f.What is the Hamiltonian?g.Using Hamilton's equations of motion, determine the four equations ofmotion in terms of the conjugate momenta and coordinates.h.Does part g. agree with parts d. and e.? 3. Suppose that you have a particle of mass m subjected to gravity pointing downward thatis constrained to move along the inside of a frictionless surface of a cone with angle a asshown below. Assume +z-axis is through center axis of cone pointing upward.а.Using spherical coordinates [x = r sin 0 cos o ,y =r sin 0 sin o, z =r cos 0], what is the equation of constraint?b.What is the kinetic energy expression of the particle after incorporatingconstraint?с.What is the potential energy expression of the particle afterincorporating constraint?d.What are the resulting two equations of motion from Euler's equationsand using answers in parts b. and c.?е.What are the conjugate momenta?

Answered: f. What is the Hamiltonian? g. Using…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Question

f.
What is the Hamiltonian?
g.
Using Hamilton's equations of motion, determine the four equations of
motion in terms of the conjugate momenta and coordinates.
h.
Does part g. agree with parts d. and e.?

3. Suppose that you have a particle of mass m subjected to gravity pointing downward that
is constrained to move along the inside of a frictionless surface of a cone with angle a as
shown below. Assume +z-axis is through center axis of cone pointing upward.
а.
Using spherical coordinates [x = r sin 0 cos o ,y =r sin 0 sin o, z =
r cos 0], what is the equation of constraint?
b.
What is the kinetic energy expression of the particle after incorporating
constraint?
с.
What is the potential energy expression of the particle after
incorporating constraint?
d.
What are the resulting two equations of motion from Euler's equations
and using answers in parts b. and c.?
е.
What are the conjugate momenta?

Expert Solution

Step 1

$G i v e n d a t a P a r t i c l e o f m a s s m i s s u b j e c t e d t o g r a v i t y p o i n t i n g d o w n w a r d s t h a t i s c o n s t r a i n e d t o m o v e a l o n g a f r i c t i o n l e s s s u r f a c e o f a c o n e i t h a n g l e α .$

Step 2

$1] E q u a t i o n o f c o n t s t r a i n t F o r t h e m o t i o n o n t h e s u r f a c e o f c o n e c o n s t r a i n t e q u a t i o n i s θ = κ$

$b] C a l c u l a t i o n o f k i n e t i c e n e r g y o f t h e p a r t i c l e L e t t h e p a r t i c l e i s a t a d i s t a c e r f r o m i t' s a x i s t a n α = \frac{r}{h} h = \frac{r}{t a n α} \sin α = \frac{r}{s} s = \frac{r}{s i n α} V e l o c i t y V^{2} = {V_{r}}^{2} + {V_{θ}}^{2} V^{2} = {\dot{V}}^{2} + r^{2} {(\dot{θ})}^{2} K i n e t i c e n e r g y K E = \frac{1}{2} m [{\dot{V}}^{2} + r^{2} {(\dot{θ})}^{2}] K E = \frac{1}{2} m [{(\frac{\dot{r}}{s i n α})}^{2} + r^{2} {(\dot{θ})}^{2}]$