Friction and Moment of Inertia

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Friction and Moment of Inertia

• ENGR 221
• March 26, 2003

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Lecture Goals

• Friction Sliding vs Tipping
• Moment of Inertia
• Radius of Gyration

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Example Problem Friction III
The movable bracket shown may be placed at any
height on 3-in. diameter pipe. If the
coefficient of static friction between the pipe
and bracket is 0.25, determine the minimum
distance x at which the load W can be supported.
Neglect the weight of the bracket.
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Example Problem Friction III
Determine the value of the friction force
required to maintain equilibrium.
FA mNA and FBmNB ,which from sum in the x
direction will give FA FB
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Example Problem Friction III
Determine the value of the friction force
required to maintain equilibrium.
Use the moment about A to determine the
distance,x.
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Example Problem Friction III
Using the moment about A to find x.
The x12 in.
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Example Problem Friction IV
A large rectangular shipping crate of height h
and width b is at rest on the floor. It is acted
on by a horizontal force P. Assume that the
material in the crate is uniformly distributed so
that the weights acts at the centroid of the
crate.
(a) Determine the conditions for which the crate
is on the verge of sliding. (b) Determine the
conditions under which the crate will tip about
point A.
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Example Problem Friction IV
Draw the free-body diagram, look at the equations
of equilibrium.
If FP and NW substitute into the moment equation
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Example Problem Friction IV
If the crate is on the verge of sliding FmsN
where the coefficient of static friction .
If the crate is on the verge on tipping it is on
the verge of rotation about point A, that is the
crate and the floor are in contact only at A.
Therefore x0
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Example Problem Friction IV
Note that tipping will occur before sliding,
provided that Psliding gt Ptipping, therefore if P
increases until motion occurs tipping will occur
before sliding if
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Class Problem Friction III
The 200-lb block is sitting of a 30o inclined
surface. The coefficient of friction between the
block and the surface is 0.50. A light
inextensible cord is attached to the block,
passes around a friction-less pulley, and is
attached to a second block of mass M. Determine
the minimum and maximum masses, Mmin and Mmax,
such that the system is in equilibrium. Is
impending motion by slipping or by tipping?
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Moment of Inertia
Consider the force acting in a beam. The force
components are DFky DA where y is the distance
from the centroid, k is constant, and DA is a
segment of the area.
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Moment of Inertia
From the properties of the beam, one can find
The integral is the first moment of area used
to calculate the location of the centroid
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Moment of Inertia
The second major characteristic of the beam is
the moment of inertia. If we look at pure
bending moment in a beam about the centroid.
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Moment of Inertia
The second area moment or the moment of
inertia is the integral.
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Moment of Inertia
The moment of inertia is used in finding the
stress in the beam.
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Moment of Inertia
The moment of inertia is found by integrating
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Radius of Gyrations
The radius of gyration relates the moment of
inertia to the cross-sectional area of the
member. In that
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Example Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration.
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about x axis
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about y axis
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Example Moment of Inertia
The area of the block is bh so that the radii of
gyration are
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Example Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration for n2.
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Example Moment of Inertia
Determine the k value, is n2
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about x axis
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about y axis
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Example Moment of Inertia
Compute the area under the curve
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Example Moment of Inertia
Compute the radii of gyration
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Class Problem Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration for a triangle.
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Class Problem Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration for a triangle.
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Example Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration.
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Example Moment of Inertia
Determine the k values
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Example Moment of Inertia
Determine the k values
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about x axis
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Example Moment of Inertia
Compute the moment of inertia about center of the
block about y axis
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Example Moment of Inertia
Compute the area
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Example Moment of Inertia
Compute the radii of gyration
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Class Problem Moment of Inertia
Compute the moment of inertia in the x and y
planes and find the corresponding radius of
gyration for a triangle.
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Homework (Due 4/2/03)
Problems
10-1, 10-5, 10-7, 10-18