Final answer:
To calculate the force constant k of a spring that brings a train to a stop, use the work-energy principle equating the train's kinetic energy to the work done by the spring. After substituting values for the train's mass, initial velocity, and spring compression into the formula, the spring constant is determined to be approximately 10802 N/m.
Step-by-step explanation:
The force constant k of a spring can be determined using the work-energy principle. The work done by the spring when it brings a subway train to a stop can be equated to the kinetic energy of the train before it stops. Since the spring does negative work and absorbs the energy of the train, we can express this as:
½mv² = ½kx²
where m is the mass of the object, v is the initial velocity, k is the spring constant, and x is the displacement of the spring when the object comes to a stop. Solving for k, the equation becomes:
k = mv² / x²
In this case, m = 3.50 × 10µ kg (the mass of the subway train), v = 0.500 m/s (the initial speed of the train), and x = 0.900 m (the amount the spring is compressed). Plugging these into the equation:
k = (3.50 × 10µ kg × (0.500 m/s)²) / (0.900 m)²
k = (3.50 × 10µ kg × 0.25 m²/s²) / 0.81 m²
k = (8750 kg · m²/s²) / 0.81 m²
k = 10802.469 N/m (approximately)
Therefore, the force constant k of the spring is approximately 10802 N/m.