unit 6: energy and momentum of rotating systems

what is the formula for rotational kinetic energy?

½Iω²

what is the total kinetic energy?

½ mv² + ½ Iω²

is kinetic energy scalar or vector?

scalar

features of the motion of a system that is rolling without slipping

• the translational motion is directly related to the rotational motion of the system

  • Δx=rΔΘ ; v=rω ; a = rα

• frictional forces do not dissipate energy

what causes the motion of a system to roll while slipping?

kinetic friction

does a system that rolls while slipping dissipate energy?

yes!!!!

how can work (W) be found through torque?

• W = τΔΘ

• area under a τ-ΔΘ graph

• W = ΔKE_rot

what is the relationship between rotational and translational motion in a system of an object that is rolling while slipping?

they are NOT directly related

angular momentum of a rigid body

L = Iω

angular momentum of a point mass

L = rmvsinΘ

define angular impulse

the product of the torque exerted on an object or rigid system and the time interval during which the torque is exerted

• angular impulse = τΔt

impulse-momentum theorem

τ_net= ΔL/Δt = Iω/Δt = Iα

graphical relation of the change in angular momentum and change in angular impulse

• net torque is the slope of a momentum-time graph

• angular impulse is the area of a torque-time graph

what causes a change in angular momentum?

an external torque acting on the system

why does an ice skater’s rotational velocity increase as they pull their arms + legs in?

their rotational inertia decreases so, in order for momentum to be conserved, the angular velocity must decrease

what is conserved in circular orbits?

everything! (angular momentum L, total mechanical energy TME, potential energy U, kinetic energy K)

what is conserved in elliptical orbits?

angular momentum L and total mechanical energy TME

what is not conserved in elliptical oribts?

potential energy U and kinetic energy K

what is always conserved in all orbits?

angular momentum L and total mechanical energy TME

what is the gravitational potential energy equation?

-(GMm)/r

describe the motion of a massive central object and an orbiting satellite

when the satellite has a negligible mass compared to the massive central object, the motion of the central object is negligible

what is escape velocity?

the velocity is the minimum velocity at which an object needs to go in order to leave the orbit of a planet OR it’s when the TME of the system is zero

what is the velocity of an object that is an infinite distance from the central body?

zero (when r = infinity)

what formulas do you need to derive the escape velocity?

• U_g + K = 0

• U_g = (GMm)/r

• K = 1/2mv²