Question

(a) A circular ring has radius r and mass M. Let L be the axis of the ring (the line that is perpendicular to the plane of the ring and that passes through the center of the ring). What is the force on a mass m that is located on L, at a distance x from the center of the ring?

(b) A hole with radius R is cut out from an innite at sheet with mass density (kg=m2). What is the force on a mass m that is located on the analogously dened line L, at a distance x from the center of the hole? Hint: Consider the plane to consist of many concentric rings, and integrate over the rings.

(c) If a particle is released from rest on L, very close to the center of the hole, show that it undergoes oscillatory motion. Find the frequency of these oscillations.

(d) If a particle is released from rest on L, at a distance x from the sheet, what is its speed when it passes through the center of the hole? What does your answer reduce to for large x (or equivalently, small R)?

Answer #1

A sphere of mass M, radius r, and rotational inertia I is
released from rest at the top of an inclined plane of height h as
shown above. (diagram not shown)
If the plane has friction so that the sphere rolls without
slipping, what is the speed vcm of the center of mass at the bottom
of the incline?

a uniform ring of mass m an radius r rolls don an inclined
plane
starting from newtons second law find the acceleration of the
ring
PLEASE INCLUDE ANY RELEVANT CALCULUS CONCEPTS

A particle of mass m moves about a circle of radius R from the
origin center, under the action of an attractive force from the
coordinate point P (–R, 0) and inversely proportional to the square
of the distance.
Determine the work carried out by said force when the point is
transferred from A (R, 0) to B (0, R).

A ring of charge with radius R = 1.5 m is centered on the origin
in the x-y plane. A positive point charge is located at the
following coordinates: x = -10.1 m y = 16.8 m z = 17.1 m The point
charge and the total charge on the ring are the same, Q = +22 C.
Find the net electric field along the z-axis at z = 1.6 m.
Enet x=?
Enet y=?
Enet z=?
Thanks!!

A solid sphere of uniform density has a mass of 6.1 × 104 kg and
a radius of 2.7 m. What is the magnitude of the gravitational force
due to the sphere on a particle of mass 2.0 kg located at a
distance of (a) 5.9 m and (b) 2.1 m from the center of the sphere?
(c) Write a general expression for the magnitude of the
gravitational force on the particle at a distance r ? 2.7 m from...

A uniform solid sphere of mass m1 and radius R1 is inside and
concentric with a spherical shell of mass m2 and radius R2. Find
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Total charge q2 is uniformly placed on a ring of radius R. The
magnitude of the electric field at position z on the axis of the
ring is given by ((kq2z)/(R^2+z^2)^(3/2))
A uniformly charged rod of total charge q1 and length L is now
placed on the z axis. The nearest end of the rod is at distance L
from the center of the ring, i.e. the rod extends from z = L to z =
2L (see figure on...

A uniform spherical shell of mass M = 17.0 kg and radius R =
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Consider a satellite of mass m in a circular orbit of radius r
around the Earth of mass ME and radius RE.
1.
What is the gravitational force (magnitude and direction) on
the satellite from Earth?
2.
If we define g(r) to be the force of gravity on a mass m at a
radial distance r from the center of the Earth, divided by the mass
m, then evaluate the ratio g(r)/g(RE)to see how g varies with
radial distance. If...

2. A circular ring with a radius R of 1 cm carries a charge
density of ?L = R sin ? (? is an azimuthal angle) µC/cm. The ring
is then placed on the xy plane with its axis the same as the
z-axis. Find the electric field intensity E and the electric
potential V on point A on z-axis 2 cm from the xy plane.

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