Classical Mechanics -
Let us consider the following kinetic (T) and potential (U)
energies of a...
Classical Mechanics -
Let us consider the following kinetic (T) and potential (U)
energies of a two-dimensional oscillator :
?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²)
?(?,?)= ?/2 (?²+?² )+???
where x and y denote, respectively, the cartesian displacements
of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of
the displacements; m the mass of the oscillator; K the stiffness
constant of the oscillator; A is the coupling constant.
1) Using the following coordinate transformations,
?= 1/√2 (?+?)
?= 1/√2...
For each of the following vector fields F , decide whether it is
conservative or not...
For each of the following vector fields F , decide whether it is
conservative or not by computing curl F . Type in a potential
function f (that is, ∇f=F). If it is not conservative, type N. A.
F(x,y)=(−4x+3y)i+(3x+16y)j f(x,y)= B. F(x,y)=−2yi−1xj f(x,y)= C.
F(x,y,z)=−2xi−1yj+k f(x,y,z)= D. F(x,y)=(−2siny)i+(6y−2xcosy)j
f(x,y)= E. F(x,y,z)=−2x2i+3y2j+8z2k
Letu=2i−3j+k,v=i+4j−k,andw=j+k.
(a) Find u × v and v × u, and show that each of those...
Letu=2i−3j+k,v=i+4j−k,andw=j+k.
(a) Find u × v and v × u, and show that each of those vectors
is orthogonal to both u and v.
(b) Find the area of the parallelogram that has u and v as
adjacent sides.
(c) Use the triple scalar product to find the volume of the
parallelepiped having adjacent
edges u, v, and w.
(1 point) For each of the following vector fields F , decide
whether it is conservative...
(1 point) For each of the following vector fields F , decide
whether it is conservative or not by computing curl F . Type in a
potential function f (that is, ∇f=F). If it is not conservative,
type N. A. F(x,y)=(10x+7y)i+(7x+10y)j f(x,y)= 10 B. F(x,y)=5yi+6xj
f(x,y)= N C. F(x,y,z)=5xi+6yj+k f(x,y,z)= D.
F(x,y)=(5siny)i+(14y+5xcosy)j f(x,y)= E.
F(x,y,z)=5x2i+7y2j+5z2k
For
F= ((1+2xy)ze^(2xy)+(1/x)+2x)i+(2x^2ze^(2xy)j+(xe^(2xy)+(1/z)k
show that F is conservative and then find work on object
moving from...
For
F= ((1+2xy)ze^(2xy)+(1/x)+2x)i+(2x^2ze^(2xy)j+(xe^(2xy)+(1/z)k
show that F is conservative and then find work on object
moving from (1,0,1) to (2,0,5).
For each of the
following vector fields F , decide whether it is
conservative or not...
For each of the
following vector fields F , decide whether it is
conservative or not by computing curl F . Type in
a potential function f (that is, ∇f=F∇f=F
). If it is not conservative, type N.
A.
F(x,y)=(−10x+3y)i+(3x+10y)jF(x,y)=(−10x+3y)i+(3x+10y)j
f(x,y)=f(x,y)=
B.
F(x,y)=−5yi−4xjF(x,y)=−5yi−4xj
f(x,y)=f(x,y)=
C.
F(x,y,z)=−5xi−4yj+kF(x,y,z)=−5xi−4yj+k
f(x,y,z)=f(x,y,z)=
D.
F(x,y)=(−5siny)i+(6y−5xcosy)jF(x,y)=(−5siny)i+(6y−5xcosy)j
f(x,y)=f(x,y)=
E.
F(x,y,z)=−5x2i+3y2j+5z2kF(x,y,z)=−5x2i+3y2j+5z2k
f(x,y,z)=f(x,y,z)=
Note: Your answers should be either expressions of x, y and z
(e.g. "3xy + 2yz"), or the letter "N"
classical Mechanics problem:
Suppose some particle of mass m is confined to move, without
friction, in...
classical Mechanics problem:
Suppose some particle of mass m is confined to move, without
friction, in a vertical plane, with axes x horizontal and y
vertically up. The plane is forced to rotate with angular velocity
of magnitude Ω about the y axis. Find the equation of motion for x
and y, solve them, and describe the possible motions. This is not
meant to be a lagrangian problem.
1. (a) Determine whether or not F is a conservative vector
field. If it is, find...
1. (a) Determine whether or not F is a conservative vector
field. If it is, find the potential function for F.
(b) Evaluate R C1 F · dr and R C2 F · dr where C1 is the
straight line path from (0, −1) to (3, 0), while C2 is the union of
two straight line paths: first piece from (0, −1) to (0, 0) and
then second piece from (0, 0) to (3, 0). (When applicable, use the
Fundamental...