A 1.4 kg mass is attached to a spring with a force constant of 56 N/m. If the mass is released with a speed of 0.26 m/s at a distance of 8.6 cm from the equilibrium position of the spring, what is its speed when it is halfway to the equilibrium position?
To find the speed of the mass when it is halfway to the equilibrium position, we need to calculate the potential energy stored in the spring at that position and convert it to kinetic energy.
First, let's find the potential energy stored in the spring when the mass is at the equilibrium position. The potential energy stored in a spring is given by the equation:
Potential energy = (1/2) * force constant * displacement^2
Given:
Force constant (k) = 56 N/m
Displacement (x) = 8.6 cm = 0.086 m
Potential energy at the equilibrium position = (1/2) * 56 N/m * (0.086 m)^2
Calculate the potential energy at the equilibrium position.
Potential energy at the equilibrium position = (1/2) * 56 N/m * (0.086 m)^2 = 0.21216 J
Now, let's find the potential energy stored in the spring when the mass is halfway to the equilibrium position. Since the spring is symmetric, halfway to the equilibrium position corresponds to a displacement of half the initial displacement.
Displacement halfway to the equilibrium position = 0.086 m / 2 = 0.043 m
Potential energy halfway to the equilibrium position = (1/2) * 56 N/m * (0.043 m)^2
Calculate the potential energy halfway to the equilibrium position.
Potential energy halfway to the equilibrium position = (1/2) * 56 N/m * (0.043 m)^2 = 0.0256248 J
The difference in potential energy between the equilibrium position and halfway to the equilibrium position is given by:
Potential energy difference = Potential energy at the equilibrium position - Potential energy halfway to the equilibrium position
Potential energy difference = 0.21216 J - 0.0256248 J
Calculate the potential energy difference.
Potential energy difference = 0.1865352 J
According to the law of conservation of energy, this potential energy difference is converted entirely to kinetic energy at the halfway position. Therefore, the kinetic energy halfway to the equilibrium position is:
Kinetic energy halfway to the equilibrium position = Potential energy difference
Kinetic energy halfway to the equilibrium position = 0.1865352 J
Finally, convert the kinetic energy to speed using the equation:
Kinetic energy = (1/2) * mass * velocity^2
Given:
Mass (m) = 1.4 kg
Solve for velocity halfway to the equilibrium position.
0.1865352 J = (1/2) * 1.4 kg * velocity^2
Simplify the equation.
0.1865352 J = 0.7 kg * velocity^2
Divide both sides by 0.7 kg.
velocity^2 = 0.1865352 J / 0.7 kg
velocity^2 = 0.26647886 J/kg
Take the square root of both sides to find the velocity halfway to the equilibrium position.
velocity = √(0.26647886 J/kg)
velocity = 0.516 m/s
Therefore, the speed of the mass when it is halfway to the equilibrium position is 0.516 m/s.