Question

Two identical simple pendulums (mass m, length L) are connected by a spring with spring constant k, as shown in the diagram. A second identical spring connects the right-hand pendulum to a wall. 1) If there are no springs, and if the displacement x of a pendulum is small, the net force on the pendulum bob is approximately F = m * d^2x / dt^2 = - (mg / L) * x Starting with this (you do not have to derive it), show that with the springs connected, the equation of motion for the left-hand pendulum is d^2x_A / dt^2 + (g / L + k / m) * x_A - (k / m) * x_B = 0 Then, find the equation of motion for the right-hand pendulum. 2) Find the angular frequencies of the normal modes in terms of m, g, L and k. (Assume the amplitude is small, so that the equations you found in part (a) are valid.)

          Two identical simple pendulums (mass m, length L) are connected by a spring with spring constant k, as shown in the diagram. A second identical spring connects the right-hand pendulum to a wall.

1) If there are no springs, and if the displacement x of a pendulum is small, the net force on the pendulum bob is approximately
F = m * d^2x / dt^2 = - (mg / L) * x
Starting with this (you do not have to derive it), show that with the springs connected, the equation of motion for the left-hand pendulum is
d^2x_A / dt^2 + (g / L + k / m) * x_A - (k / m) * x_B = 0
Then, find the equation of motion for the right-hand pendulum.

2) Find the angular frequencies of the normal modes in terms of m, g, L and k.
(Assume the amplitude is small, so that the equations you found in part (a) are valid.)

Two identical simple pendulums (mass m, length L) are connected by a spring with spring constant k, as shown in the diagram. A second identical spring connects the right-hand pendulum to a wall.

1) If there are no springs, and if the displacement x of a pendulum is small, the net force on the pendulum bob is approximately
F = m * d^2x / dt^2 = - (mg / L) * x
Starting with this (you do not have to derive it), show that with the springs connected, the equation of motion for the left-hand pendulum is
d^2xA / dt^2 + (g / L + k / m) * xA - (k / m) * xB = 0
Then, find the equation of motion for the right-hand pendulum.

2) Find the angular frequencies of the normal modes in terms of m, g, L and k.
(Assume the amplitude is small, so that the equations you found in part (a) are valid.)

Added by Joanna J.

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