A crate of mass 0.812 kg is placed on a rough incline of angle
35.3 degrees. Near the base of the incline is a spring of spring
constant 1140 N/m. The mass is pressed against the spring a
distance x and released. It moves up the slope
0.154 meters from the compressed position before
coming to a stop. If the cofficient of kinetic friction is 0.195,
how far (m) was the spring compressed?
To determine the distance the spring was compressed, we can use the energy conservation principle. By considering the work done by various forces, we can equate the initial potential energy stored in the spring to the sum of the final potential energy and the work done against friction.
Let's break down the problem and the steps needed to solve it:
Step 1: Calculate the gravitational force on the crate.
The force due to gravity can be found using the formula: F_gravity = m * g, where m is the mass of the crate and g is the acceleration due to gravity. In this case, m = 0.812 kg and g = 9.8 m/s^2.
F_gravity = 0.812 kg * 9.8 m/s^2 = 7.9616 N
Step 2: Calculate the normal force.
The normal force acting on the crate is equal to the component of the weight perpendicular to the incline. In this case, the normal force is equal in magnitude but opposite in direction to the force due to gravity (F_gravity).
Normal force = F_gravity = 7.9616 N
Step 3: Calculate the force of kinetic friction.
The force of kinetic friction can be determined using the formula: F_kinetic_friction = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force.
F_kinetic_friction = 0.195 * 7.9616 N = 1.5512 N
Step 4: Calculate the work done against friction.
The work done against friction is equal to the force of kinetic friction multiplied by the distance traveled up the incline (0.154 m).
Work done against friction = F_kinetic_friction * distance
= 1.5512 N * 0.154 m = 0.2382 J
Step 5: Calculate the spring potential energy.
The potential energy stored in the spring can be determined using the formula: PE_spring = (1/2) * k * x^2, where k is the spring constant and x is the distance the spring was compressed.
PE_spring = (1/2) * 1140 N/m * x^2
Step 6: Apply the principle of energy conservation.
According to the principle of energy conservation, the initial potential energy in the compressed spring is equal to the sum of the final potential energy and the work done against friction.
(1/2) * 1140 N/m * x^2 = 0.2382 J
Step 7: Solve for x.
Rearranging the equation, we get:
x^2 = (2 * 0.2382 J) / (1140 N/m)
x^2 = 0.000417
x ≈ √0.000417
x ≈ 0.0204 m
Therefore, the spring was compressed approximately 0.0204 meters.