Answer:
To find the spring constant of the spring in this scenario, we can apply the principle of conservation of mechanical energy. The initial potential energy stored in the compressed spring will be converted into kinetic energy when the object leaves the spring. This allows us to equate the two energy forms.
The potential energy stored in a spring is given by the formula:
PE = (1/2) * k * x^2,
where PE is potential energy, k is the spring constant, and x is the displacement (compression or extension) of the spring.
The kinetic energy for an object is given by the formula:
KE = (1/2) * m * v^2,
where KE is kinetic energy, m is the mass of the object, and v is the velocity of the object.
Using these equations, we can set the initial potential energy equal to the final kinetic energy:
(1/2) * k * x^2 = (1/2) * m * v^2.
Given:
Mass (m) = 0.500 kg
Displacement (x) = 12.0 cm = 0.12 m
Velocity (v) = 20.0 cm/s = 0.20 m/s
Substituting the given values into the equations:
(1/2) * k * (0.12 m)^2 = (1/2) * (0.500 kg) * (0.20 m/s)^2.
Simplifying:
0.06 * k = 0.050 J.
Dividing both sides of the equation by 0.06:
k = 0.050 J / 0.06.
Calculating:
k ≈ 0.833 J/m.
Therefore, the spring constant of the spring is approximately 0.833 J/m.