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Vector Mechanics for Engineers: Dynamics
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Curvilinear Motion: Position, Velocity & Acceleration
• Particle moving along a curve other than a straight line is in curvilinear motion.
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Curvilinear Motion: Position, Velocity & Acceleration
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Vector Mechanics for Engineers: Dynamics
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Rectangular Components of Velocity & Acceleration• When position vector of particle P is given by its
rectangular components,
kzjyixrrrrr
++=
• Velocity vector,
kvjviv
kzjyixkdt
dzj
dt
dyi
dt
dxv
zyxrrr
r&
r&
r&
rrrr
++=
++=++=
• Acceleration vector,
kajaia
kzjyixkdt
zdj
dt
ydi
dt
xda
zyxrrr
r&&
r&&
r&&
rrrr
++=
++=++=2
2
2
2
2
2
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Vector Mechanics for Engineers: Dynamics
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Rectangular Components of Velocity & Acceleration• Rectangular components particularly effective
when component accelerations can be integrated independently, e.g., motion of a projectile,
00 ==−==== zagyaxa zyx &&&&&&
with initial conditions,( ) ( ) ( ) 0,,0 000000 ==== zyx vvvzyx
Integrating twice yields
( ) ( )( ) ( ) 0
02
21
00
00
=−==
=−==
zgtyvytvx
vgtvvvv
yx
zyyxx
• Motion in horizontal direction is uniform.
• Motion in vertical direction is uniformly accelerated.
• Motion of projectile could be replaced by two independent rectilinear motions.
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Excerpt from Hibbeler
Plan: Establish a fixed x,y coordinate system (in the solution here, the origin of the coordinate system is placed at A). Apply the kinematic relations in x and y-directions.
Given: Skier leaves the ramp at θA = 25o and hits the slope at B.
Find: The skier’s initial speed vA.
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Homework: Projectile motion
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Motion Relative to a Frame in Translation• Designate one frame as the fixed frame of reference.
All other frames not rigidly attached to the fixed reference frame are moving frames of reference.
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Example:Relative motion
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Tangential and Normal Components
• Velocity vector of particle is tangent to path of particle. In general, acceleration vector is not. Wish to express acceleration vector in terms of tangential and normal components.
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Tangential and Normal Components
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Sample Problem 11.10
A motorist is traveling on curved section of highway at 60 km/h. The motorist applies brakes causing a constant deceleration rate.
Knowing that after 8 s the speed has been reduced to 66 m/s, determine the acceleration of the automobile immediately after the brakes are applied.
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Sample Problem 11.10
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Exercise: n-t component curvilinear translation
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Homework: n-t component curvilinear translation
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Radial and Transverse Components• When particle position is given in polar coordinates,
it is convenient to express velocity and acceleration with components parallel and perpendicular to OP.
rerrrr
=
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Radial and Transverse Components• When particle position is given in cylindrical
coordinates, it is convenient to express the velocity and acceleration vectors using the unit vectors . and ,, keeR
rrrθ
• Position vector,
kzeRr Rrrr
+=
• Velocity vector,
kzeReRdt
rdv R
r&
r&r&
rr
++== θθ
• Acceleration vector,
( ) ( ) kzeRReRRdt
vda R
r&&
r&&&&r&&&
rr
+++−== θθθθ 22
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Sample Problem 11.12
Rotation of the arm about O is defined by θ = 0.15t2 where θ is in radians and tin seconds. Collar B slides along the arm such that r = 0.9 - 0.12t2 where r is in meters.
After the arm has rotated through 30o, determine (a) the total velocity of the collar, (b) the total acceleration of the collar, and (c) the relative acceleration of the collar with respect to the arm.
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Sample Problem 11.12
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Example: r-theta curvilinear translation
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Homework: r-theta curvilinear translation