Loading...

PPT – Physics 207, Lecture 5, Sept. 17 PowerPoint presentation | free to download - id: 6d38d-ZDc1Z

The Adobe Flash plugin is needed to view this content

Physics 207, Lecture 5, Sept. 17

- Goals
- Solve problems with multiple accelerations in

2-dimensions (including linear, projectile and

circular motion) - Discern different reference frames and

understand how they relate to particle motion in

stationary and moving frames - Recognize different types of forces and know how

they act on an object in a particle

representation - Identify forces and draw a Free Body Diagram

Assignment HW3, (Chapters 4 5, due 9/25,

Wednesday) Read through Chapter 6, Sections 1-4

Exercise (2D motion with acceleration)

Relative Trajectories Monkey and Hunter

A hunter sees a monkey in a tree, aims his gun at

the monkey and fires. At the same instant the

monkey lets go. Does the bullet

- go over the monkey?
- hit the monkey?
- go under the monkey?

Schematic of the problem

- xB(Dt) d v0 cos q Dt
- yB(Dt) hf v0 sin q Dt ½ g Dt2
- xM(Dt) d
- yM(Dt) h ½ g Dt2
- Does yM(Dt) yB(Dt) hf?

(x,y) (d,h)

Monkey

Does anyone want to change their answer ?

What happens if g0 ? How does introducing g

change things?

hf

g

v0

q

Bullet

(x0,y0) (0 ,0) (vx,vy) (v0 cos q, v0 sin q)

Uniform Circular Motion (UCM) is common so we

have specialized terms

- Arc traversed s q r
- Tangential velocity vt
- Period, T, and frequency, f
- Angular position, q
- Angular velocity, w

s

vt

r

q

Period (T) The time required to do one full

revolution, 360 or 2p radians

Frequency (f) 1/T, number of cycles per unit time

Angular velocity or speed w 2pf 2p/T, number

of radians traced out per unit time (in UCM

average and instantaneous will be the same)

Example Question (note the commonality with

linear motion)

- A horizontally mounted disk 2 meters in diameter

spins at constant angular speed such that it

first undergoes 10 counter clockwise revolutions

in 5 seconds and then, again at constant angular

speed, 2 counter clockwise revolutions in 5

seconds. - 1 What is the period of the initial rotation?
- 2 What is initial angular velocity?
- 3 What is the tangential speed of a point on the

rim during this initial period? - 4 Sketch the angular displacement versus time

plot. - 5 What is the average angular velocity?
- 6 If now the turntable starts from rest and

uniformly accelerates throughout and reaches the

same angular displacement in the same time, what

must the angular acceleration be? - 7 What is the magnitude and direction of the

acceleration after 10 seconds?

Example Question

- A horizontal turntable 2 meters in diameter

spins at constant angular speed such that it

first undergoes 10 counter clockwise revolutions

in 5 seconds and then, again at constant angular

speed, 2 counter clockwise revolutions in 5

seconds. - 1 What is the period of the turntable during the

initial rotation - T (time for one revolution) Dt / of

revolutions/ time 5 sec / 10 rev 0.5 s - 2 What is initial angular velocity?
- w angular displacement / time 2 p f 2 p / T

12.6 rad / s - 3 What is the tangential speed of a point on the

rim during this initial period? - We need more..

Relating rotation motion to linear velocity

- In UCM a particle moves at constant tangential

speed vt around a circle of radius r (only

direction changes). - Distance tangential velocity time
- Once around 2pr vt T
- or, rearranging (2p/T) r vt
- w r vt

- Definition If UCM then w constant
- So vT w r 4 p rad/s 1 m 12.6 m/s

4 A graph of angular displacement (q) vs. time

Angular displacement and velocity

- Arc traversed s q r
- in time Dt then Ds Dq r
- so Ds / Dt (Dq / Dt) r
- in the limit Dt ? 0
- one gets
- ds / dt dq / dt r
- vt w r
- w dq / dt
- if w is constant, integrating w dq / dt,
- we obtain q qo w Dt
- Counter-clockwise is positive, clockwise is

negative

Sketch of q vs. time

5 Avg. angular velocity Dq / Dt 24 p /10

rad/s

Next part..

