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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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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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Homework: Projectile motion

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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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Tangential and Normal Components

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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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Sample Problem 11.10

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Vector Mechanics for Engineers: Dynamics

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Exercise: n-t component curvilinear translation

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Vector Mechanics for Engineers: Dynamics

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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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Vector Mechanics for Engineers: Dynamics

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Example: r-theta curvilinear translation

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Homework: r-theta curvilinear translation