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I am sorry!! There is no time to cover: Chapter 10 Please Read it!!! Chapter 10. Rotation of a Rigid Object About Fixed Axis 10.1 Angular Position, Velocity and Acceleration Topic of Chapter: Bodies rotating First, rotating, without translating. Then, rotating AND translating together.

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Slide1 l.jpg

I am sorry!!

There is no time to cover:

  • Chapter 10

    Please Read it!!!



10 1 angular position velocity and acceleration l.jpg
10.1 Angular Position, Velocity and Acceleration

  • Topic of Chapter: Bodies rotating

    • First, rotating, without translating.

    • Then, rotating AND translating together.

  • Assumption:Rigid Object


Rigid object l.jpg
Rigid Object

  • A rigid object is one that is non-deformable

    • 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

  • Rigid Object motion≡Translational motion of center of mass (everything we’ve done up to now) + Rotational motion about axis through center of mass (this chapter!). Can treat two parts of motion separately.


Angular position l.jpg
Angular Position

  • Axis of rotation is the center of the disc

  • Choose a fixed reference line

  • Point P is at a fixed distance rfrom the origin

  • 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


Angular position 2 l.jpg
Angular Position, 2

  • Polar coordinates:

    • Convenient to use to represent the position of P (or any other point)

    • P is located at (r, q) where ris the distance from the origin to P and qis the measured counterclockwise from the reference line

  • Motion:

    • 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 = qr(10.1a)


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Radian

  • This can also be expressed as

    (10.1b)

  • q

    • is a pure number, ratio of 2 lengths (dimensionless)

    • commonly is given the artificial unit, radian

    • Solving problems in this chapter, need calculators inRADIAN MODE!!!!

  • One radian is the angle subtended by an arc length equal to the radius of the arc

  • When srq1 Radian


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Conversions

  • Comparing degrees and radians

  • Converting from degrees to radians

  • Converting from radians to degrees


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Example 10.1Birds of Pray- in Radians

  • Bird’s eye can distinguish objects that subtended an angle no smaller than 3x10- 4 rad

  • (a) 310-4 rad =  º?

    = (310-4 rad)(180º/πrad)  =0.017º

  • (b)r = 100 m,  = ?

     = r = (100)  (310-4)

     = 0.03 m = 3 cm

    MUST be in radians in

    part (b)!


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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 qbetween the reference line on the object and the fixed reference line in space

    • The fixed reference line in space is often the x-axis


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Angular Displacement

  • The angular displacement of the rigid object is defined as the angle the object rotates in a time interval t

  • This is the angle that the reference line of length r sweeps out


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Average Angular Speed

  • The average angular speed, (Greek omega), of a rotating rigid object is the ratio of the angular displacement() to the time interval (t)

    (10.2)


Instantaneous angular speed l.jpg
Instantaneous Angular Speed

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

    (10.3)


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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 l.jpg
Average Angular Acceleration

  • The average angular acceleration,

    (Greek alpha) of a rotating 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:

    (10.4)


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Instantaneous Angular Acceleration

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

    (10.5)


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Angular Acceleration, final

  • 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


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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 objectas well as the individual particles in the object


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Directions, details

  • 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

  • The direction of a follows from its definition a = d/dt

    • Same aswif increases

    • Opposite toif decreases


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Hints for Problem-Solving

  • Similar to the techniques used in linear motion problems

    • Constantangularacceleration techniques ≡ Constantlinearacceleration techniques

  • 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


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10. 2 Rotational Kinematics: Rotational Motion with Constant Angular Acceleration

  • 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


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Rotational Kinematics, final Angular Acceleration

  • We’ve seen analogies between quantities:

    LINEAR and ANGULAR

    Displacement Angular Displacement

    x ≈ 

    Velocity Angular Velocity

    v ≈ 

    Acceleration Angular Acceleration

    a ≈ 


Rotational equations l.jpg
Rotational Equations Angular Acceleration

  • For αconstant,we can use the equations from Chapter 2 with this replacements!!

    (10.6) (10.7)

    (10.8) (10.9)


Comparison between rotational and linear equations l.jpg
Comparison Between Rotational and Linear Equations Angular Acceleration

These are ONLY VALID if all angular quantities are in radian units!!


