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Kinetics of a particle

Objectives

• To analyze the accelerated motion of a particle using the equation of motion with different coordinate systems.

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Newton’s Laws of Motions (1687)

First Law

A particle originally at rest, or moving in a straight line with a constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force.

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Newton’s Laws of Motions (1687)

First Law

A particle originally at rest, or moving in a straight line with a constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force.

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Newton’s Laws of Motions (1687)

Second Law

The unbalanced force acting on the particle is proportional to the time rate of change of the particle’s linear momentum.

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Newton’s Laws of Motions (1687)

Third Law

The mutual forces of action and reaction between two particles are equal, opposite, and collinear.

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Equation of MotionRectangular Coordinates

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amF

zz

yy

xx

maF

maF

maF

Procedure for Analysis

• Select the inertial coordinate system

• Draw FBD

• Establish direction and sense of acceleration

• Solve for unknowns

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Problem 13-12

• Determine the normal force the 10-kg crate A exerts on the smooth cart if the cart is given an acceleration of a = 2 m/s2 down the plane. Also, what is the acceleration of the crate?

(θ=30⁰)

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Problem 13-23

A force F = 150 N is applied to the cord.

Determine how high the 300-N (~ 30 kg)

block A rises in 2 s starting

from rest. Neglect the weight

of the pulleys and cord.

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Equation of MotionNormal and Tangential Coordinates

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bb

nn

tt

maF

maF

maF

Problem 13-67

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Determine the constant speed of the passengers on the amusement-park ride if it is observed that the supporting cables are directed at θ=30⁰ from the vertical. Each chair including its passenger has a mass of 80 kg. Also, what are the components of force in the n, t and b directions which the chair exerts on a 50-kg passenger during the motion?

Equation of MotionCylindrical Coordinates

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zz

rr

maF

maF

maF

Useful Formulas

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2 rrar rra 2 zaz

ddr

r

/tan

Problem 13-88

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The boy of mass 40-kg is sliding down the spiral slide at a constant speed such that his position, measured from the top of the chute, has components r=1.5 m, θ=0.7t rad, and z=-0.5t m, where t is in seconds. Determine the components of force Fr, Fθ and Fz

which the slide exerts on him at the instant t=2 s. Neglect the size of the boy.

Example 1

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A can C, having mass 0.5-kg, moves along a grooved horizontal slot. The slot is in a form of spiral defined by equation r = (0.1θ) m, where θis in radians. If the arm OA is rotating at a constant rate in the horizontal plane, determine the force it exerts on the can at the instant θ = π rad.