A 0.4 kg mass attached to a spring is pulled back horizontally across a table so that the potential energy of the system is increased from zero to 200 J. Ignoring friction, what is the kinetic energy of the system after the mass is released and has moved to a point where the potential energy has decreased to 75 J?
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To find the kinetic energy of the system after the potential energy has decreased to 75 J, we need to understand the relationship between potential energy and kinetic energy in this system.
In this scenario, the potential energy of the system is stored in the spring due to its displacement from the equilibrium position. The potential energy stored in a spring is given by the formula:
PE = (1/2) k x^2
where PE is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
Given that the potential energy has increased from zero to 200 J, we can set up the following equation:
200 J = (1/2) k x^2 ----(Equation 1)
Similarly, when the potential energy decreases to 75 J, we can set up another equation:
75 J = (1/2) k y^2 ----(Equation 2)
To solve these equations, we need to find the values of k, x, and y.
Since the mass of the object is given as 0.4 kg, we can use the equation for the spring constant:
k = (4π^2 m) / T^2
where m is the mass and T is the period of the spring. In this case, the period of the spring is not given, so we'll assume it as 1 second for simplicity.
k = (4π^2 * 0.4) / (1^2)
k ≈ 15.72 N/m
Now let's solve Equation 1 for x:
200 J = (1/2) * 15.72 N/m * x^2
400 J = 15.72 N/m * x^2
x^2 = 400 J / (15.72 N/m)
x ≈ 3.19 m
Next, we can calculate the value of y using Equation 2:
75 J = (1/2) * 15.72 N/m * y^2
150 J = 15.72 N/m * y^2
y^2 = 150 J / (15.72 N/m)
y ≈ 2.71 m
Now that we have the displacements (x and y) from the equilibrium position, we can calculate the velocities after the mass is released.
The total mechanical energy (potential energy + kinetic energy) in the system remains constant. Therefore, the total mechanical energy at the initial state (200 J potential energy) must be equal to the total mechanical energy at the final state (75 J potential energy + kinetic energy).
Initial mechanical energy = Final mechanical energy
200 J = 75 J + kinetic energy
To find the kinetic energy, subtract the final potential energy from the initial mechanical energy:
kinetic energy = 200 J - 75 J
kinetic energy = 125 J
Therefore, the kinetic energy of the system after the mass is released and has moved to a point where the potential energy has decreased to 75 J is approximately 125 J.