10 3 angular and linear quantities l.jpg

Displacements Angular Acceleration

Speeds

(10.10)

Accelerations

(10.11)

Every point on the rotating object has the same angular motion

Every point on the rotating object does not have the same linear motion

10.3 Angular and Linear Quantities


Speed comparison l.jpg
Speed Comparison Angular Acceleration

  • The linear velocity is always tangent to the circular path

    • called the tangential velocity

  • The magnitude is defined by the tangential speed

    (10.10)


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Acceleration Comparison Angular Acceleration

  • The tangential acceleration is the derivative of the tangential velocity

    (10.11)


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Speed and Acceleration Remarks Angular Acceleration

  • All points on the rigid object will have the same angular speed, but not the same tangential speed

  • All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration

  • The tangential quantities depend on r, and r is not the same for all points on the object


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Centripetal Acceleration Angular Acceleration

  • An object traveling in a circle, even though it moves with a constant speed, will have an acceleration

    • Therefore, each point on a rotating rigid object will experience a centripetal acceleration

      (10.12)


Resultant acceleration l.jpg
Resultant Acceleration Angular Acceleration

  • The tangential component of the acceleration is due to changing speed

  • The centripetal component of the acceleration is due to changing direction

  • Total acceleration can be found from these components

    (10.13)


Translational rotational analogues connections l.jpg
Translational-Rotational Angular AccelerationAnalogues & Connections

ANALOGUES

TranslationRotation

Displacementx 

Velocityv 

Accelerationa 

CONNECTIONS

s = r v = rat = r 

ac = v2/r = 2 r


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Rotational Motion Example Angular Acceleration

  • For a compact disc player to read a CD, the angular speedmust vary to keep the tangential speed constant (vt = wr )

  • At the inner sections,the angular speed isfasterthan at the outer sections


Example 10 2 bicycle l.jpg
Example 10.2 Angular AccelerationBicycle

  • A bicycle slows down uniformly from vi = 8.4 m/s.

    To rest (f = 0) over a distance of 115 m.

    Diameter of wheel = 0.69 m ( r = 0.34 m).

  • FIND:

  • (a) i : v = r i =vi /r = 8.4 m/s/0.34 m 

    i =24.7 rad/s


Example 10 2 bicycle final l.jpg
Example 10.2 Angular AccelerationBicycle, final

  • (b)Total revolutions of the wheels before come to rest:

     s/r = 115 m/0.34 m = 338.2 rad/2πrad/rev

    53.8 rev

  • (c).= (f2 – i2)/2 = (0 – 24.72)/2(53.8)rad/s2 =0.902 rad/s2

  • (d). Time to stop:

    t = (f – i)/= (0 – 24.7)/0.902 s 

    t =27.4 s


10 4 rotational kinetic energy l.jpg
10.4 Rotational Kinetic Energy Angular Acceleration

  • An object rotating about some axis with an angular speed, ω, has rotational kinetic energy even though it may not have any translational kinetic energy

  • Each particle has a kinetic energy of

    Ki = ½ mivi2

  • Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = wi r


Rotational kinetic energy cont l.jpg
Rotational Kinetic Energy, cont Angular Acceleration

  • The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles

    (10.14)

    (10.16)

  • I≡ Σmi ri2 (10.15)MOMENT OF INERTIA


Rotational kinetic energy final l.jpg
Rotational Kinetic Energy, final Angular Acceleration

  • There is an analogy between the kinetic energies associated with linear motion (K = ½ mv 2) and the kinetic energy associated with rotational motion (KR= ½ Iw2)

  • Rotational kinetic energyis not a new type of energy, the form is different because it is applied to a rotating object

  • The units of rotational kinetic energy are Joules (J)


Example 10 3 four rotating objects l.jpg
Example 10.3 Angular AccelerationFour Rotating Objects

  • Find Iy and KR for this system.

  • Radii of the spheres <<a or b.

  • Iy≡ Σmi ri2 

    Iy= m(0) + m(0) + Ma2 +Ma2 Iy= 2Ma2

  • KR = ½Iyw2 = ½ (2Ma2)w2

  • KR = Ma2w2


Example 10 3 four rotating objects final l.jpg
Example 10.3 Angular AccelerationFour Rotating Objects, final

  • Find Iz and KR for this system.

  • Radii of the spheres <<a or b.

  • Iz≡ Σmi ri2 

    Iz= mb2 + mb2 + Ma2 +Ma2 Iz= 2mb2 +2Ma2

  • KR = ½Izw2 

    KR = ½ (2mb2 +2Ma2)w2

    KR = (2mb2 +Ma2)w2


10 5 moment of inertia l.jpg
10.5 Moment of Inertia Angular Acceleration

  • The definition of moment of inertia is

  • The dimensions of moment of inertia are

    ML2 and its SI units are kg.m2

  • We can calculate the moment of inertia of an object more easily by assuming it is divided into many small volume elements, each of mass Dmi


Moment of inertia cont l.jpg
Moment of Inertia, cont Angular Acceleration

  • We can rewrite the expression for Iin terms of Dm

    (10.17)

  • Since r = m/V (Volumetric Mass Density).