- 6 If now the turntable starts from rest and

uniformly accelerates throughout and reaches the

same angular displacement in the same time, what

must the tangential acceleration be?

Well, if w is linearly increasing

- Then angular velocity is no longer constant so

dw/dt ? 0 - Define tangential acceleration as at dvt/dt r

dw/dt - So s s0 (ds/dt)0 Dt ½ at Dt2

and s q r - We can relate at to dw/dt
- q qo wo Dt Dt2
- w wo Dt
- Many analogies to linear motion but it isnt

one-to-one - Note Even if the angular velocity is constant,

there is always a radial acceleration.

Tangential acceleration?

- 6 If now the turntable starts from rest and

uniformly accelerates throughout and reaches the

same angular displacement in the same time, what

must the tangential acceleration be?

- q qo wo Dt Dt2
- (from plot, after 10 seconds)
- 24 p rad 0 rad 0 rad/s Dt ½ (at/r) Dt2
- 48 p rad 1m / 100 s2 at

- 7 What is the magnitude and direction of the

acceleration after 10 seconds?

Circular motion also has a radial (perpendicular)

component

Uniform circular motion involves only changes in

the direction of the velocity vector, thus

acceleration is perpendicular to the trajectory

at any point, acceleration is only in the radial

direction. Quantitatively (see text)

Centripetal Acceleration ar

vt2/r Circular motion involves continuous

radial acceleration

Non-uniform Circular Motion

For an object moving along a curved trajectory,

with non-uniform speed a ar at (radial and

tangential)

at

ar

Tangential acceleration?

- 7 What is the magnitude and direction of the

acceleration after 10 seconds?

- at 0.48 p m / s2
- and w r wo r r Dt 4.8 p m/s

vt - ar vt2 / r 23 p2 m/s2

Tangential acceleration is too small to plot!

Angular motion, signs

- Also note if the angular displacement, velocity

and/or accelarations are counter clockwise then

this is said to be positive. - Clockwise is negative

Exercise

A Ladybug sits at the outer edge of a

merry-go-round, and a June bug sits halfway

between the outer one and the axis of rotation.

The merry-go-round makes a complete revolution

once each second. What is the June bugs

angular velocity?

A. half the Ladybugs. B. the same as the

Ladybugs. C. twice the Ladybugs. D. impossible

to determine.

J

L

Circular Motion

- UCM enables high accelerations (gs) in a small

space - Comment In automobile accidents involving

rotation severe injury or death can occur even at

modest speeds. - In physics speed doesnt kill.acceleration

does (i.e., the sudden change in velocity).

Mass-based separation with a centrifuge

Before

After

arvt2 / r and f 104 rpm is typical with r

0.1 m and vt w r 2p f r

How many gs?

ca. 10000 gs

bb5

gs with respect to humans

- 1 g Standing
- 1.2 g Normal elevator acceleration (up).
- 1.5-2g Walking down stairs.
- 2-3 g Hopping down stairs.
- 1.5 g Commercial airliner during takeoff run.
- 2 g Commercial airliner at rotation
- 3.5 g Maximum acceleration in amusement park

rides (design guidelines). - 4 g Indy cars in the second turn at Disney World

(side and down force). - 4 g Carrier-based aircraft launch.
- 10 g Threshold for blackout during violent

maneuvers in high performance aircraft. - 11 g Alan Shepard in his historic sub orbital

Mercury flight experience a maximum force of 11

g. - 20 g Colonel Stapps experiments on acceleration

in rocket sleds indicated that in the 10-20 g

range there was the possibility of injury because

of organs moving inside the body. Beyond 20 g

they concluded that there was the potential for

death due to internal injuries. Their

experiments were limited to 20 g. - 30 g The design maximum for sleds used to test

dummies with commercial restraint and air bag

systems is 30 g.

