PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 11. Angular velocity, acceleration Rotational/ Linear analogy (angle in radians) Centripetal acceleration:

PHYSICS 231

INTRODUCTORY PHYSICS I

PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 11

• Angular velocity, acceleration

• Rotational/ Linear analogy

• (angle in radians)

• Centripetal acceleration:

(to center)

Last Lecture

€

ω =ΔθΔt=θ f −θ it f − ti

α =ω f −ωit

Δθ ↔ Δxω0 ↔ v0ω f ↔ v f

α ↔ a

t ↔ t

€

at = rα

€

v t = rω

€

Δs = rΔθ

€

acent =ω2r =

v2

r

€

⇒

Newton’s Law of Universal Gravitation

• Always attractive• Proportional to both masses

• Inversely proportional to separation squared

F =Gm1m2

r2

G=6.67×10−11m3

kg⋅s2⎛

⎝⎜⎞

⎠⎟

Gravitation Constant

• Determined experimentally• Henry Cavendish, 1798• Light beam / mirror amplify motion

Weight

• Force of gravity on Earth

• But we know

€

Fg =GMEm

RE2

€

Fg = mg

€

g =GME

RE2

€

⇒

Example 7.14

8.81 m/s2

(0.90 g)

Often people say astronauts feel weightless, because there is no gravity in space.

This explanation is wrong!

What is the acceleration due to gravity at the height of the space shuttle (~350 km above the earth surface)?

Example 7.14 (continued)

Correct explanation of weightlessness:

• Everything (shuttle, people, bathroom scale, etc.) also falls with same acceleration

• No counteracting force (earth’s surface)

• “Accelerating Reference Frame”

• Same effect would be felt in falling elevator

Example 7.15aAstronaut Bob stands atop the highest mountain of planet Earth, which has radius R.Astronaut Ted whizzes around in a circular orbit at the same radius.Astronaut Carol whizzes around in a circular orbit of radius 3R.Astronaut Alice is simply falling straight downward and is at a radius R, but hasn’t hit the ground yet. Bob

Alice

Ted

Carol

Which astronauts experience weightlessness?

A.) All 4B.) Ted and CarolC.) Ted, Carol and Alice

Example 7.15bAstronaut Bob stands atop the highest mountain of planet Earth, which has radius R.Astronaut Ted whizzes around in a circular orbit at the same radius.Astronaut Carol whizzes around in a circular orbit of radius 3R.Astronaut Alice is simply falling straight downward and is at a radius R, but hasn’t hit the ground yet. Bob

Alice

Ted

Carol

Assume each astronaut weighs w=180 lbs on Earth.

The gravitational force acting on Ted is

A.) wB.) ZERO

Example 7.15cAstronaut Bob stands atop the highest mountain of planet Earth, which has radius R.Astronaut Ted whizzes around in a circular orbit at the same radius.Astronaut Carol whizzes around in a circular orbit of radius 3R.Astronaut Alice is simply falling straight downward and is at a radius R, but hasn’t hit the ground yet. Bob

Alice

Ted

Carol

Assume each astronaut weighs w=180 lbs on Earth.

The gravitational force acting on Alice is

A.) wB.) ZERO

Example 7.15dAstronaut Bob stands atop the highest mountain of planet Earth, which has radius R.Astronaut Ted whizzes around in a circular orbit at the same radius.Astronaut Carol whizzes around in a circular orbit of radius 3R.Astronaut Alice is simply falling straight downward and is at a radius R, but hasn’t hit the ground yet.

Bob

Alice

Ted

Carol

Assume each astronaut weighs w=180 lbs on Earth.

The gravitational force acting on Carol is A.) wB.) w/3C.) w/9D.) ZERO

Example 7.15eAstronaut Bob stands atop the highest mountain of planet Earth, which has radius R.Astronaut Ted whizzes around in a circular orbit at the same radius.Astronaut Carol whizzes around in a circular orbit of radius 3R.Astronaut Alice is simply falling straight downward and is at a radius R, but hasn’t hit the ground yet.

