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Simple Harmonic Motion: Coupled spring and pendulum system

Please go to the word document, which includes a graph and explains better. It should be much easier to understand it on word.

A mass M1 on a frictionless plane is connected to a support O by a spring of stiffness k. Mass M2 is supported by a string of length l from M1.
a. What are the forces acting on M1 and M2?
b. By a consideration of these forces, derive the equations of motion of M1 and M2 using the small angle approximation
c. For M1=M2, find the normal frequencies and the ratio of the amplitudes of each mass for each mode.

M1 - mass on frictionless plane
M2 - mass at the end of the pendulum which is attached at its pivot to M1.
X1 is the displacement from equilibrium of mass M1
X2 is the displacement from equilibrium position of mass M2
θ is the angle of the pendulum from equilibrium position
l is the length of the pendulum

*normal frequencies refer to frequencies in the normal mode of vibration which use variables A=X1+X2 and B= X1-X2. The letters don't matter. X1 and X2 refer to the horizontal displacement of the two masses, X1 for the mass on the frictionless plane, X2 for the mass on the pendulum.


Solution Preview

Please see the attached file.

The forces acting on M1 are
1. The weight M1g of the block and the normal reaction of the plane, both in vertical direction not in consideration in horizontal motion of M1.
2. The force due to stretched spring - Kx1 in horizontal direction, negative because in negative x direction.
3. The tension T in the string making an angle  with the downward vertical.

The forces acting on the pendulum bob are
1. The weight M2g acting vertically downward.
2. The tension T in the string at an angle  to the upward vertical.


The equation of motion for M1 is given by
{If  is small and measured in radians sin = 1.}
The equation of motion for M2 in x direction is given by
and the equation of motion for M2 in x ...

Solution Summary

A good problem to understand coupled SHM. Simple harmonic motion for coupled spring and pendulum systems are analyzed..