Rotation

Copyright © 2012 Pearson Education Inc.

Rotation

Physics 7C lecture 10

Thursday October 31, 8:00 AM – 9:20 AMEngineering Hall 1200


External forces and center-of-mass motion

• When a body or collection of particles is acted upon by external forces, the center of mass moves as though all the mass were concentrated there.


External forces and center-of-mass motion

• Fragments of a firework shell would fly at 100 m/s for 5 seconds before they burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance.


External forces and center-of-mass motion• Fragments of a firework shell would fly at 100 m/s for 5 seconds before they

burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance.

motion of center of mass:

motion of fragments relative to center of mass:


Rocket propulsion

• As a rocket burns fuel, its mass decreases, as shown in Figure below.

• What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0?


Rocket propulsion

between time t and t + dt, according to momentum conservation:

(m+dm) v = m (v+dv) + dm (v-vex)



Rocket propulsion

between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm (v-vex)

m dv – v dm + (v-vex) dm = 0

(m0- λ t) dv –vex λ dt = 0

dv - λ vex dt /(m0- λ t)= 0

v + vex ln(m0- λ t) = constant

v = v0 + vex ln (m0/(m0- λ t)) = v0 + vex ln (m0/m)



Introduction

• The north star is Polaris today, but 5000 years ago it was Thuban. What caused the change?

• What causes bodies to start or stop spinning?

• We’ll introduce some new concepts, such as torque and angular momentum, to deepen our understanding of rotational motion.


Introduction

• How do we quantify the spinning of wind turbine?


Angular displacement

• angular displacement: θ

• unit: radian

• direction: (right hand rule!)


Radian

2 π radian = 360 degree


Angular displacement

• Motion of a spinning wheel


Angular displacement and velocity

• How do we quantify the spinning of wind turbine?

ω = dθ /dt


Angular velocity is a vector!

• signs of angular displacement


Right hand rule

• Angular displacement is a vector, use right hand rule to determine the direction.


Angular acceleration

• α = dω/dt


Angular acceleration

• calculate ω from α


These are very similar to linear motion

• linear and angular motion:


Linear vs. angular motion

• v =?



• what is the acceleration?



• radian vs. degree


Example

• calculate the acceleration of the black point in the disk.


Example



Example



Speed of propeller

• calculate the speed of the tip of the propeller.


Speed of propeller

• calculate the speed of the tip of the propeller.

Documents

Rotation