physics 207: lecture 5, pg 1 physics 207, lecture 5, sept. 17 l goals: solve problems with multiple...

Physics 207: Lecture 5, Pg 1

Physics 207, Physics 207, Lecture 5, Sept. 17Lecture 5, Sept. 17 Goals:Goals:

Solve problems with multiple accelerations in 2-Solve problems with multiple accelerations in 2-dimensions dimensions (including linear, projectile and circular (including linear, projectile and circular motion)motion)

Discern different reference frames and understand Discern different reference frames and understand how they relate to particle motion in stationary and how they relate to particle motion in stationary and moving framesmoving frames

Recognize different types of forces and know how they Recognize different types of forces and know how they act on an object in a particle representationact on an object in a particle representation

Identify forces and draw a Identify forces and draw a Free Body DiagramFree 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. go over the monkey?

B. hit the monkey?

C. go under the monkey?

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 …


Schematic of the problem

xB(t) = d = v0 cos t yB(t) = hf = v0 sin t – ½ g t2

xM(t) = d yM(t) = h – ½ g t2

Does yM(t) = yB(t) = hf?

Does anyone want to change their answer ?

(x0,y0) = (0 ,0)

(vx,vy) = (v0 cos , v0 sin

Bullet v0

g

(x,y) = (d,h)

hf

What happens if g=0 ?

How does introducing g change things?

Monkey


Uniform Circular Motion (UCM) is common so we have specialized terms

Arc traversed s = r Tangential velocity vt

Period, T, and frequency, f Angular position, Angular velocity,

Period (T): The time required to do one full revolution, 360° or 2 radians

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

Angular velocity or speed = 2f = 2/T, number of radians traced out per unit time (in UCM average and instantaneous will be the same)

r vt

s


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 QuestionExample 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 rotationT (time for one revolution) = t /# of revolutions/ time = 5 sec / 10 rev = 0.5 s

2 What is initial angular velocity?

= angular displacement / time = 2 f = 2 / 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… 2r = vt T

or, rearranging (2/T) r = vt

r = vt

Definition: If UCM then constant

3 So vT = r = 4 rad/s 1 m = 12.6 m/s

r vt

s

4 A graph of angular displacement () vs. time


Angular displacement and velocity

Arc traversed s = r

in time t then s = r

so s / t = ( / t) r

in the limit t 0

one gets

ds / dt = d/ dt r

vt = r

= d/ dt

if is constant, integrating = d/ dt,

we obtain: = + t

Counter-clockwise is positive, clockwise is negative

r vt

s


Sketch of Sketch of vs. timevs. time

time (seconds)

(rad

ians

)

= + t

= + 5rad

= + t

= rad + rad

5 Avg. angular velocity = / t = 24 /10 rad/s


Next part…..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?


Then angular velocity is no longer constant so d/dt ≠ 0 Define tangential acceleration as at = dvt/dt = r d/dt So s = s0 + (ds/dt)0 t + ½ at t2 and s = r We can relate at to d/dt

= o + o t + t2

= o + t Many analogies to linear motion but it isn’t one-to-one Note: Even if the angular velocity is constant, there is

always a radial acceleration.

Well, if is linearly increasing …

at

r

1

2

at

r


Tangential acceleration? Tangential acceleration?

= o + o t + t2

(from plot, after 10 seconds)

24 rad = 0 rad + 0 rad/s t + ½ (at/r) t2

48 rad 1m / 100 s2 = at

r vt

s1

2

at

r

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?



Circular motion also has a radial (perpendicular) componentCircular motion also has a radial (perpendicular) component

Centripetal Acceleration

ar = vt2/r

Circular motion involves continuous radial acceleration

vt

r

ar

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)


Non-uniform Circular Motion

For an object moving along a curved trajectory, with non-uniform speeda = ar + at (radial and tangential)

ar

at

ar = v2

r

dt

d| |vat =


Tangential acceleration? Tangential acceleration?

at = 0.48 m / s2

and r = o r + r t = 4.8 m/s = vt

ar = vt2 / r = 23 2 m/s2

r vt

s

at

r


Tangential acceleration is too small to plot!


Angular motion, signsAngular 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


Circular MotionCircular Motion

UCM enables high accelerations (g’s) in a small space

Comment: In automobile accidents involving rotation severe injury or death can occur even at modest speeds.

[In physics speed doesn’t kill….acceleration does (i.e., the sudden change in velocity).]


Mass-based separation with a centrifuge

How many g’s?

Before After

ar=vt2 / r and f = 104 rpm is

typical with r = 0.1 m and vt = r = 2 f r

bb5

ca. 10000 g’s


g’s with respect to humansg’s 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 Stapp’s 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.


Relative motion and frames of referenceRelative 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 VelocityRelative 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 t

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: “Central concept for problem solving: “xx” and “” and “yy” ” components of motion treated independently.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:

Reference frame

on the ground.

Reference frame

on the moving cart.

y(t) motion governed by 1) a = -g y

2) vy = v0y – g t3) y = y0 + v0y – g t2/2

x motion: x = vxt

Net motion: R = x(t) i + y(t) j (vector)


What causes motion?What causes motion?(Actually changes in motion)(Actually changes in motion)

What are forces ?What are forces ?

What kinds of forces are there ?What kinds of forces are there ?

How are forces and motion related ?How are forces and motion related ?


Physics 207, Physics 207, Lecture 5, Sept. 17Lecture 5, Sept. 17

Assignment: HW3, (Chapters 4 & 5, due 9/25, Wednesday)

Read Chapter 5 through Chapter 6, Sections 1-4

physics 207: lecture 5, pg 1 physics 207, lecture 5, sept. 17 l goals: solve problems with multiple...

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