Download 1 Torque (Moment) - Definition Torque (Moment) Torque (Moment

Torque (Moment) - Definition
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When a force is applied to a body, the moment of the
force about a moment center (axis, point) is the tendency
of the force to rotate the body about the moment center.
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Torque (Moment)
Moment is a vector so it has both magnitude and direction.
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Depends on:
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Magnitude of moment is magnitude of force times
perpendicular distance from moment center to
line of action of force.
Units are force times length: ft-lb, N-m
Size of force
Direction of force
Location of force
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Direction determined by the RIGHT hand rule.
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Find the moment about A
Scalar formulation:
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Magnitude: τO = Frsinθ, where
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Place right hand on moment center with fingers pointing
toward line of action of the force.
Curl fingers in direction force is pointing.
Direction of moment is direction of thumb.
Example:
Torque (Moment) – Calculation
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Although units are the same as those for work or
energy, Joule is not used for moments.
r is ANY vector from O (moment center) to line of
action of the force,
θ is angle between
r is length of
r
and the force, and
r.
Direction: determined by right hand rule.
Vector formulation:
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Use vector product (cross-product):
r ×F
M A = Fr sin θ kˆ = (200 N )r sin θ kˆ = 14.1 Nm kˆ
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Example:
Example:
Find the moment about A
M A = Fr sin θ kˆ = (200 N )r sin θ kˆ = 14.1 Nm kˆ
M A = Fr sin θ kˆ = (200 N )r sin θ kˆ = 14.1 Nm kˆ
Fixed Axis Rotation
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Find the moment about A
Fixed Axis Rotation
Given a single particle
‰ Mass = m
‰ Attached to a “massless” string
‰ Undergoing circular motion
‰ Subjected to the force Ft tangent to the path
Ft
∑ Ft = mat
Ft = m(αR )
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A Rigid Body
‰ Example, a rod
‰ mi = small element of mass
‰ Fi = Force on mi
Fi = mi ati = mi riα
Fi ri = (mi riα )ri = mi ri 2α
∑ F r = (∑ m r )α
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Multiply by R
Ft R = m(αR )R
τ = (mR 2 )α
Torque (moment) about axis of rotation
Mass Moment of Inertia
i i
i i
τ = Iα
Torque about axis of rotation
Mass Moment of Inertia
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Fixed Axis Rotation
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Mass Moment of Inertia of a Rigid Body
‰ dm = differential mass
‰ ρ = mass density
‰ V = volume
Mass
Moments
of
Inertia
I = ∫ r 2 dm
m
I = ∫ r 2 ρ dV
V
I = ρ ∫ r 2 dV
V
If ρ is constant
Eight Step Process
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Decide what needs to be isolated. (may be the hardest part).
Draw the isolated Free Body & Kinetic Diagrams (complete with
all external boundaries) and set them equal to each other.
Choose a Coordinate System (C.S.).
a)
Add all EXTERNALLY APPLIED forces & moments
acting ON the Free Body Diagram:
a1)
Given forces and moments including weight.
a2)
Support reactions (where the body is cut from
the rest of the world).
b)
Add all mass*acceleration terms to the Kinetic Diagram.
Add all necessary dimensions.
Enforce Newton’s 2nd Law:
a)
If necessary, set up any required Kinematic equations.
b)
Solve ALL equations for ALL unknowns.
Check work and answers for units, directions, proper notation,
S.F., reasonableness, etc.
The 30-kg uniform disk is pin supported at its center. If it
starts from rest, determine the number of revolutions it
must make to attain an angular velocity of 20 rad/s. Also,
what are the reaction forces at the pin O? Assume 3 SF.
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