Constant Angular Acceleration, Rotational Kinematic Energy, Conservation of Angular Momentum

rotational kinematics

a rigid object is rotating about a fixed axis

angular displacement

delta theta

how much has the object been rotated

angular velocity

w

how fast does the object rotate

angular acceleration

a

how fast does the angular velocity change

SI unit of angle

radian (rad) which is dimensionless

1 rev = 360 = 2pi rad

generally positive direction

counterclockwise

period (time for one revolution around circle)

T = 2pi/w

if angular acceleration is constant you can use the

rotational kinematics equations

vf = vi + at

wf = wi + at

x(t) = xo + vt + 1/2at^2

0f = 0i + wit + 1/2at^2

vf^2 = vi^2 + 2a(x)

wf^2 = wi^2 + 2a(0)

the tangential speed of an object is

vt= 2pir/T = rw

another way of writing centripital acceleration

ar = vt^2/r = rw^2

in translational motion, an object has kinetic energy K = 1/2mv^2

in rotational motion, the different parts of an object move at different linear speeds BUT all the parts move at the same rotational speed (w)

moment of inertia

I

measures how hard it is to change the angular velocity of an object

units: kg\*m^2

moment of inertia equation

summation of m(r^2)

an objects moment of inertia depends on

its axis of rotation

Krot

1/2\*I\*w^2

moment of inertia changes depending on different body types

rod

hoop

cylinder/disk

solid sphere

hollow sphere

maximum torque

force perpendicular to the radial line

zero torque

force 180 degrees from the pivot

torque equation

r\*F

r

vector from the axis of rotation to the point where force is applied

theta in torque

angle between r and F

to find torque magnitude

use formula

to find torque direction

right hand rule

Newton’s Second Law for Rotation

Tnet = summation T = I\*a

torque, moment of inertia, and angular acceleration must all be with respect to the same axis of rotation

static equilibrium

if an object is at rest - with no rotation nor translation - the object is in static equilibrium

• net force on object is zero ( no translation )

• net torque on object is zero ( no rotation )

if the net torque on an object is zero

it doesn’t matter which axis of rotation we choose