Chapter 9 Rotational Motion Rotational Motion n n

Chapter 9 Rotational Motion

Rotational Motion n n Many interesting physical phenomena are not “linear” Ball at the end of a string revolving Planets around Sun Moon around Earth New Physical Quantities: • Angular Displacement • Angular velocity • Angular acceleration

The Radian n n The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r n 57. 3°

More About Radians n Whole circle Comparing degrees and radians n Converting from degrees to radians n

Example q=49. Express it in radians, revolutions.

Angular Displacement n n n Axis of rotation is the center of the disk Need a fixed reference line During time t, the line on rotating object moves through angle θ

Angular Displacement, cont. n The angular displacement is defined as the angle the object rotates through during some time interval n n n The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement

Average Angular Velocity n The average angular velocity, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Angular velocity, cont. n n The instantaneous angular velocity Units of angular velocity are radians/sec • rad/s n n Velocity will be positive if θ is increasing (counterclockwise) Velocity will be negative if θ is decreasing (clockwise)

Average Angular Acceleration n The average angular acceleration of an object is defined as the ratio of the change in the angular velocity to the time it takes for the object to undergo the change:

Angular Acceleration, cont n n Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular velocity and the same angular acceleration Angular speed

Angular Acceleration, final n n n The sign of the acceleration does not have to be the same as the sign of the angular velocity The instantaneous angular acceleration Only consider rotations with uniform angular acceleration

Example A bicycle wheel makes 50 revolutions in 0. 5 min. Compute the angular speed in rev/s, rad/s, deg/s.

Example The bicycle wheel traveling at 1. 67 rev/s decelerates uniformly to a stop in 3 seconds. Compute the angular deceleration in rev/s², rad/s², deg/s²

Analogies Between Linear and Rotational Motion Linear Motion with constant acc. (x, v, a) Rotational Motion with fixed axis and constant a (q, , a)

Example A car coasts to a stop with a uniform deceleration of 1. 2 rad/s² for its wheel. If its initial angular speed was 1000 rev/min, how many revolutions does the wheel make before coming to a stop?

Relationship Between Angular and Linear Quantities n Displacements n Speeds n n n Accelerations Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion

Examples n n n Pulley Rolling wheel A belt runs on a pulley with diameter 5 cm which revolves at a speed of 1000 rev/min. What length of belt passes over the pulley each second?

Example A 60 cm diameter wheel is rotating @ speed of 3. 0 rev/s. How fast is this car going?

Example If the masses of Atwood’s machine are accelerating at 4. 8 m/s² and radius of 0. 3 m, what is a?

Centripetal Acceleration n n An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration is due to the change in the direction of the velocity

Centripetal Acceleration, cont. n n n Centripetal refers to “center-seeking” The direction of the velocity changes The acceleration is directed toward the center of the circle of motion

Centripetal Acceleration, final n The magnitude of the centripetal acceleration is given by • This direction is toward the center of the circle

Centripetal Acceleration and Angular Velocity n n n The angular velocity and the linear velocity are related (v = rω) The centripetal acceleration can also be related to the angular velocity must be in rad/s

Forces Causing Centripetal Acceleration n Newton’s Second Law says that the centripetal acceleration is accompanied by a force • F = ma • F stands for any force that keeps an object following a circular path n n n Tension in a string Gravity Force of friction

Examples n n Ball at the end of revolving string (m=500 grams, r=120 cm, =0. 8 rev/s). Tension? Fast car rounding a curve

More on circular Motion n n Length of circumference = 2 R Period T (time for one complete circle)

Newton’s Law of Universal Gravitation n Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

Universal Gravitation, 2 n n n G is the constant of universal gravitational G = 6. 673 x 10 -11 N m² /kg² This is an example of an inverse square law

Motion of Satellites n n n Consider only circular orbit Radius of orbit r: Gravitational force is the centripetal force.

Motion of Satellites n Period Kepler’s 3 rd Law n Synchronous Orbits

Communications Satellite n A geosynchronous orbit • Remains above the same place on the earth • The period of the satellite will be 24 hr n n r = h + RE Still independent of the mass of the satellite

Satellites and Weightlessness n n n weighting an object in an elevator Elevator at rest: mg Elevator accelerates upward: m(g+a) Elevator accelerates downward: m(g+a) with a<0 Satellite: a=-g!!

Chapter 10 Dynamics of Rotation

Force vs. Torque n n Forces cause accelerations Torques cause angular accelerations • rotation n Force and torque are related

Torque, cont n Torque, , is the tendency of a force to rotate an object about some axis n n is the torque F is the force • symbol is the Greek tau n n n l is the lever arm SI unit is N. m Lever Arm: Perpendicular distance from the pivot point to the line of force

Direction of Torque n Torque is a vector quantity • We will treat only 2 -d torque so no need for vector notion. • If the turning tendency of the force is counterclockwise, the torque will be positive (+) • If the turning tendency is clockwise, the torque will be negative (-)

Multiple Torques n When two or more torques are acting on an object, the torques are added • with the signs n If the net torque is zero, the object’s rate of rotation doesn’t change

General Definition of Torque n n The applied force is not always perpendicular to the position vector The component of the force perpendicular to the object will cause it to rotate F is the force n L is distance between pivot and point of action n q is the angle n

Moment of Inertia n n The angular acceleration is inversely proportional to the analogy of the mass in a rotating system This mass analog is called the moment of inertia, I, of the object • SI units are kg m 2

Newton’s Second Law for a Rotating Object n n The angular acceleration is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object

More About Moment of Inertia n n There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation

Moment of Inertia of a Uniform Ring n n Image the hoop is divided into a number of small segments, m 1 … These segments are equidistant from the axis

Other Moments of Inertia

Example Wheel of radius R=20 cm and I=30 kg·m^2. Force F=40 N acts along the edge of the wheel. 1. Angular acceleration? 2. Angular speed 4 s after starting from rest? 3. Number of revolutions for the 4 s?

Rotational Kinetic Energy n n An object rotating about some axis with an angular speed, ω, has rotational kinetic energy Ekr=½Iω2 Energy concepts can be useful for simplifying the analysis of rotational motion

Total Energy of a System n Conservation of Mechanical Energy • Remember, this is for conservative forces, no dissipative forces such as friction can be present • Potential energies of any other conservative forces could be added

Rolling down incline n Energy conservation Linear velocity and angular speed are related v=R n Smaller I, bigger v, faster!! n

Work-Energy in a Rotating System n n In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy (Ekt+Ekr+Ep)i+W=(Ekt+Ep)f

Angular Momentum n n Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Angular momentum is defined as • L = I ω • and

Angular Momentum, cont n n If the net torque is zero, the angular momentum remains constant Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. • That is, when

Conservation Rules, Summary n In an isolated system, the following quantities are conserved: • Mechanical energy • Linear momentum • Angular momentum

Conservation of Angular Momentum, Example n With hands and feet drawn closer to the body, the skater’s angular speed increases • L is conserved, I decreases, increases

Example A 500 grams uniform sphere of 7. 0 cm radius spins at 30 rev/s on an axis through its center. n Moment of inertia n Rotational kinetic energy n Angular momentum

Example A turntable is a uniform disk of metal of mass 1. 5 kg and radius 13 cm. What torque is required to drive the turntable so that it accelerates at a constant rate from 0 to 33. 3 rpm in 2 seconds?