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.
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To determine the spring constant, we can make use of the conservation of mechanical energy. At the maximum height, the kinetic energy of the pellet is zero, and all the energy is in the form of potential energy.
The potential energy stored in the compressed spring can be calculated using the equation:
PE_spring = (1/2) k Δx^2
Where:
- PE_spring is the potential energy stored in the spring
- k is the spring constant
- Δx is the displacement of the spring (compressed length)
Given:
- Δx = 7.87 x 10^-2 m
- PE_spring = mgh (potential energy stored in the spring is equal to the potential energy of the pellet at maximum height)
- 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)
Substituting the values into the equation:
mgh = (1/2) k Δx^2
(1.27 x 10^-2 kg)(9.8 m/s^2)(7.90 m) = (1/2) k (7.87 x 10^-2 m)^2
Simplifying:
k = [(1.27 x 10^-2 kg)(9.8 m/s^2)(7.90 m)] / [(1/2)(7.87 x 10^-2 m)^2]
Now we can calculate the value of k.
To determine the spring constant, we can use the concept of conservation of mechanical energy. In this case, the mechanical energy of the pellet-spring system is conserved since we are ignoring air resistance.
The mechanical energy of the system is given by the sum of the potential energy and the kinetic energy:
Mechanical Energy = Potential Energy + Kinetic Energy
At the maximum height reached by the pellet, all of its initial kinetic energy is converted into potential energy:
Potential Energy = m * g * h
where m is the mass of the pellet, g is the acceleration due to gravity, and h is the maximum height.
When the pellet is released from the compressed spring, it has potential energy equal to the potential energy stored in the compressed spring:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement of the spring from its unstrained length.
Equating the two expressions for potential energy:
m * g * h = (1/2) * k * x^2
Rearranging the equation to solve for k:
k = (2 * m * g * h) / x^2
Plugging in the given values:
m = 1.27 x 10^-2 kg
g = 9.8 m/s^2
h = 7.90 m
x = 7.87 x 10^-2 m
k = (2 * 1.27 x 10^-2 kg * 9.8 m/s^2 * 7.90 m) / (7.87 x 10^-2 m)^2
Calculating the expression:
k ≈ 127.6 N/m
Therefore, the spring constant of the compressed spring is approximately 127.6 N/m.