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johnq2k7
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Hamilton's Principle Equations! Work Shown Please Help!
A particle of mass m moves under the influence of gravity alng the helix z=k(theta), and r=R, where R and k are constants and z is vertical.
a.) Using cartesian co-ordinates, write down the expressions for the kinetic energy of the system.
b.) Change to cylindrical co-ordinate system using x= rcos(theta) y= rsin(theta)
express your eq for the kinetic energy as a function of the new co-ordinates. Give also expressions for the potential energy and the Langrangian of the system (in cylindrical co-ordinates).
c.) How many degrees of freedom do you have for this system? Name them (or it)
d.) Calculate the equation(s) of motion using the Langrange equation for this system.
my work:
a.) T= 1/2 m(x*time-deriv.)^2 + 1/2(y*time-deriv)^2 + 1/2(k(theta)*time-deriv.)^2
b.) if x= rcos(theta) and y=rsin(theta)
equation for kinetic energy in terms of cylin. co-ord is:
x= rcos(theta)
y= rsin(theta)
therefore, x-time deriv.= r*angle-timederiv. cos(theta)
y-time deriv.= r*angle-timederiv.sin(theta)
therefore, kinetic energy in cylin. co-ord.
T= (1/2)*(m)[(r*time-deriv)+(r^2*angletimederiv^2)+(ztimederiv^2)]
and therefore potential energy equals:
U=mgz
therefore the langrangian of the system in cylin. co-ord equals:
since z=k(theta) and r=R
therefore langrangian equals (L)= (1/2)*(m)[(R^2/k^2)(ztimederiv^2)+(ztimederiv^2)] -mgz
c.) need some help here, i believe the degree of freedom is six:
since in three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names:
Moving up and down (heaving);
Moving left and right (swaying);
Moving forward and backward (surging);
Tilting forward and backward (pitching);
Turning left and right (yawing);
Tilting side to side (rolling).
not sure if this is correct need help
d.) using langrangian equation:
z-2ndtime deriv= g/[(r^2/k^2)+1]
need help proving that
PLEASE HELP A LOT OF WORK SHOWN NEED HELP HERE!
A particle of mass m moves under the influence of gravity alng the helix z=k(theta), and r=R, where R and k are constants and z is vertical.
a.) Using cartesian co-ordinates, write down the expressions for the kinetic energy of the system.
b.) Change to cylindrical co-ordinate system using x= rcos(theta) y= rsin(theta)
express your eq for the kinetic energy as a function of the new co-ordinates. Give also expressions for the potential energy and the Langrangian of the system (in cylindrical co-ordinates).
c.) How many degrees of freedom do you have for this system? Name them (or it)
d.) Calculate the equation(s) of motion using the Langrange equation for this system.
my work:
a.) T= 1/2 m(x*time-deriv.)^2 + 1/2(y*time-deriv)^2 + 1/2(k(theta)*time-deriv.)^2
b.) if x= rcos(theta) and y=rsin(theta)
equation for kinetic energy in terms of cylin. co-ord is:
x= rcos(theta)
y= rsin(theta)
therefore, x-time deriv.= r*angle-timederiv. cos(theta)
y-time deriv.= r*angle-timederiv.sin(theta)
therefore, kinetic energy in cylin. co-ord.
T= (1/2)*(m)[(r*time-deriv)+(r^2*angletimederiv^2)+(ztimederiv^2)]
and therefore potential energy equals:
U=mgz
therefore the langrangian of the system in cylin. co-ord equals:
since z=k(theta) and r=R
therefore langrangian equals (L)= (1/2)*(m)[(R^2/k^2)(ztimederiv^2)+(ztimederiv^2)] -mgz
c.) need some help here, i believe the degree of freedom is six:
since in three dimensions, the six DOFs of a rigid body are sometimes described using these nautical names:
Moving up and down (heaving);
Moving left and right (swaying);
Moving forward and backward (surging);
Tilting forward and backward (pitching);
Turning left and right (yawing);
Tilting side to side (rolling).
not sure if this is correct need help
d.) using langrangian equation:
z-2ndtime deriv= g/[(r^2/k^2)+1]
need help proving that
PLEASE HELP A LOT OF WORK SHOWN NEED HELP HERE!