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January 27, 2015

January 27, 2015

Posted by **shasha** on Saturday, March 12, 2011 at 7:07am.

the resistance is proportional to the velocity. Obtain the differential equation of its motion

and discuss the motion.

- Diff eqn -
**MathMate**, Saturday, March 12, 2011 at 10:16amThis is not an initial value problem!

External forces are zero, so the governing equation is homogeneous:

mx"+Bx'+kx=0

x"=d²x/dt².

x'=dx/dt

x=displacement, positive to the right

m=mass

B=resistance proportional to the velocity

k=stiffness (of a spring), which is resistance proportional to the displacement.

For a pendulum of length L and mass m,

making an angle θ with the vertical, three forces act on the mass m.

The vertical force due to gravity, mg.

A horizontal restoring force (towards the equilibrium position) of mgsin(θ), and the tension, which is equal and opposite to the resultant of the two forces.

The restoring force mgsin(θ) is often approximated by

kx=mgLsin(θ)≅mgLθ for small angles (when θ is in radians).

- Diff eqn- IVP -
**shasha**, Monday, March 14, 2011 at 8:00amwhy stiffness (of a spring) ?

- Diff eqn- IVP -
**MathMate**, Monday, March 14, 2011 at 8:16amIt is an analogy.

The general equation of motion make an analogy with a mass attached to the end of a spring:

mx"+Bx'+kx=0

where k represents the restoring force per unit displacement.

- Diff eqn- IVP -
**shasha**, Monday, March 14, 2011 at 8:54amoww...ok...

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