A spring has a length of 0.248 m when a 0.300 kg mass hangs from it, and a length of 0.750 m when a 2.42 kg mass hangs from it.
(a) What is the force constant of the spring?
(b) What is the unloaded length of the spring?
To find the force constant of the spring and the unloaded length of the spring, 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.
(a) To find the force constant of the spring:
1. Identify the given values:
- Length of the spring when a 0.300 kg mass hangs from it: 0.248 m
- Length of the spring when a 2.42 kg mass hangs from it: 0.750 m
2. Determine the displacements of the spring:
- Displacement for the 0.300 kg mass: (0.248 m - unloaded length)
- Displacement for the 2.42 kg mass: (0.750 m - unloaded length)
3. Apply Hooke's Law:
The force exerted by a spring is given by F = -kx, where F is the force, k is the force constant, and x is the displacement.
For the 0.300 kg mass:
F = -k * (0.248 m - unloaded length) --> Equation 1
For the 2.42 kg mass:
F = -k * (0.750 m - unloaded length) --> Equation 2
4. Calculate the force for each mass:
For the 0.300 kg mass, the force is equal to the weight of the mass (mg):
F_0.300 = (0.300 kg) * (9.8 m/s^2) --> Equation 3
For the 2.42 kg mass:
F_2.42 = (2.42 kg) * (9.8 m/s^2) --> Equation 4
5. Equate the forces gained from Equations 1-4:
-k * (0.248 m - unloaded length) = (0.300 kg) * (9.8 m/s^2) --> Equation 5
-k * (0.750 m - unloaded length) = (2.42 kg) * (9.8 m/s^2) --> Equation 6
6. Solve the system of equations (Equations 5 and 6) simultaneously to find the force constant (k).
(b) To find the unloaded length of the spring:
- Rearrange Equation 5 or 6 to solve for the unloaded length.
- Substitute the force constant (k) obtained from the previous step into the equation and solve for the unloaded length.
Note: The force constant (k) is given in N/m, and the unit of length is meters (m).