a) To calculate the system's total mechanical energy, we need to consider the potential energy stored in the springs and the kinetic energy of the stone.
The potential energy stored in each spring can be calculated using the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position. Since the cup is pulled to x = 0.7 m to the left of the vertical, the displacement for each spring is 0.7 m.
Potential Energy of each spring = (1/2) * 23.7 N/m * (0.7 m)^2 = 4.13 J
Since there are two identical springs, the total potential energy stored in the springs is:
Total Potential Energy = 2 * Potential Energy of each spring = 2 * 4.13 J = 8.26 J
The kinetic energy of the stone when it is released can be calculated using the formula:
Kinetic Energy = (1/2) * m * v^2
where m is the mass of the stone and v is its velocity. The stone's mass is given as 1.97 kg.
The equilibrium position is when the springs line up vertically, so at x = 0 (after it was released), the stone has displaced downwards by a distance equal to the equilibrium length of the springs, which is 50 cm or 0.5 m.
Since the stone was pulled to x = 0.7 m and released, it will oscillate between x = -0.7 m to x = 0.7 m (crossing through the equilibrium position twice). Therefore, at x = 0, it has displaced upwards by a distance of 0.7 m.
Using the principle of conservation of mechanical energy, we can equate the potential energy at the max displacement (0.7 m) to the kinetic energy at x = 0.
Potential Energy at 0.7 m = 8.26 J
Kinetic Energy at x = 0 = (1/2) * 1.97 kg * v^2
8.26 J = (1/2) * 1.97 kg * v^2
Solving for v^2:
v^2 = (2 * 8.26 J) / 1.97 kg
v^2 = 8.37 m^2/s^2
Taking the square root of both sides:
v ≈ 2.89 m/s
Therefore, the speed of the stone at x = 0 is approximately 2.89 m/s.