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# Week 13

By Pedro James,2014-03-01 07:39
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Week 13

Week 13

Chapter13 Kinetics of a Particle: Force and Acceleration Sec. 13.1 ~ 13.6 . Homework Prob. 15, 25, 58, 79, 101, 107

13.1 Newtons Laws of Motion

1st Law: If a particle, originally at rest or moving with a constant velocity, is not subjected to an

unbalanced force, then it will remain in this state.

2nd Law:

3rd Law: The mutual forces of action and reaction between two particles are equal, opposite, and

collinear.

13.4 Motion in Rectangular Coordinates

ˆˆˆˆˆˆFiFjFkmaiajak;;？;;()；；；xyzxyz

FmaFmaFma？？？; ; ；；；xxyyzz

Procedure for analysis

Select the inertial coordinate system.

Draw the free-body diagram.

Identify the acceleration to show the kinetic diagram.

List the equations of motion.

Solve the kinematics problem.

Important Points for Solving Problem

Equations of motion

Friction: Ff = k N. Ff always opposes the motion of the particle relative to the

contacting surface.

Spring: Fs = k (l l0).

Kinematics

a = a(t); v(t) = ?a(t)dt s(t) = ?v(t)dt

a = a(s); ?a(s)ds = ?v(s)dv

Make sure the positive inertial coordinate directions used for writing the kinematic

equations are the same as those used for writing the equations of motion. Example 3:

The 2-lb collar C fits loosely on the smooth shaft. If the spring is unstretched when s

= 0 and the collar is given a velocity of 15 ft/s, determine the velocity of the collar

when s = 1 ft. (Prob. 13-38)

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13.5 Motion in N-T Coordinates

ˆˆˆFuFuFumama;;？;；；；ttnnbbtn

FmaFmaF？？？; ; 0 ；；；ttnnb

3 22~?dy，? 1;?，：(2dx ；)dvv?，??aa？？？ and tn2 dtdy

2 dx

Example 6:

The package has a weight of 5 lb and slides down the chute. When it reaches the

curved portion AB, it is traveling at 8 ft/s ( = 0o). If the chute is smooth, determine the speed of the package when it reaches the intermediate point C ( = 30o) and when

it reaches the horizontal plane ( = 45o). Also, find the normal force on the package at C. (Prob. 13-70)

Example Prob. 13-77;p133

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13.6 Motion in Cylindrical Coordinates

ˆˆˆFuFuFumamama;;？;;；；；rrzzrz！！！

2 ！！！FmrrFmrrFmz？；？;？(); (2); ；；；rz

A positive ( means that it’s measured in the positive direction of .

rdrtan？？(drdrd/

Example 7:

Rod OA rotates counterclockwise with a constant angular velocity of 5 rad/s. The double collar B is pin-connected together such that one collar slides over the rotating rod and the other slides over the horizontal curved rod, of which the shape is described by the equation r = 1.5(2 - cos) ft. If both collars weight 0.75 lb, determine

the normal force which the curved rod exerts on one collar at the instant = 120o. Neglect friction. (Prob. 13-89)

Example 9:

Using air pressure, the 0.5-kg ball is forced to move through the tube lying in the horizontal plane and having the shape of a logarithmic spiral. If the tangential force exerted on the ball due to the air is 6 N, determine the rate of increase in the ball’s speed at the instant = /2. What direction does it act in? (Prob. 13-106)

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13.3 Equation of Motion for a System of Particles

The Equation of motion will be extended to a System of Particles. For ith particle:

maFFf？？; iiiii

FandfiiHere represent the resultant external force and internal force respectively.

All these equations are added together vectorially, we obtain: ；？；;；maFf ；？；maF iiiiiii

and no mass is leaving the system, we have: And consider the relation ；？mrmriiG

and maF？； ；？mamaGiiG

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