A pendulum consists of a mass m suspended from a fixed point by means of string

Question

A pendulum consists of a mass m suspended from a fixed point by means of string of length b and is immersed in a viscious medium. the medium provides a retardign force proportional to the velocity, the constant of proportionality being R. Using either torques or tangential forces write the equation of motion

i.e.

I = sum of torques (theta has a double dot on top of it)

or

matan= Sum of forces

Assuming small so sin ~ (radians) write the solution for =(t)

Here: General slution since initial , dot not given.

Explanation / Answer

The equation of motion by balancing torque will be

'' = -(g/b)sin - (R/m)' ( cancelling mass (m) from both sides )

as sin

''=-(g/b) - (R/m)'

Solving the given differential equation gives a solution,

Taking e^t as a solution of homogeneous equation

2 + (R/m) +(g/b)=0

Solving we get 1 and 2

1= (-(R/m)+((R/m)^2 -4g/b))/2

2= (-(R/m)-((R/m)^2 -4g/b))/2

Depending on the determinant we get 1=2 or 1,2 real and distinct or 1,2 complex conjugates.

So the complementary solution will be (t)=c1e^1t + c2e^2t ( for 1,2 real and distinct)

(c1+c2t)e^1t   (1=2 )

or,

Asin1t + Bcos2t (  1,2 complex conjugates.)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.