Consider the modified Atwood machine, As you can verify, for every 1 cm that mas
ID: 1359397 • Letter: C
Question
Consider the modified Atwood machine, As you can verify, for every 1 cm that mass 1 drops, mass 2 will be raised up by just 0.5 cm. Consider the pulleys and ropes to be frictionless and of negligible weight.
a)What is the potential energy of this system, as a function of the coordinate ?
b)What is the kinetic energy of this system, as a function of x'?
c)Put these expressions together to find the total mechanical energy.
d)Find the equation of motion, and solve for the acceleration x''.
e)Under what circumstances is the system stable?
Explanation / Answer
(a) just consider that initially mass m2 is at rest and mass m1 is at the top
so on pulling the mass m1 by X the mass m2 will pull up by amount by X/2
let us say the distance of the pulley from ground is L
therefore
Potential energy = m1g (L-X) + m2g((X/2))
(b) Kinetic energy
if mass m is moving with velocity V then then m2 is also moving with same as on same string
therefore
kinetic energy = (1/2)(m1+m2)V2 = m1 g(L-X) + m2(g(X/2))
(c) Total energy = K.E + P.E.
(d) When we differetiate the energy equation with respect to time then we get acceleration
dE/dt = (m1+m2)Va - m1V + m2(V/2) (dX/dt = V)
putting dE/dt = 0
a = m1-m2 / (m1+m2)