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# Dynamics II Chapter 6 - PowerPoint PPT Presentation

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

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## Dynamics IIChapter 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

• Such a displacement requires 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 Joule (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

• Gets us back to conservation of linear momentum

### 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

### 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:

• 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:

### Oscillations cont’d

• Strain energy from total work done is:

### 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

### Simple Harmonic Motion

• Frequency of oscillations: 1/T

• T = period, given by:

### Simple Harmonic Motion cont’d

• KE + PE + SE = Constant

### Damped Oscillations

• Remember, there’s almost always friction

• Sometimes friction is deliberately introduced

### Damping Coefficient (c) determines behavior

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

• Damping ratio:

• Underdamped: <1

• Overdamped: >1

### 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

### Pendulums cont’d

• SHM applies for small displacement

• Period of a simple pendulum:

### 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)

• Work

• Energy

• Power

• Momentum

• Impulse

• Collisions

• Oscillation