A spring has a force constant of 590.0 N/m.

(a) Determine the potential energy stored in the spring when the spring is stretched 3.77 cm from equilibrium.

(b) Determine the potential energy stored in the spring when the spring is stretched 3.26 cm from equilibrium.

(c) Determine the potential energy stored in the spring when the spring is unstretched.

PE=1/2 k x^2

and c is obviously zero, since x is zero.

To determine the potential energy stored in a spring, we need to use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position.

Hooke's Law can be expressed as:
F = -kx

Where:
F is the force exerted by the spring,
k is the force constant (also known as the spring constant), and
x is the displacement from the equilibrium position.

(a) To determine the potential energy stored in the spring when it is stretched 3.77 cm from equilibrium, we first need to convert the displacement from centimeters to meters:
x = 3.77 cm = 0.0377 m

Using Hooke's Law, we can determine the force exerted by the spring:
F = -kx
F = -(590.0 N/m)(0.0377 m)
F ≈ -22.25 N

The negative sign indicates that the force is in the opposite direction of the displacement, as the spring is stretched.

The potential energy stored in a spring can be calculated using the equation:
PE = 0.5kx²

Substituting the values:
PE = 0.5(590.0 N/m)(0.0377 m)²
PE ≈ 0.415 J

Therefore, the potential energy stored in the spring when it is stretched 3.77 cm from equilibrium is approximately 0.415 Joules.

(b) To determine the potential energy stored in the spring when it is stretched 3.26 cm from equilibrium, we follow the same steps as in part (a):

x = 3.26 cm = 0.0326 m

Using Hooke's Law:
F = -kx
F = -(590.0 N/m)(0.0326 m)
F ≈ -19.20 N

The potential energy stored in the spring:
PE = 0.5kx²
PE = 0.5(590.0 N/m)(0.0326 m)²
PE ≈ 0.324 J

Therefore, the potential energy stored in the spring when it is stretched 3.26 cm from equilibrium is approximately 0.324 Joules.

(c) When the spring is unstretched, the displacement (x) is equal to zero. In this case, the potential energy stored in the spring is also zero since the spring is in its equilibrium position and not storing any energy.