Bob

Alice

Ted

Carol

Which astronaut(s) undergo an acceleration g=9.8 m/s2?

A.) AliceB.) Bob and AliceC.) Alice and TedD.) Bob, Ted and AliceE.) All four

Kepler’s Laws

• Tycho Brahe (1546-1601)• Extremely accurate astronomical observations

• Johannes Kepler (1571-1630)• Worked for Brahe• Used Brahe’s data to find mathematical description of planetary motion

• Isaac Newton (1642-1727)• Used his laws of motion and gravitation to derive Kepler’s laws

Kepler’s Laws

1) Planets move in elliptical orbits with Sun at one of the focal points.

2) Line drawn from Sun to planet sweeps out equal areas in equal times.

3) The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

Kepler’s First Law

• Planets move in elliptical orbits with the Sun at one focus.

• Any object bound to another by an inverse square law will move in an elliptical path

• Second focus is empty

Kepler’s Second Law

• Line drawn from Sun to planet will sweep out equal areas in equal times

• Area from A to B equals Area from C to D.

True for any central force due to angular momentum conservation (next chapter)

Kepler’s Third Law

• The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

• The constant depends on Sun’s mass, but is independent of the mass of the planet

€

R3

T 2= Constant

Derivation of Kepler’s Third Law

m

M

€

Fgrav =GMm

R2= macent = mω

2R

€

ω =2π

T

€

R3

T 2=GM

4π 2

€

⇒

Example 7.16

Data: Radius of Earth’s orbit = 1.0 A.U. Period of Jupiter’s orbit = 11.9 years

Period of Earth’s orbit = 1.0 years

Find: Radius of Jupiter’s orbit

5.2 A.U.

Example 7.17

Given: The mass of Jupiter is 1.73x1027 kg and Period of Io’s orbit is 17 daysFind: Radius of Io’s orbit

r = 1.85x109 m

Gravitational Potential Energy• PE = mgh valid only near Earth’s surface

• For arbitrary altitude

• Zero reference level isat r=

PE =−GMmr

Example 7.18

You wish to hurl a projectile from the surface of the Earth (Re= 6.38x106 m) to an altitude of 20x106 m above the surface of the Earth. Ignore rotation of the Earth and air resistance.

a) What initial velocity is required?

b) What velocity would be required in order for the projectile to reach infinitely high? I.e., what is the escape velocity?

c) (skip) How does the escape velocity compare to the velocity required for a low earth orbit?

a) 9,736 m/s

b) 11,181 m/s

c) 7,906 m/s

Chapter 8

Rotational Equilibrium and

Rotational Dynamics

Wrench Demo

Torque

• Torque, , is tendency of a force to rotate object about some axis

• F is the force• d is the lever arm (or moment arm)

• Units are Newton-meters

=Fd

Door Demo

Torque is vector quantity

• Direction determined by axis of twist• Perpendicular to both r and F• Clockwise torques point into paper. Defined as negative

• Counter-clockwise torques point out of paper. Defined as positive

r r FF

- +

Non-perpendicular forces

Φ is the angle between F and r

=Fr sinφ

Torque and Equilibrium

• Forces sum to zero (no linear motion)

• Torques sum to zero (no rotation)

ΣFx = 0 and ΣFy = 0

Σ =0

Meter Stick Demo

Axis of Rotation

• Torques require point of reference• Point can be anywhere

• Use same point for all torques• Pick the point to make problem least difficult (eliminate unwanted Forces from equation)

Example 8.1

Given M = 120 kg.Neglect the mass of the beam.

a) Find the tensionin the cable

b) What is the forcebetween the beam andthe wall

a) T=824 N b) f=353 N

Another Example

Given: W=50 N, L=0.35 m, x=0.03 m

Find the tension in the muscle

F = 583 N

xL

W