A 4 kg block is pushed by an external force against a spring with spring constant 137 N/m until the spring is compressed by 2.1 m from its uncompressed length (x = 0). The block rests on a horizontal plane that has a coefficient of kinetic friction of 0.58 but is NOT attached to the spring.
To find the magnitude of the external force required to compress the spring, we need to consider the forces acting on the block.
Let's break down the problem step by step:
1. Determine the gravitational force:
The gravitational force acting on the block is given by the formula F_gravity = m * g, where m is the mass of the block (4 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2). Therefore, F_gravity = 4 kg * 9.8 m/s^2 = 39.2 N.
2. Calculate the spring force:
The spring force is given by Hooke's law: F_spring = -k * x, where F_spring is the force exerted by the spring, k is the spring constant (137 N/m), and x is the displacement from the equilibrium position (2.1 m). Since the direction of the force is opposite to the displacement, we include a negative sign. Therefore, F_spring = -137 N/m * 2.1 m = -287.7 N.
3. Consider the friction force:
The friction force is given by F_friction = μ * N, where F_friction is the friction force, μ is the coefficient of kinetic friction (0.58), and N is the normal force. The normal force is equal to the gravitational force in this case, so N = F_gravity = 39.2 N. Thus, F_friction = 0.58 * 39.2 N = 22.736 N.
4. Determine the net force:
The net force acting on the block is the sum of all the forces. Since the block is not moving at a constant velocity, the net force must be zero. Therefore, the external force needed to compress the spring is equal to the sum of the spring force and the friction force, but with the opposite sign. Thus, F_external = F_spring + F_friction = -287.7 N + (-22.736 N) = -310.436 N.
Therefore, the magnitude of the external force required to compress the spring is approximately 310.436 N.