ISNS 3371 - Phenomena of Nature Angular Momentum Momentum associated with rotational or orbital motion angular momentum = mass x velocity x radius

ISNS 3371 - Phenomena of Nature

Angular Momentum

Momentum associated with rotational or orbital motionangular momentum = mass x velocity x radius


Torque and Conservation of Angular Momentum

Conservation of angular momentum - like conservation of momentum -in the absence of a net torque (twisting force), the total angular momentum of a system remains constant

Torque - twisting force


Four forces: weight (mg), upward normal force (N), tension in paper (T), and friction force (N).

If spool not yet moving, net horizontal force is zero or:

Tcos() = N

Only two of the forces produce a torque about the center of the spool (T and N). Equating the torques gives:

r1T = r2 N

Dividing into previous equation gives

cos() = r1/ r2 This gives the critical angle which determines which way the spool will rotate

The Moving Spool


Conservation of angular momentum - like conservation of momentum - in the absence of a net torque (twisting force), the total angular momentum of a system remains constant.

Newton’s Third Law of Rotation Motion: For every torque that one object exerts on a second object, there is an equal but oppositely directed torque that the second object exerts on the first object.

Conservation of Angular Momentum


A spinning skater speeds up as she brings her arms in and slows down as she spreads her arms because of conservation of angular momentum

ISNS 3371 - Phenomena of Nature Angular Momentum

Momentum associated with rotational or orbital motion:

angular mom = mass x velocity x radius.

The angular momentum vector is pointed along the axis of rotation - right-hand rule: curl the fingers of your right hand into a fist and point your thumb up. If the direction of your fingers is the direction of rotation, the angular momentum vector is pointed along your thumb

Note: The angular momentum of a rigid body (a hoop, cylinder, etc…) is the sum of the angular momentums of the particles composing the body

ISNS 3371 - Phenomena of Nature Moment of InertiaThe property of a body that is a measure of its rotational inertia - resists a change in angular (rotational) velocity (and thus angular momentum) -analogous to mass - a measure of body’s translational inertia which resists a change in translational velocity/momentum

- determined by mass and distribution of mass - how far the mass is from center of rotation

Torque = moment of inertia X angular acceleration

This is analogous to F = mavt, at

r

= vt/r is the angularvelocity

= at/r is the angular acceleration

so = r

Angular acceleration measures how fast angular velocity changes



Matter and Energy


DEFINITION:

• Anything that occupies space and has mass

PROPERTIES OF MATTER:

• Mass - a measure of a body’s resistance to a change in its state of motion - its inertia

• Density - mass per unit volume

• Dimensions - height, length, width

• Electric charge - positive/negative/neutral

• Heat content - everything above absolute 0 (-459.67º F) has heat - no such quantity as cold - only absence of heat

• Resistance to flow of electric current - flow of charged particles - electrons

• Pressure - exerted by moving molecules in all directions - resists compression

Matter

ISNS 3371 - Phenomena of NatureEnergy

Definition of Energy:• Anything that can change the condition of matter• Ability to do work – the mover of substance (matter)• Work is a force acting over a distance• Force: The agent of change – push or pull on a body

Hence: Work is the change in the energy of a system resulting from the application of a force acting over a distance.

Work = force X distance

Units of Energy:Joule = amount of work done when a force of 1 Newton is applied over 1 meter1 J = 1N - m = 1 kg m2/s2 -

1 Joule = 1/4184 Calorie, so2500 Cal = 1 x 107 J (average daily requirement for a human)


Solar energy striking Earth’s surface per second = 2.5 x 1017 J.Energy released by burning 1 liter of oil = solar energy striking square 100 m on a side in 1 second

Energy Comparisons


Four Types of Forces:

• Gravitational – holds the world together

• Electromagnetic – attraction/repulsion of charged matter

• Strong Nuclear – holds nucleus together

• Weak Nuclear – involved in reactions between subatomic particles

Fundamental Forces of Nature


Energy

Three basic categories:

Kinetic energy = energy of motion

KE = 1/2mv2

Potential energy = stored energy

gravitational, chemical, elastic,electrostatic, etc…

Radiative - energy carried by light

{MechanicalEnergy


Potential Energy

One form of potential energy is gravitational potential energy - the energy which an object stores due to its ability to fall

•It depends on:– the object’s mass (m)– the strength of gravity (g)– the distance which it falls (h)

PE = mgh

Before the sun was formed - matter contained in cloud diffuse gas cloud - most far from the center - large gravitational energy. As cloud contracted under its own gravity - gravitational energy converted to thermal energy until hot enough to ignite nuclear fusion

m

h

g


Potential Energy

• energy is stored in matter itself• this mass-energy is what would be released if an amount of

mass, m, were converted into energy

E = mcE = mc22

[ c = 3 x 108 m/s is the speed of light; m is in kg, then E is in joules]

The mass energy in a 1-kg rock is equal to as much energy as 7.5 billion liters of oil = enough to run all the cars in the U.S. for a weekA 1-megaton hydrogen bomb converts only about 3 ounces of mass into energy.


Conservation of Energy

• Energy can be neither created nor destroyed.

• It merely changes it form or is exchanged between objects.

• This principle (or law) is fundamental to science.

• The total energy content of the Universe was determined in the Big Bang and remains the same today.


Types of Energy

Energy cannot be created or destroyed, only changed

– Mechanical –

• Potential - stored energy

• Kinetic- energy of motion KE=1/2mv2

– Electrical

– Chemical

– Elastic

– Gravitational

– Thermal

– Radiant

– Nuclear


Conversion of Energy

Throwing a baseball

Nuclear energy (nuclear fusion on sun) - Radiative energy (sunlight) - Chemical energy (photosynthesis) - Chemical energy in pitcher’s body (from eating plants) - Mechanical kinetic energy (motion of arm) - Mechanical kinetic energy (movement of the baseball). Thus, ultimate source of KE in baseball is mass energy stored in hydrogen of Sun - created in Big Bang.

Hydroelectric dam

Gravitational - mechanical - electrical

Nuclear reactor

Nuclear - thermal - mechanical - electrical

CarChemical - thermal - mechanical


Power:Rate of change of energy

Power = work done/time interval = E/t

(remember: means a change in a quantity)

Power:1 watt = 1J/sThus for every second a 100 W light bulb is on, the electric company charges for 100 J of energy.The average daily power requirement for a human is about the same as for a 100-W light bulb.

Power


Applications of Conservation of Energy


Machines

Machines can be used to multiply force:

(force X distance)input = (force X distance)output

Decrease the distance and the force will increase.

Work/Energy is not changed!


Levers

Fulcrum is in the center:d1 = d2

so

F1 = F2

Fulcrum is closer to one end:

d1 > d2

So

F2 > F1

Give me a long enough lever and a place to put the fulcrum and I can move the world (Archimedes, 250 BC).


Pulleys


€

Fnet = −mgsinθ

For small angles, sin =

€

Fnet = −mgθ = ma

€

−mgθ = mαl

vt, at

r

= vt/r is the angularvelocity

= at/r is the angular acceleration

so = r

This becomes the differential equation:

€

d2θ

dt 2+g

lθ = 0

Pendulum solution (you are not expected to know this)

With solution

€

=max cosg

lt

For a complete oscillation:

€

g

lP = 2π so

€

P = 2πl

g

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ISNS 3371 - Phenomena of Nature Angular Momentum Momentum associated with rotational or orbital motion angular momentum = mass x velocity x radius