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## Dynamics II Chapter 6

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**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 • Such a displacement requires Energy**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**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**Work Done by Constant Force cont’d**• Work done at an angle θ is: W = Fd cosθ**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.**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**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**One more trick in numerical intergration**• Consider the force as a trapezoid, not a rectangle**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**Kinetic Energy**• Energy possessed because of motion • Kinetic energy is expressed as: • Given as SI unit Joules (J)**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 =**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**Gravitational Potential Energy**• Energy possessed by a system due to position of a body • Also called potential energy PE = mgh**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.**Gravitational Potential Energy cont’d**• Example: h = 6m g = 9.81m/s2 PE = (70 kg)(9.81 m/s2)(6 m) = 4120 J**Conservation of Mechanical Energy**• Kinetic energy • Gravitational potential energy • Strain potential energy**Conservation of Mechanical Energy cont’d**• Total energy in a system does not change**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**Power**• The rate of doing work • The rate of energy transfer • Given as SI units Watts (W) or J/s**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.**Newton’s 2nd Law:**Realizing Force is variable and using smallest intervals possible: Change in momentum = Impulse**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**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**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**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**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**In Between**• Coefficient of Restitution 0 1 Inelastic Perfectly Plastic Perfectly Elastic**Elastic and Inelastic Collisions cont’d**• Inelastic collisions: • Total kinetic energy is decreased but some other energy is increased • 0 < e < 1**Coefficient of Restitution**• Depends on BOTH bodies: • Basketball and gym floor • Baseball and bat • Tennis ball and racquet**Velocities before impact**Velocities after impact**V2**V1 V3 V4**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**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**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**Oscillations cont’d**• Extension of spring is: