part d 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 thez-axis at a constant angular velocity w. Gravity acts downward along the z-axis with aconstant acceleration g.a) Using Newton's second law in an inertial frame, derive an expression for radius ro ofa fixed circular orbit (i.e. a solution with r = ro = const.). What is the normal forcethe 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)isF(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 conditionon 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).dodtObtain the answer for an arbitrary motion of the bead,z = f (r)r

Given datamass of bead=mshape of curved wire=z=f(r)r=x2+y2angular velocity=ωusing newton second law…

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?

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?

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Concept explainers

Properties of Aluminium

Aluminum (or Aluminum) is a silvery-white in color metal. The chemical symbol of aluminum is Al and it has an atomic number (Z) of 13 in the periodic table. The aluminum name for this metal is derived from the Latin word ‘alumen’, which means bitter salt. Aluminum was discovered in the year 1825 by Hans Oersted, and later Friedrich Wöhler isolated aluminum. It is the most commonly found metal in the earth’s core, which is around 8% of the total core mass. Moreover, it is the third most commonly found element on earth after oxygen and silicon. Electron distribution of aluminum per shell is 2, 8, and 3 with a valency of 3. Atomic weight is nearly 26.9815 u.

Question

part d

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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