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Engineering Mechanics

Chapter 12

Kinetics of Particles: Newton’s Second Law

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Contents

Introduction

Newton’s second law of motion

Linear momentum of a particle. Rate of change of linear

momentum

Equations of motion

Dynamic equilibrium

Angular momentum of a particle. Rate of change of angular

momentum

Equations of motion in terms of radial and transverse components

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12.1 Introduction

Newton’s second law is used in dynamics to analyze the motion of

particles.

The mass of the particle is defined as the ratio of the magnitudes of the

resultant force and of the acceleration.

The linear momentum of a particle is L = mv

The angular momentum of a particle about a point O is Ho = r x mv

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12.2 Newton’s second law of motion

If the resultant force acting on a particle is not zero, the particle will have

an acceleration proportional to the magnitude of the resultant and in

the direction of this resultant force.

𝐹1

𝑎1=

𝐹2

𝑎2=

𝐹3

𝑎3= constant = mass

F = ma

SF = ma

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12.3 Linear momentum of a particle. Rate of

change of linear momentum

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𝐅 = 𝑚𝑑𝐕

𝑑𝑡=

𝑑 𝑚𝐯

𝑑𝑡

linear momentum or momentum of the particle is

L = mV

𝐅 = 𝐋

12.5 Equations of motion

Rectangular Components.

Tangential and Normal Components.

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SF = ma

zzyyxx

zyxzyx

maFmaFmaF

kajaiamkFjFiF

SSS

S )()(

• 𝐅 = 𝑚𝐚 = 𝑚𝑑𝑣

𝑑𝑡𝐞𝑡 +𝑚

𝑣2

𝜌𝐞𝑛

12.6 Dynamic equilibrium

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SF – ma = 0

where – ma is an inertia vector.

Friction

F = msN Impending slip

F = mkN Slip

where

mk = kinetic friction coefficient

ms = static friction coefficient

Springs

Fs = kd = k (L - L0)

Sample Problem 12.1

The crate is sliding down an incline. Its initial position and initial speed are 1.5 m and 8 m/s, respectively. Determine the

acceleration of the crate and the force in the spring when the crate first stops.

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y = 0 = constant, ay = 0

SFy: mg cosq – N = 0, N = mg cosq

F = mkN = mkmg cosq

SFx: mg sinq – mkmg cosq – k (x - L0) = max ________________ (1)

when the crate first stops v = 0; v2 = v02 + 2ax(x-x0) _____________ (2)

Solve for ax and x

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SF = ma

SFx: mg sinq – F – Fs = max

SFy: mg cosq – N = may

Fs = kd = k (x - L0)

F = mkN

Sample Problem 12.2

The bob of a 2-m pendulum describes an arc of a circle in a

vertical plane. If the tension in the cord is 2.5 times the weight

of the bob for the position shown, find the velocity and the acceleration of the bob in that position.

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Sample Problem 12.3

A race car moves at a constant speed v along a banked turn on the

track shown. Let the bank angle and turn radius of curvature be those

of the Talladega Superspeedway in East Aboga, Alabama, which

means that r is 350 m and the turn bank angle is 33o. For this turn, determine the maximum value of v such that the car does not slide.

Assume that the static friction coefficient between the car and track is ms = 0.9.

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12.7 Angular momentum of a particle. Rate of change of angular momentum

The moment of the vector mv about O is called the moment of

momentum, or the angular momentum Ho

Ho = ro x mv

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Ho = rmv sin f

= rmvq = mr2𝜃

12.7 Angular momentum of a particle. Rate of change of angular momentum

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• 𝐇 𝑂 = 𝐫 × m𝐯 + 𝐫 ×𝑚𝐯

• = 𝐯 × m𝐯 +𝐫 ×𝑚𝐚

• = 𝐫 × 𝐅

• 𝐇 𝑂 = 𝐌𝑂

Newton’s second law states that the sum of the moments about

O of the forces acting on the particle is equal to the rate of

change of the moment of momentum, or angular momentum, of the particle about O.

12.8 Equations of motion in terms of radial and transverse components

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• 𝐅 = 𝑚𝐚 = 𝑚 𝑟 − 𝑟𝜃 2 𝐞𝑟 +𝑚 𝑟𝜃 + 2𝑟 𝜃 𝐞𝜃

Sample Problem 12.4

A block B of mass m can slide freely on a frictionless arm OA

which rotates in a horizontal plane at a constant rate 𝜃 𝑜 . Knowing that B is released at a distance ro from O, express as a

function of r, (a) the component vr of the velocity of B along

OA, (b) the magnitude of the horizontal force F exerted on B by the arm OA.

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Sample Problem 12.5

A small sphere is at rest at the top of a frictionless semicylindrical

surface. The sphere is given as light nudge to the right so that it

slides along the surface. Determine the angle q at which the sphere separates form the surface.

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Sample Problem 12.6

Consider a disk of mass m at one

end of a linear elastic spring, the

other end of which is pinned. The

disk is free to move in the smooth

horizontal plane.

Determine 𝑟 and 𝜃 when r = 0.35

m, q = 0 rad, 𝑟 = 0.35 m/s, 𝜃 = 0.5 rad/s, the unstretched length of

the spring is 0.25 m, and the ratio

of spring constant and the mass

of the disk, k/m, is 5 s-2.

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Sample Problem 12.7

A child was playing on a merry-

go-round as shown in the figure.

In walking over the platform while

spinning, not only did he feel

thrown radially outward but also he felt thrown sideways.

Investigate this motion, and

determine the forces required to walk radially at a constant rate vo

on a platform of radius r, while

spinning at constant angular rate wo.