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Dynamics II Chapter 6 Chapter Objectives Discuss dynamics in terms of: Work Energy Power Momentum Conservation laws Oscillations Work and Energy Consider a force F. Sometimes, the object moves in the same direction as F That means a parallel displacement vector d exists

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Dynamics ii chapter 6 l.jpg

Dynamics IIChapter 6


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Chapter Objectives

  • Discuss dynamics in terms of:

    • Work

    • Energy

    • Power

    • Momentum

    • Conservation laws

    • Oscillations


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Work and Energy

  • Consider a force F.

  • Sometimes, the object moves in the same direction as F

    • That means a parallel displacement vector d exists

  • Such a displacement requires Energy


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A word about Energy

  • Uses of energy:

    • Lifting a weight to a given height

    • Charging a battery

    • Boiling water

    • Heating an object

  • Types of energy:

    • Mechanical energy

    • Electrical energy

    • Heat


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Work Done by Constant Force cont’d

  • Work done in a straight line is:

    • F = force

    • d = distance traveled

  • Work expressed as N·m.

  • 1 Nm=1 J oule (J)

W=Fd


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Work Done by Constant Force cont’d

  • Work done at an angle θ is:

W = Fd cosθ


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Work Done by Variable Force

  • Example: throwing a javelin

    • Force is zero, then rises to a maximum, then decreases back to zero as javelin is released

    • F varies with x

  • Consider Figure 6-4.




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Work Done by Variable Force cont’d

  • Variable force can be divided into intervals

    • Interval 1 has force value F1 and so on.

  • Work done on interval 1 is:

    • Δx = (xf-xi)/10

    • Width of each interval on axis

W=F1Δx


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Work Done by Variable Force cont’d

Error decreases as number of intervals increases…

Integral gets the size of each interval down to the infinitesimal


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One more trick in numerical intergration

  • Consider the force as a trapezoid, not a rectangle


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Energy

  • Energy can be viewed as the capacity to do work

  • When we deal with “mechanical work”, we’re talking about

    • Kinetic Energy (KE)

    • Gravitational Potential Energy (PE)

    • Strain Potential Energy


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Kinetic Energy

  • Energy possessed because of motion

  • Kinetic energy is expressed as:

    • Given as SI unit Joules (J)


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Work and Energy

  • Consider the work required to accelerate and object

  • Distance moved:

  • v=at, so t=v/a, so:

  • F=ma, so a=F/m, so:

  • Work = Fd =


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Kinetic Energy cont’d

  • Example:

    • Calculate the kinetic energy of a sprinter of mass 70 kg moving at 10 m/s.

    • KE = 0.5(70 kg)(10 m/s)2

    • = 3500 J


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Gravitational Potential Energy

  • Energy possessed by a system due to position of a body

  • Also called potential energy

PE = mgh


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Gravitational Potential Energy cont’d

  • Example:

    • A pole vaulter of body mass 70 kg succeeds in clearing a bar that is 6 m above the ground. Calculate his potential energy at the top of the vault.


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Gravitational Potential Energy cont’d

  • Example:

    h = 6m

    g = 9.81m/s2

    PE = (70 kg)(9.81 m/s2)(6 m)

    = 4120 J


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Conservation of Mechanical Energy

  • Kinetic energy

  • Gravitational potential energy

  • Strain potential energy


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Conservation of Mechanical Energy cont’d

  • Total energy in a system does not change


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We’re lying to you again…

  • A real system inevitably involves friction

  • Presence of friction means energy is converted to heat or sound

  • Thermal energy: energy created when heat is generated


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Power

  • The rate of doing work

  • The rate of energy transfer

  • Given as SI units Watts (W) or J/s


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Impulse–Momentum Relationship

  • Impulsive forces:

    • Forces that occur over very short time intervals

      • Usually milliseconds

      • Bodies deform

      • Example: bat striking ball

  • Change of momentum of ball = Impulse

  • Consider Figure 6-8.


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Newton’s 2nd Law:

Realizing Force is variable and using smallest intervals possible:

Change in momentum = Impulse


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Collisions in One Dimension

  • Consider the following collisions:

    • Hockey puck with stick

    • Golf ball and club

    • Ball and bat in baseball

    • Foot and ball in soccer



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Collisions in One Dimension cont’d

  • Force between colliding partners exists for very short time

  • Impulse on objects is the same but with opposite signs

  • Change in momentum is the same but in the opposite direction


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Collisions in One Dimension cont’d

  • m1 and m2 = masses of objects 1 and 2

  • v1f and v1i = final and initial velocities of object 1

  • v2f and v2i = final and initial velocities of object 2



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Elastic and Inelastic Collisions

  • Elastic collisions:

    • Kinetic energy is conserved

    • Consider a ball dropped on a floor from height h0 that bounces all the way back up


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Perfectly Elastic Impact

Velocity of system is conserved

Drop a Superball (close to perfect)

Perfectly Plastic Impact

Permanent deformation of at least one body

Bodies do not separate

Drop a lump of clay

Extremes


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In Between

  • Coefficient of Restitution

0

1

Inelastic

Perfectly Plastic

Perfectly Elastic


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Elastic and Inelastic Collisions cont’d

  • Inelastic collisions:

    • Total kinetic energy is decreased but some other energy is increased

    • 0 < e < 1


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Coefficient of Restitution

  • Depends on BOTH bodies:

    • Basketball and gym floor

    • Baseball and bat

    • Tennis ball and racquet


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Velocities before impact

Velocities after impact


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V2

V1

V3

V4


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One Simplification

  • If one body is stationary (e.g. a floor), the equation becomes simpler

  • Example: ball dropped to floor from height hd will bounce to a height of hb


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Bat-Ball Games and Spin

  • Speed of ball after impact is increased by:

    • Increasing the mass of the bat

    • Decreasing the mass of the ball

    • Increasing the initial velocity of the bat or ball

    • Increasing the angle of incidence

    • Increasing the value of e





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Oscillations

  • Hooke’s law:

    • Extension x of a spring is proportional to applied force F

    • k is spring constant/stiffness

      • given as N/m

F=-kx


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Oscillations cont’d

  • Extension of spring is:



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Oscillations cont’d

  • Strain energy from total work done is:


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Oscillatory Motion

  • Examples in sports:

    • Vibrations of racquets

    • Oscillations of trampolines

    • Response of ligaments to sudden movements

    • Response of gold clubs to impact with a ball


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Simple Harmonic Motion

  • Frequency of oscillations: 1/T

  • T = period, given by:


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Simple Harmonic Motion cont’d

  • KE + PE + SE = Constant


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Damped Oscillations

  • Remember, there’s almost always friction

  • Sometimes friction is deliberately introduced


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Damping Coefficient (c) determines behavior

  • Critically Damped: Return to zero in shortest possible time with no oscillation, cc

  • Damping ratio:

  • Underdamped: <1

  • Overdamped: >1


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Pendulums

  • Simple pendulum:

    • An object suspended from the end of a cord attached to a rigid support

    • Weight is resolved along the direction of the cord and perpendicular

    • Cord is assumed massless and inextensible


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Pendulums cont’d

  • SHM applies for small displacement

  • Period of a simple pendulum:


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Using a physical pendulum to determine I

  • Use Newton’s second law applied to angular motion (M=I), determine what  must be for the object oscillating as a pendulum, and approximate for small angles (sinθ≈0 when θ is small)


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Topics Introduced

  • Work

  • Energy

  • Power

  • Momentum

  • Impulse

  • Collisions

  • Oscillation


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