Final answer:
To find the initial separation distance d between the mass and the spring, apply conservation of energy principles relating gravitational potential energy and elastic potential energy, involving mass m, gravitational acceleration g, spring constant k, spring compression x, and relate distance d to these variables.
Step-by-step explanation:
The student is asking to find the initial separation distance d between a mass and a spring. To solve this, we need to apply the principles of conservation of energy. The mass slides down a frictionless incline and compresses a spring, momentarily coming to rest. The gravitational potential energy at the top of the incline is converted into elastic potential energy in the spring at maximum compression. The key formula here is:
Gravitational Potential Energy (initial) = Elastic Potential Energy (at max compression)
mgh = ½ kx²
where m is the mass, g is the acceleration due to gravity, h is the height related to distance d by the sine of the incline's angle, k is the spring constant, and x is the compression of the spring.
To find d, we solve the equation for h in terms of x and d, using trigonometry to relate the vertical height h to the distance d down the incline. We calculate the distance d using the known values for m, g, k, and x.