A bead of mass m slides without friction along a curved wire with shape z = f(r) x² + y?, i.e. the distance from the z-axis. The wire is rotated around the where r = 2-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a constant acceleration g. a) Using Newton's second law in an inertial frame, derive an expression for radius ro of a fixed circular orbit (i.e. a solution with r = ro = the wire applies to the bead to keep it in a circular orbit? b) Show that the equation of motion for r(t) (general equation not the circular motion) const.). What is the normal force is F(1+ f'(r)²) + i² f' (r)f"(r) + gf'(r) – w?r = 0. Using this verify your answer to part (a). c) Consider small displacements from the circular orbit, r = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable. d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire. The angular velocity is w = . Obtain the answer for an arbitrary motion of the bead, i.e. do not assume r = ro or r = ro + e(t). dø dt:
A bead of mass m slides without friction along a curved wire with shape z = f(r) x² + y?, i.e. the distance from the z-axis. The wire is rotated around the where r = 2-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a constant acceleration g. a) Using Newton's second law in an inertial frame, derive an expression for radius ro of a fixed circular orbit (i.e. a solution with r = ro = the wire applies to the bead to keep it in a circular orbit? b) Show that the equation of motion for r(t) (general equation not the circular motion) const.). What is the normal force is F(1+ f'(r)²) + i² f' (r)f"(r) + gf'(r) – w?r = 0. Using this verify your answer to part (a). c) Consider small displacements from the circular orbit, r = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable. d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire. The angular velocity is w = . Obtain the answer for an arbitrary motion of the bead, i.e. do not assume r = ro or r = ro + e(t). dø dt:
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Can you please solve just D part? Thanks.
Transcribed Image Text:f (r)
Vx2 + y2, i.e. the distance from the z-axis. The wire is rotated around the
z-axis at a constant angular velocity w. Gravity acts downward along the z-axis with a
A bead of mass m slides without friction along a curved wire with shape z
where r =
constant acceleration
g.
a) Using Newton's second law in an inertial frame, derive an expression for radius ro of
a fixed circular orbit (i.e. a solution with r =
the wire applies to the bead to keep it in a circular orbit?
b) Show that the equation of motion for r(t) (general equation not the circular motion)
ro =
const.). What is the normal force
is
ř(1+ f'(r)²) + i² f'(r)f"(r) +gf'(r) - w?r = 0.
Using this verify your answer to part (a).
c) Consider small displacements from the circular orbit, r =
on the function f(r) such that a circular orbit at r = ro is stable.
d) Find the force on the bead in the o direction, i.e. perpendicular to the plane of wire.
The angular velocity is w =
i.e. do not assume r = ro or r = ro + e(t).
ro+e(t). Derive a condition
. Obtain the answer for an arbitrary motion of the bead,
dt ·
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