 0,300.
What is the maximum elongation of the spring?
To find the maximum elongation of the spring, we need to consider the forces acting on the block on the inclined plane and apply Newton's second law of motion.
1. First, let's consider the forces acting on the block:
- The weight of the block acts vertically downwards and is given by W = m * g, where m is the mass of the block (10 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- The normal force (N) acts perpendicular to the surface of the inclined plane and counterbalances the weight of the block. Since the block is on an inclined plane, the normal force can be calculated using N = m * g * cos(theta), where theta is the angle of the inclined plane (45 degrees).
- The force of friction (F_f) acts opposite to the direction of motion and is given by F_f = coefficient of kinetic friction * N. Here, the coefficient of kinetic friction is 0.300, and we already calculated the value for N.
2. Next, let's break down the weight force into its components:
- The weight force can be divided into two components:
- The component acting parallel to the inclined plane is W_parallel = m * g * sin(theta). This force tends to pull the block down the plane.
- The component acting perpendicular to the inclined plane is W_perpendicular = m * g * cos(theta). This force is counteracted by the normal force.
3. Now, let's calculate the net force acting on the block:
- The net force is the sum of all forces acting on the block. Since the block is moving up the inclined plane, we consider the frictional force to be acting in the opposite direction to motion. Therefore, the net force is given by F_net = W_parallel - F_f.
4. Finally, let's use Hooke's law to find the maximum elongation of the spring:
- Hooke's law states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position. Mathematically, this relationship can be written as F_spring = -k * x, where F_spring is the force exerted by the spring, k is the spring constant (250 N/m), and x is the displacement or elongation of the spring.
- At maximum elongation, the spring force equals the net force acting on the block. Therefore, we can equate F_spring and F_net: -k * x = F_net.
Now, we have all the necessary information and equations to solve for the maximum elongation of the spring.