A bad day at the lab.

- In 1998, a Cornell campus laboratory was

seriously damaged when the rotor of an

ultracentrifuge failed while in use. - Description of the Cornell Accident -- On

December 16, 1998, milk samples were running in a

Beckman. L2-65B ultracentrifuge using a large

aluminum rotor. The rotor had been used for this

procedure many times before. Approximately one

hour into the operation, the rotor failed due to

excessive mechanical stress caused by the

g-forces of the high rotation speed. The

subsequent explosion completely destroyed the

centrifuge. The safety shielding in the unit did

not contain all the metal fragments. The half

inch thick sliding steel door on top of the unit

buckled allowing fragments, including the steel

rotor top, to escape. Fragments ruined a nearby

refrigerator and an ultra-cold freezer in

addition to making holes in the walls and

ceiling. The unit itself was propelled sideways

and damaged cabinets and shelving that contained

over a hundred containers of chemicals. Sliding

cabinet doors prevented the containers from

falling to the floor and breaking. A shock wave

from the accident shattered all four windows in

the room. The shock wave also destroyed the

control system for an incubator and shook an

interior wall.

Relative motion and frames of reference

- Reference frame S is stationary
- Reference frame S is moving at vo
- This also means that S moves at vo relative to

S - Define time t 0 as that time when the origins

coincide

Relative Velocity

- The positions, r and r, as seen from the two

reference frames are related through the

velocity, vo, where vo is velocity of the r

reference frame relative to r - r r vo Dt
- The derivative of the position equation will give

the velocity equation - v v vo
- These are called the Galilean transformation

equations - Reference frames that move with constant

velocity (i.e., at constant speed in a straight

line) are defined to be inertial reference frames

(IRF) anyone in an IRF sees the same

acceleration of a particle moving along a

trajectory. - a a (dvo / dt 0)

Central concept for problem solving x and y

components of motion treated independently.

- Example Man on cart tosses a ball straight up in

the air. - You can view the trajectory from two reference

frames

Example (with frames of reference) Vector addition

An experimental aircraft can fly at full throttle

in still air at 200 m/s. The pilot has the nose

of the plane pointed west (at full throttle) but,

unknown to the pilot, the plane is actually

flying through a strong wind blowing from the

northwest at 140 m/s. Just then the engine fails

and the plane starts to fall at 5 m/s2.

What is the magnitude and directions of the

resulting velocity (relative to the ground) the

instant the engine fails?

Calculate A B

Ax Bx -200 140 x 0.71 and Ay

By 0 140 x 0.71

Home Exercise, Relative Motion

- You are swimming across a 50 m wide river in

which the current moves at 1 m/s with respect to

the shore. Your swimming speed is 2 m/s with

respect to the water. - You swim across in such a way that your path is a

straight perpendicular line across the river. - How many seconds does it take you to get across?

Home Exercise

Choose x axis along riverbank and y axis across

river

- The time taken to swim straight across is

(distance across) / (vy )

- Since you swim straight across, you must be

tilted in the water so that your x component of

velocity with respect to the water exactly

cancels the velocity of the water in the x

direction

vy 1 m/s

rivers frame

Home Exercise 2

Where do you land if the river flows at 2 m/s

while swimming at the same heading in the river

(i.e., q asin ½) ?

- The time taken to swim straight across

(distance across) / (vy ) - time 50 m / ( 2 m/s cos q) 50/3½ seconds

- Dist in river vx t -2 m/s sin q t -2

(50/3½) 1/2 m -29 m - (upstream)

- Dist river flows vr t 2 m/s t -2 (50/3½) m

58 m - Final position -29 m 58 m 29 m down the

shore.

What causes motion? (Actually changes in

motion) What are forces ? What kinds of forces

are there ? How are forces and motion related ?

Physics 207, Lecture 5, Sept. 17

Assignment HW3, (Chapters 4 5, due 9/25,

Wednesday) Read Chapter 5 through Chapter 6,

Sections 1-4