Angular motions in revolutions, degrees, and radians

•  One complete cycle of 360° is one revolution.

•  One complete revolution is 2π radians.

•  Relating the two, 360° = 2 π radians or 1 radian = 57.3°.

Rotational System

Angular displacement is the angle being swept out

•  Like a second hand sweeping around a clock, a radius vector will travel through a displacement of degrees, radians, or revolutions.

•  We denote angular displacement as Θ (theta). It is the angular equivalent of x or y in earlier chapters.

Angular velocity •  The angular velocity is the angle swept out divided by the

time it took to sweep out the angular displacement.

•  Angular velocity is denoted by the symbol ω (omega).

•  Angular velocity is measured in radians per second (SI standard) as well as other measures such as r.p.m. (revolutions per second).

Angular velocity is a vector •  You can visualize the position of the vector by sweeping out

the angle with the fingers of your right hand. The position of your thumb will be the position of the angular velocity vector. This is called the “right-hand rule.”

Angular acceleration •  The angular acceleration is the change of angular

velocity divided by the time interval during which the change occurred.

•  Use the symbol α (alpha) to denote radians per second2.

Angular acceleration is a vector

•  The angular acceleration vector will be parallel or antiparallel to the angular velocity vector.

Linear and angular quantities related

Bicycle pedals and gears

Rotational energy

•  Just like linear kinetic energy is ½ mv2, the angular energy will be determined by ½ Iω2.

Finding the moment of inertia for common shapes

Rotation of a uniform sphere about a center axis

•  Is this a good model for our planet rotating?

Torque

•  A force applied at a right angle to a lever will generate a torque.

•  The distance from the pivot to the point of force application will be linearly proportional to the torque produced.

A rigid body in motion about a moving axis •  One might imagine playing with a yo-yo toy while riding a bicycle down a

hillside.

Rolling with and without slipping •  Rolling with slipping may be calculated. Slipping

makes things worse (for driving and calculations).

The race of objects with different moments

•  This problem is a “classic” and appears on most professors’ exams on this material.

Consider the acceleration of a rolling sphere

•  Figures will introduce an effect of friction.

Gyroscopic precession

•  The precession of a gyroscope shows up in many “common” situations.

Vocabulary: • Slender rod =İnce Çubuk • Rectangular plate =Dikdörtgen plaka • Conservative force =Korunumlu Kuvvet • Edge =Kenar • Impulse= İtme • Hollow=Oyuk • Hollow cylinder=İçi boş silindir • Solid=Katı • Thin walled hollow cylinder =İnce kenarlı içi boş silindir • Solid Sphere=İçi dolu küre • Uniform Sphere=Homojen küre • Generate = Meydana getirmek • Pivot=Mil • Rigid Body=Katı cisim • Rolling=Yuvarlanma • Slipping=Kayma • Appear =gözükmek • Gyroscopic precession =Jiroskopik Devinim • Quantity=nicelik

• Angular=Açısal • Revolution=Devir • Degree=Derece • Cycle=Tur • Sweep=Süpürmek • Visualize =Gözünde canlandırmak • Thumb =Başparmak • Finger=Parmak • Right-hand -rule =Sağ el kuralı • Angular displacement =Açısal yerdeğiştirme • Experience = Deneyim • Gear= Dişli, vites • Rear=Arka • Front=Ön • Moment of inertia =Eylemsizlik Momenti • Apparatus =Cihaz • Common shapes =Ortak şekiller • Flywheel=Çark • Angular Momentum=Açısal Momentum • Sprocket=Zincir Dişlisi