Final answer:
To find the spring's force constant, calculate the parallel component of the block's weight along the incline using the mass, gravity, and the angle of the incline. Then, apply Hooke's Law by equating the spring force to the parallel force component and dividing by the stretch of the spring to solve for the spring constant.
Step-by-step explanation:
To determine the force constant of the spring that holds the 30.0-kg block on a frictionless ramp inclined at 60°, we need to consider the forces acting on the block. Due to the ramp being frictionless, the only forces to consider are the weight of the block and the restoring force from the spring. The weight of the block can be resolved into two components: one perpendicular to the ramp and one parallel to the ramp.
The component of the weight parallel to the incline is what stretches the spring. The spring force, which equals the parallel component of the weight, can be described by Hooke's Law F_spring = k × Δx, where 'k' is the spring constant, and 'Δx' is the stretch of the spring. The parallel component can be calculated using the formula F_parallel = m × g × sin(θ), where 'm' is the mass of the block, 'g' is the acceleration due to gravity (9.8 m/s²), and 'θ' is the angle of the incline.
By setting the force from the spring equal to the parallel component (F_spring = F_parallel), we can solve for the spring constant 'k'. Hence, k = F_parallel / Δx. Substituting the given values, we have k = (30.0 kg × 9.8 m/s² × sin(60°)) / 0.05 m.
After calculating, we will obtain the force constant of the spring in Newtons per meter (N/m).