Chapter 9 Angular Motion in a Plane ANGULAR

Chapter 9 Angular Motion in a Plane

ANGULAR DISPLACEMENT • Angular displacement (q ) is usually expressed in radians, in degrees, or in revolutions.

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. 2 3 1 57. 30 4 6 segments gets to here. 2 p segments gets completely around. 6 5 1 rev = 3600 = 2 p radians (rad)

Thus the angle q in radians is given in terms of the arc length l it subtends on a circle of radius r by The radian measure of an angle is a dimensionless number.

THE ANGULAR SPEED The angular speed (w ) of an object whose axis of rotation is fixed is the rate at which its angular coordinate, the angular displacement q, changes with time. If q changes from qi to qf in a time t, then the average angular speed is

• The units of are exclusively rad/s. Since each complete turn or cycle of a revolving system carries it through 2 p rad w = 2 p f. • f is the frequency in revolutions per second, rotations per second, or cycles per second. • Accordingly, w is called the angular frequency. We can associate a direction with w and thereby create a vector quantity.

THE ANGULAR ACCELERATION • The angular acceleration (a ) of an object whose axis of rotation is fixed is the rate at which its angular speed changes with time. • If the angular speed changes uniformly from wi to wf in the time t, then the angular acceleration is constant and

The units of a are typically rad/s 2, rev/min 2, and such. It is possible to associate a direction with w, and therefore with a, thereby specifying the angular acceleration vector a, but we will have no need to do so here.

Equations for uniformly accelerated angular motion are exactly analogous to those for uniformly accelerated linear motion. In the usual notation we have:

RELATIONS BETWEEN ANGULAR AND TANGENTIAL QUANTITIES: • When a wheel of radius r rotates about an axis whose direction is fixed, a point on the rim of the wheel is described in terms of the circumferential distance l it has moved, its tangential speed v, and its tangential acceleration a. T. • These quantities are related to the angular quantities q, w, and a, which describe the rotation of the wheel, through the relations:

• provided radian measure is used for q, w, and a.

Let’s derive these equations. l q w r

If w gets bigger then v gets bigger. a. T is in the same direction of v. If w gets smaller then v gets smaller. a. T is in the opposite direction of v. q w r

• provided radian measure is used for q, w, and a. • By simple reasoning, l can be shown to be the length of belt wound on the wheel or the distance the wheel would roll (without slipping) if free to do so. • In such cases, v and a. T refer to the tangential speed and acceleration of a point on the belt or of the center of the wheel.

Cycloid q r rq

Translation plus Rotation

Cycloid Remember this slide q r rq What is happening to the velocity of this point?

ROTATIONAL INERTIA • Law of inertia for rotating systems An object rotating about an axis tends to remain rotating at the same rate about the same axis unless interfered with by some external influence. • Examples: bullet, arrow, and earth • • • Demo – Football and spinning basketball Demo - Whirly Tube (Zinger) Demo – Whirly Shooter Demo - Disc Gun Demo - Rubber Bands

• Demo - Inertia Bars • Moment of inertia (rotational inertia) The sluggishness of an object to changes in its state of rotational motion • Distribution of mass is the key. • Example: Tightrope walker

CENTRIPETAL ACCELERATION • Centripetal acceleration (ac): • A point mass m moving with constant speed v around a circle of radius r is undergoing acceleration. • The direction of the velocity is continually changing. • This gives rise to an acceleration ac of the mass, directed toward the center of the circle. • We call this acceleration the centripetal acceleration; its magnitude is given by

q w r

Because v = rw, we also have where w must be in rad/s.

THE CENTRIPETAL FORCE The centripetal force (F ( c) is the force that must act on a mass m moving in a circular path of radius r to give it the centripetal acceleration v 2/r. From F = ma, we have Where FC is directed toward the center of the circular path.

• Centripetal force - center seeking force • Examples: tin can and string, sling, moon and earth, car on circular path • Demo - Coin on clothes hanger • Demo - String, ball, and tube • Demo - Loop the loop

CENTRIFUGAL FORCE • Centrifugal force - center fleeing force • Often confused with centripetal • Examples: sling and bug in can • Demo - Walk the Line • Centrifugal force is attributed to inertia.

CENTRIFUGAL FORCE IN A ROTATING REFERENCE FRAME • A frame of reference can influence our view of nature. • For example: we observe a centrifugal force in a rotating frame of reference, yet it is a fictitious (pseudo) force. • Centrifugal force stands alone (there is no action-reaction pair) - it is a fictitious force.

• Another pseudo force - Coriolis