A 0.59 kg mass is placed at the end of a vertical spring that has a spring constant of 68 N/m and eased down into its equilibrium position.
A. Determine the change in the elastic potential energy of the system.
B.Determine the system's change in gravitational potential energy.
To determine the change in the elastic potential energy of the system, we can use the formula for elastic potential energy:
Elastic potential energy (PE) = 0.5 * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
A. Determine the change in the elastic potential energy of the system.
To find the change in elastic potential energy, we need to find the initial and final positions of the mass. Let's assume the equilibrium position of the spring is taken as the reference point (x = 0).
Since the mass is initially eased down into its equilibrium position, it has zero potential energy. Therefore, the initial elastic potential energy is zero.
The final position is the equilibrium position, so the displacement from the equilibrium position (x) is also zero.
Therefore, the change in elastic potential energy (ΔPE) = PE_final - PE_initial = 0 - 0 = 0
Hence, the change in the elastic potential energy of the system is zero.
B. Determine the system's change in gravitational potential energy.
The formula for gravitational potential energy is:
Gravitational potential energy (PE) = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the vertical height.
Since the mass is placed at the end of a vertical spring and eased down into its equilibrium position, the height (h) is the displacement from the equilibrium position.
As the spring elongates, the mass goes higher, increasing its height.
To find the height, we can use Hooke's law equation:
F = k * x
The force (F) exerted by the spring is equal to the weight (mg) acting in the opposite direction, where m is the mass and g is the acceleration due to gravity.
mg = k * x
Rearranging the equation:
x = (mg) / k
Substituting the given values: m = 0.59 kg, g = 9.8 m/s^2, and k = 68 N/m, we can solve for x.
x = (0.59 kg * 9.8 m/s^2) / 68 N/m
x ≈ 0.085 meters
Now we can find the change in gravitational potential energy:
ΔPE = PE_final - PE_initial = m * g * h_final - m * g * h_initial
At the initial position, the height is zero, so the initial gravitational potential energy is zero.
ΔPE = m * g * h_final - 0
ΔPE = m * g * (h_final - h_initial)
ΔPE = m * g * (0.085 - 0)
Substituting the given values: m = 0.59 kg and g = 9.8 m/s^2, we can calculate the change in gravitational potential energy.
ΔPE ≈ 0.59 kg * 9.8 m/s^2 * 0.085 meters
Hence, the system's change in gravitational potential energy is approximately: ΔPE ≈ 0.49 J.