Energy and Momentum - Full Theory

Is the elastic potential energy stored in a spring greater when the spring is stretched by 1.5 cm or when it is compressed by 1.5 cm?

is en a scalar or vector

scalar

max deformation =

2 x deformation

at the bottom Fx =

2Fg

acceleration in SHM

max, 0, max

velocity

0, max (constant), 0

mass sliding into the spring w/o an applied force, Fnet =

Fx

if you don’t know if it’s max compression/stretch, there is a ________

net force

for elas pot en, the en is ___________ inside the material due to its _________ and has the pot to do _____

stored; state; work

examples of elas pot en

rubber balls, diving boards, car bumpers, springs in a mattress… etc

hooke’s law

• linear relash btwn F and the extension/compression of the spring

• slope of such graph is the force or spring constant

relating hooke’s law to newton’s 3rd law

the force exerted by a mass on a spring is equal and opposite (in direction) to the force the spring exerts on the mass

restorative force

the force exerted by the spring on the obj as the result of newton’s 3rd law and the spring’s natural inclination to return to its original form

restorative force (Fx) is actually ____ bc it is:

neg; opposing the applied force

higher spring constant

stiffer spring

lower spring constant

easier to extend/compress spring

dir of the restorative force is always opposite and proportional to the dir of the ________________

displacement

elastic limit

when stretched beyond such limit, the material will no longer return to its original shape; no material is perfectly elastic

natural equilibrium position

posi of a spring without being stretched or compressed by a force or mass

forces are only balanced at:

equilibirum positions (don’t assume)

work equals to the change in

elas pot en : the area under a force and deformation graph gives both the elas pot en and the work done

elas pot en can be converted to other forms, like

sound en of a guitar string, person jumping on trampoline, kin en of an arrow shot

simple harmonic motion (shm)

the periodic motion of a moving object where the acceleration is proportional to the displacement

• motion is sinusoidal in time and demonstrates a single resonant frequency as the mass oscillates back and forth.

resonant freq

natural freq of an obj

SHM is similar to _____ ___________

wave motion

relating SHM to the motion of a wave and sinoisodal patterns

one cycle = one sine wave

periodic motion examples

clocks, EM spectrum, human body vibrations, sound waves, playground swings

examples of situations where SHM is not wanted

vehicles, weighing scales, tall buildings, bridges

• vibration would need to be damped → motion is stopped

damped harmonic motion

repeated motion where the A of vibration ↓ and energy ↓ with time

the more amplitude, the more _____

en

relating SHM to UCM

• amp = radius

• Fc = Fx

• ac is proportional w deformation

• from the sideview, the obj in UCM looks like it’s moving back and forth

derivation of period (time for one cycle to complete)

see notes; draw diagrams

T and f do not depend on _______ , only _________________

deformation; mass and spring constant

momentum

the product of a moving object’s mass and its velocity

• “quantity of motion” coined by newton

factors affecting momentum

• mass → affects Ff

• velocity

scalar x vector =

vector

vector x vector =

scalar

two objs where one has double the other’s speed but half their mass, the momentum would be the _____but the one w the higher speed would have _____ kin en

same; more

w/o an unbalanced force, the obj’s speed would be constant, making the momentum _____________

constant

if an obj experiences an unblanced force (aka acceleration), the speed would change as well as the _______________

momentum

impulse

change in momentum of a system

• is a VECTOR → same dir as the unbalanced force

base units of momentum and impulse

N x s

derivation of formula for impulse

see notes

real world applications of momentum and impulse

sports safety, vehicle safety and collision systems like airbags, bumpers, sensors, footwear

a large force applied over a short period of time vs a small force applied over a longer period of time could

potentially produce the same change in momentum

impulse momentum theory

analyzing the momentum before and after the two obj’s interact bc making precie measurements of force and time is hard to determine as the contact usually lasts for less than milliseconds

purpose of goalie pads in hockey (impulse)

1. to provide protection through shock absorbance

2. to slow and stop the puck

physics principles of goalie pads in hockey (impulse)

• pads slow the puck’s speed as it makes contact, making it lose momentum

• DISTRIBUTES THE FORCE OVER A GREATER AREA

• bc as the time increases for the force to be transferred, the F decreases, lowering the force being propelled onto the person’s body

shifting ref frames steps

1. shift the red frame by choosing the velocity that is closest to 0 and add the same mag of speed but oppo dir on both velocities

2. apply the special equations where one of the velocities is 0

3. solve for the other velocity

4. SHIFT REF FRAME BACK (add the orig speed back to your answers and the initial velocities)

change in momentum in one obj must be equal to the ______________________________ of another obj

change in momentum

if kin en is not conserved, the en was transformed into

non-conservative forces like therm en / sound en , resulting in the deformation of the bodies involved in collisions

elastic collisions

• mom is conserved

• kin en is conserved

• Ek = Ek ‘ (rep kin en before and after collision, not during)

inelastic collisions

• mom is conserved

• kin en is NOT CONSERVED But the total en of system is always conserved

• usually Ek initial > Ek final except for explosions

perfectly inelastic collisions

• ideal collision where the 2 obj’s are stuck together after the collision

• final velocities of the 2 obj’s are the same

work done by non-conservative forces

• neg value

• final mech en - initial mech en

perfectly elastic collisions almost _______

never happen in the real world except on LARGE macroscopic obj’s or microscopic obj’s

derivation of conservation of linear momentum

see worksheet

at max compression, the obj’s experience the min separation and

move together MOMENTARILY w the same v

as Ee ___________ Ek ______________ and vice versa

decreases, increases

before collision mech en =

after collision mech en = during collision mech en (energies jst take diff forms)

special cases of elastic collisions

1. when obj’s have the same mass, their velocities switch

2. when a mass collides w a much heavier and stationary obj (like the earth lol), the v of the lighter mass is reversed while the heavy mass is stationary

proof of special case 1 of elas collisions

set m1and m 2 = m and use the special equation for both final v’s

proof of special case 2 of elas collisions

set m1 = 0 since it is negligible in comprison and use special eqn for both final v’s

derivation of Ee formula

see notes ; find area under graph and replace F with kx