A rifle fires a 1.27 x 10-2-kg pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by 7.87 x 10-2 m from its unstrained length. The pellet rises to a maximum height of 7.90 m above its position on the compressed spring. Ignoring air resistance, determine the spring constant.
To determine the spring constant, we can use the conservation of mechanical energy. The potential energy stored in the compressed spring is converted into the kinetic energy of the pellet as it rises. At the highest point, the kinetic energy is zero, and all the energy is in the form of potential energy.
The potential energy of a spring can be calculated using the formula:
PE = (1/2)kx^2
Where PE is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
In this case, the maximum potential energy of the spring is equal to the maximum gravitational potential energy of the pellet at its highest point.
mgh = (1/2)kx^2
Where m is the mass of the pellet, g is the acceleration due to gravity, h is the maximum height reached by the pellet, k is the spring constant, and x is the displacement of the spring.
Rearranging the equation, we can solve for the spring constant:
k = (2mgh) / x^2
Now we can substitute the given values:
m = 1.27 x 10^(-2) kg (mass of the pellet)
g = 9.8 m/s^2 (acceleration due to gravity)
h = 7.90 m (maximum height reached by the pellet)
x = 7.87 x 10^(-2) m (displacement of the spring)
k = (2 * 1.27 x 10^(-2) kg * 9.8 m/s^2 * 7.90 m) / (7.87 x 10^(-2) m)^2
Evaluating this expression will give us the spring constant.