A light spring is initially used to suspend masses from the ceiling. When a 4.0 kg mass is attached to the spring, its length is 22.0 cm. When an additional 2.3 kg mass is attached to the bottom of the 4.0 kg mass, the spring's length measures 26.8 cm. Determine the spring's length (in cm) when no mass is attached to it.
To determine the spring's length when no mass is attached to it, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
First, let's determine the spring constant, k. We can use the equation F = kx, where F is the force exerted by the spring and x is the displacement from the equilibrium position.
When a 4.0 kg mass is attached to the spring, the length is 22.0 cm. We know that the force exerted by the spring at this point is equal to the weight of the mass. The weight is given by the equation F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F = mg
F = (4.0 kg)(9.8 m/s^2)
F = 39.2 N
Using Hooke's Law, we can now determine the spring constant k:
39.2 N = k * (0.22 m) (since 22.0 cm = 0.22 m)
k = 39.2 N / 0.22 m
k = 178.18 N/m
Now let's find the length of the spring when no mass is attached to it. We can use the equation for the force exerted by the spring:
F = kx
When no mass is attached, the force exerted by the spring is zero. Solving for x (the displacement), we have:
0 = 178.18 N/m * x
x = 0 meters
So, the length of the spring when no mass is attached is zero meters.