Finish Momentum Start Spinning Around

Finish MomentumStart Spinning Around March 22, 2006

Watsup? • Quiz on Friday • Last part of energy conservation through today. • Watch for still another WebAssign • Full calendar for the remainder of the semester is on the website near the end of the last set of PowerPoint slides. • Next exam is next week!

Last Time

We therefore define the center of mass as

Center of Mass, Coordinates • The coordinates of the center of mass are • where M is the total mass of the system

Center of Mass, position • The center of mass can be located by its position vector, rCM • ri is the position of the i th particle, defined by

Center of Mass, Example • Both masses are on the x-axis • The center of mass is on the x-axis • The center of mass is closer to the particle with the larger mass

Think of the extended object as a system containing a large number of particles The particle distribution is small, so the mass can be considered a continuous mass distribution Center of Mass, Extended Object

Center of Mass, Extended Object, Coordinates • The coordinates of the center of mass of a uniform object are

Center of Mass, Extended Object, Position • The position of the center of mass can also be found by: • The center of mass of any symmetrical object lies on an axis of symmetry and on any plane of symmetry

Density r Usually, r is a constant but NOT ALWAYS!

Center of Mass, Example • An extended object can be considered a distribution of small mass elements, Dm=rdxdydz • The center of mass is located at position rCM

For a flat sheet, we define a mass per unit area … s

Linear Density m

Center of Mass, Rod • Find the center of mass of a rod of mass M and length L • The location is on the x-axis (or yCM = zCM = 0) • xCM = L / 2 Let's just do it!

A golf club consists of a shaft connected to a club head. The golf club can be modeled as a uniform rod of length L and mass m1 extending radially from the surface of a sphere of radius R and mass m2. Find the location of the club’s center of mass, measured from the center of the club head. R m2 L m1

Motion of a System of Particles • Assume the total mass, M, of the system remains constant • We can describe the motion of the system in terms of the velocity and acceleration of the center of mass of the system • We can also describe the momentum of the system and Newton’s Second Law for the system

Velocity and Momentum of a System of Particles • The velocity of the center of mass of a system of particles is • The momentum can be expressed as • The total linear momentum of the system equals the total mass multiplied by the velocity of the center of mass

Acceleration of the Center of Mass • The acceleration of the center of mass can be found by differentiating the velocity with respect to time

Newton’s Second Law for a System of Particles • Since the only forces are external, the net external force equals the total mass of the system multiplied by the acceleration of the center of mass: SFext = MaCM • The center of mass of a system of particles of combined mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system

Momentum of a System of Particles • The total linear momentum of a system of particles is conserved if no net external force is acting on the system • MvCM = ptot = constant when SFext = 0

Motion of the Center of Mass, Example • A projectile is fired into the air and suddenly explodes • With no explosion, the projectile would follow the dotted line • After the explosion, the center of mass of the fragments still follows the dotted line, the same parabolic path the projectile would have followed with no explosion

Chapter 10 Rotation of a Rigid Object about a Fixed Axis

Rigid Object • A rigid object is one that is nondeformable • The relative locations of all particles making up the object remain constant • All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible

Angular Position • Axis of rotation is the center of the disc • Choose a fixed reference line • Point P is at a fixed distance r from the origin

Angular Position, 2 • Point P will rotate about the origin in a circle of radius r • Every particle on the disc undergoes circular motion about the origin, O • Polar coordinates are convenient to use to represent the position of P (or any other point) • P is located at (r, q) where r is the distance from the origin to P and q is the measured counterclockwise from the reference line

Angular Position, 3 • As the particle moves, the only coordinate that changes is q • As the particle moves through q, it moves though an arc length s. • The arc length and r are related: • s = q r

Radian • This can also be expressed as • q is a pure number, but commonly is given the artificial unit, radian • One radian is the angle subtended by an arc length equal to the radius of the arc

Conversions • Comparing degrees and radians 1 rad = = 57.3° • Converting from degrees to radians θ [rad] = [degrees]

Angular Position, final • We can associate the angle q with the entire rigid object as well as with an individual particle • Remember every particle on the object rotates through the same angle • The angular position of the rigid object is the angle q between the reference line on the object and the fixed reference line in space • The fixed reference line in space is often the x-axis

Angular Displacement • The angular displacement is defined as the angle the object rotates through during some time interval • This is the angle that the reference line of length r sweeps out

Average Angular Speed • The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Instantaneous Angular Speed • The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

Angular Speed, final • Units of angular speed are radians/sec • rad/s or s-1 since radians have no dimensions • Angular speed will be positive if θ is increasing (counterclockwise) • Angular speed will be negative if θ is decreasing (clockwise)

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

Instantaneous Angular Acceleration • The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0

Angular Acceleration • Units of angular acceleration are rad/s² or s-2 since radians have no dimensions • Angular acceleration will be positive if an object rotating counterclockwise is speeding up • Angular acceleration will also be positive if an object rotating clockwise is slowing down

Angular Motion, General Notes • When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration • So q, w, a all characterize the motion of the entire rigid object as well as the individual particles in the object

Directions • Strictly speaking, the speed and acceleration (w, a) are the magnitudes of the velocity and acceleration vectors • The directions are actually given by the right-hand rule Using this convention, w is a VECTOR!

Hints for Problem-Solving • Similar to the techniques used in linear motion problems • With constant angular acceleration, the techniques are much like those with constant linear acceleration • There are some differences to keep in mind • For rotational motion, define a rotational axis • The choice is arbitrary • Once you make the choice, it must be maintained • The object keeps returning to its original orientation, so you can find the number of revolutions made by the body

Rotational Kinematics • Under constant angular acceleration, we can describe the motion of the rigid object using a set of kinematic equations • These are similar to the kinematic equations for linear motion • The rotational equations have the same mathematical form as the linear equations

Rotational Kinematic Equations

The derivations are similar to what we did with kinematics. Example:

Comparison Between Rotational and Linear Equations

Displacements Speeds 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 Relationship Between Angular and Linear Quantities

A dentist's drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51  104 rev/min. (a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

An airliner arrives at the terminal, and the engines are shut off. The rotor of one of the engines has an initial clockwise angular speed of 2 000 rad/s. The engine's rotation slows with an angular acceleration of magnitude 80.0 rad/s2. (a) Determine the angular speed after 10.0 s. (b) How long does it take the rotor to come to rest?

A centrifuge in a medical laboratory rotates at an angular speed of 3 600 rev/min. When switched off, it rotates 50.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.

Speed Comparison • The linear velocity is always tangent to the circular path • called the tangential velocity • The magnitude is defined by the tangential speed

Acceleration Comparison • The tangential acceleration is the derivative of the tangential velocity

Finish Momentum Start Spinning Around