A bead of mass m slides without friction along a curved wire with shape z f(r) Vx2 + y?, i.e. the distance from the z-axis. The wire is rotated around the where r = z-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 = const.). What is the normal force 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) 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 = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable.
A bead of mass m slides without friction along a curved wire with shape z f(r) Vx2 + y?, i.e. the distance from the z-axis. The wire is rotated around the where r = z-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 = const.). What is the normal force 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) 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 = ro+e(t). Derive a condition on the function f(r) such that a circular orbit at r = ro is stable.
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Transcribed Image Text:A bead of mass m slides without friction along a curved wire with shape z = f(r)
where r =
Vx2 + y?, 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
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 = const.). What is the normal force
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)
is
F(1+ f'(r)²) + r² f' (r)f" (r) + gf'(r) - w²r = 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 =
i.e. do not assume r = ro or r = ro+ e(t).
do
dt
Obtain the answer for an arbitrary motion of the bead,
z = f (r)
r
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