Question

a pendulum consists of a mass m suspended by a masslessspring with unextended length b and spring constant k. thependulum's point of support rises verticallywith constantacceleration a.
a. use the lagrangian method to find the equations of motion
b. determine the Hamilton's equation of motion
c. What is the period of small oscillations?

Answer 1

Draw the free diagram of the pendulum. First write the lagrangain equation. L=T-U ....... (1) Here, T is the kinetic energy and U is the potential energy. The kinetic energy is given as: .... (2)

From the figure. tan = And. x=(b+r) sin e y=(b+r)cos e Let b+r=r Therefore, i=rsin e +(6+r) cose() And, y = rcos 8-(+r)sin (0) Squaring and adding the above two equations. *2 + y2 =r?+(6+r) 42 Substitute equation (3) in the equation (2). T="[+++(6+)* 62]

Determine the potential energy. U=-mgy + kr ? =-mg (b+r) cos 2 + 2 ka? Substitute the value of kinetic energy and potential energy in the equation (1). L=T-U => [+(+r)? &*]+mg(b+r)cose – ? Hence, the required equations of motions are [i+(6+r)*@?]+ mg (b+r)cos e = 1 ko Write the equation of motion forr. mg cose-lor=mi Fore. dala del adae m(b+r) 8 =-mg(b+r) sin e Ö=-sin (b+r)

Hence, the required equations of motions are mg cose-lor= mil and (b) Calculate the vertical acceleration. y = (6+r)cos e = ['cos e –(6+r) sin elė)] = Pcos 8 – rsin 0 (0)-rsin (@)-(b+r)cos (e?)-(6+r) sine (0) Substitute value of r and 7 from first part in the above equation.

m 1 j=rcos 8 – rsin e()-(b+r)cos (82)-(6+r) sin e(ö) -- cos - 2r sin (0)-(6+r)cos ( 22)-(6+r)sine -7 (b+r) =g-rcos 8–2rsin e(0)–(6+r)cos e(02) a=g-Ercos 8–2rsin e(0)-(b+r)cose (82) Determine the Hamiltonian equations of motion. H=pg- Here, og Therefore,

Calculate the value of p. = m(b+r)* è=_Po m(b+r)? Therefore, H = p.r+ p. 6-1 = p (%) n (betri Pe-L t he in-mg (b +r)cose+beina The equations of motion are calculated below:

And, P. Or =-1'+mg cose Fore. •-OH P amber And, Pe = m(b+r)? 8 = m(b+r)? Ö=-mg(6+r) sin e Ö=_g sin (6+r)

Answer 2

(c) Ans:

as the point of support rises vertically with constant acceleration

effective gravity , geff = g + a

for the time constant

T = 2pi * sqrt(m/k)

the time period will be same

time period is 2pi * sqrt(m/k)