Atwood's machine may be modeled as two particles, with masses m₁ and m₂, hanging by a string supported by a massless pulley. Also, friction is neglected. Let the position vector of m₁be
r₁=xE₁−bE₂
Where b is the radius of the pulley.Let N be the tension in the string.
(a) Show that the position vector of m₂ is
r₂=(c-x)E₁+bE₂
Where c is a constant.
(b) Show that the larangian is given by
L= 1/2(m₁+m₂)x²+(m₁-m₂)gx+d,
Where d is a constant
(c) Represent Atwood's machine in a plane lying in 6-dimensional Hertizan space i.e..by a super particle P with coordinates
u¹=x, u²= -b, u³=0, u⁴=c-x, u⁵=b, u⁶=0,
And position vector
r(P,t)= xe₁-be₂+(c-x)e₄+be₅
d. What are the magnitudes of the basis vectors in part (c)?
e. Construct the contravariant basis vectors e¹ and e⁴ and show that the force acting on the superparticle is
Φ=(m₁g-N)e¹+(m₂g-N)e⁴
f. What is the configuration manifold of the superparticle? Draw a sketch.
g. Calculate the covariant basis vector a₁ of the configuration manifold
h. Show that the constraint force is perpendicular to the configuration manifold
I Show that the acceleration x is constant