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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 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

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 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

- Forces that occur over very short time intervals
- 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

ExtremesElastic 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 after impact

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)

Topics Introduced

- Work
- Energy
- Power
- Momentum
- Impulse
- Collisions
- Oscillation

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