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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

Dynamics II Chapter 6

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Dynamics IIChapter 6

- Discuss dynamics in terms of:
- Work
- Energy
- Power
- Momentum
- Conservation laws
- Oscillations

- 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 in a straight line is:
- F = force
- d = distance traveled

- Work expressed as N·m.
- 1 Nm=1 Joule (J)

W=Fd

- Work done at an angle θ is:

W = Fd cosθ

- 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.

- 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

Error decreases as number of intervals increases…

Integral gets the size of each interval down to the infinitesimal

- Consider the force as a trapezoid, not a rectangle

- 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

- Energy possessed because of motion
- Kinetic energy is expressed as:
- Given as SI unit Joules (J)

- 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 =

- 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

- Energy possessed by a system due to position of a body
- Also called potential energy

PE = mgh

- 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.

- Example:
h = 6m

g = 9.81m/s2

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

= 4120 J

- Kinetic energy
- Gravitational potential energy
- Strain potential energy

- Total energy in a system does not change

- 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

- The rate of doing work
- The rate of energy transfer
- Given as SI units Watts (W) or J/s

- 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

- Consider the following collisions:
- Hockey puck with stick
- Golf ball and club
- Ball and bat in baseball
- Foot and ball in soccer

- 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

- 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 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

- Coefficient of Restitution

0

1

Inelastic

Perfectly Plastic

Perfectly Elastic

- Inelastic collisions:
- Total kinetic energy is decreased but some other energy is increased
- 0 < e < 1

- 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

- 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

- 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

- 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

- Extension of spring is:

- Strain energy from total work done is:

- Examples in sports:
- Vibrations of racquets
- Oscillations of trampolines
- Response of ligaments to sudden movements
- Response of gold clubs to impact with a ball

- Frequency of oscillations: 1/T
- T = period, given by:

- KE + PE + SE = Constant

- Remember, there’s almost always friction
- Sometimes friction is deliberately introduced

- Critically Damped: Return to zero in shortest possible time with no oscillation, cc
- Damping ratio:
- Underdamped: <1
- Overdamped: >1

- 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

- SHM applies for small displacement
- Period of a simple pendulum:

- 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