An elastic cord is 69 cm long when a weight of 57 N hangs from it but is 87 cm long when a weight of 85 N hangs from it. What is the "spring" constant k of this elastic cord?
To find the spring constant (k) of an elastic cord, 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.
Mathematically, Hooke's Law can be expressed as:
F = -kx
Where F is the force applied to the spring, k is the spring constant, and x is the displacement from the equilibrium position.
In this case, we know the lengths of the elastic cord (69 cm and 87 cm) and the weights applied to it (57 N and 85 N). To find k, we need to determine the amount of displacement caused by each weight.
Let's calculate the displacements caused by the weights:
Displacement_1 = 87 cm - 69 cm = 18 cm = 0.18 m
Displacement_2 = 69 cm - Equilibrium length = 0 cm = 0 m (equilibrium position)
Now, we can use Hooke's Law to find k:
F_1 = -k * Displacement_1
57 N = -k * 0.18 m
To isolate k, we need to rearrange the equation:
k = -57 N / 0.18 m
Calculating this, we get:
k ≈ -316.67 N/m
The negative sign indicates that the force exerted by the cord is opposite to the direction of displacement. Thus, the spring constant for this elastic cord is approximately 316.67 N/m.