  • For the small volume segment r = dm/dV or dm = r dV, thenIwill be:

    (10.17a)

  • If ris constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known


Notes on various densities l.jpg
Notes on Various Densities Angular Acceleration

  • Volumetric Mass Density: mass per unit volume: r≡m/V

    • For the small volume segment r = dm/dV or dm = rdV

  • Surface Mass Density: mass per unit area: ≡m/A

    • For the small area segment  = dm/dA or dm = dA

    • of a sheet of uniform thickness, t : s≡m/A= mt/At= (m/V )t = rt  s≡rt

  • Linear Mass Density: mass per unit length: l≡m /L

    • For the small length segment l = dm/dLor dm = ldL

    • lof a rod of uniform cross-sectional area A: l = m /L = mA /AL = (m/V )A =rA  l = rA


Example 10 4 moment of inertia of a uniform thin hoop l.jpg
Example 10.4 Angular AccelerationMoment of Inertia of a Uniform Thin Hoop

  • Since this is a thin hoop, all mass elements are the same distance (R) from the center


Example 10 5 moment of inertia of a uniform rigid rod l.jpg
Example 10.5 Angular AccelerationMoment of Inertia of a Uniform Rigid Rod

  • The shaded area has a mass

    • dm = l dx

  • Then the I thru its CM is:


Example 10 6 moment of inertia of a uniform solid cylinder l.jpg
Example 10.6 Angular AccelerationMoment of Inertia of a Uniform Solid Cylinder

  • Divide the cylinder into concentric shells with radius r, thickness dr and length L

  • Then for I



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Parallel-Axis Theorem Angular Acceleration

  • In the previous examples, the axis of rotation coincided with the axis of symmetry of the object

  • For an arbitrary axis, the parallel-axis theorem often simplifies calculations

  • The theorem states:

    I = ICM + MD 2(10.18)

    • Iis about any axis parallel to the axis through the center of mass of the object

    • ICM is about the axis through the center of mass

    • D is the distance from the CM axis to the arbitrary axis


Parallel axis theorem example l.jpg
Parallel-Axis Theorem Example Angular Acceleration

  • The axis of rotation goes through O

  • The axis through CM is shown

  • The moment of inertia about the axis through Owould be:

    IO = ICM + MD 2


Example 10 7 moment of inertia for a rod rotating around one end l.jpg
Example 10.7 Angular Acceleration Moment of Inertia for a Rod Rotating Around One End

  • Iof the rod about its CM is

  • D is equal to ½L

  • Therefore:


Torque l.jpg
Torque Angular Acceleration

  • Torque, t, is the tendency of a force to rotate an object about some axis

    • Torque is a vector

    • t = r F sin f = F d

      • F is the force

      • f is the angle the force makes with the horizontal

      • d is the moment arm (or lever arm)


Torque cont l.jpg
Torque, cont Angular Acceleration

  • The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force

    • d = r sin Φ


Torque final l.jpg
Torque, final Angular Acceleration

  • The horizontal component of F (F cos f) has no tendency to produce a rotation

  • Torque will have direction

    • 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


Net torque l.jpg
Net Torque Angular Acceleration

  • The force F1 will tend to cause a counterclockwise rotation about O

  • The force F2 will tend to cause a clockwise rotation about O

  • St = t1 + t2 = F1d1 – F2d2


Torque vs force l.jpg
Torque vs. Force Angular Acceleration

  • Forces can cause a change in linear motion

    • Described by Newton’s Second Law

  • Forces can cause a change in rotational motion

    • The effectiveness of this change depends on the force and the moment arm

    • The change in rotational motion depends on the torque


Torque units l.jpg
Torque Units Angular Acceleration

  • The SI units of torque are N.m

    • Although torque is a force multiplied by a distance, it is very different from work and energy

    • The units for torque are reported in N.m and not changed to Joules


Torque and angular acceleration l.jpg
Torque and Angular Acceleration Angular Acceleration

  • Consider a particle of mass m rotating in a circle of radius r under the influence of tangential force Ft

  • The tangential force provides a tangential acceleration:

    • Ft = mat


Torque and angular acceleration particle cont l.jpg
Torque and Angular Acceleration, Particle cont. Angular Acceleration

  • The magnitude of the torque produced by Ft around the center of the circle is

    • t = Ft r = (mat) r

  • The tangential acceleration is related to the angular acceleration

    • t = (mat) r = (mra) r = (mr 2) a

  • Since mr 2 is the moment of inertia of the particle,

    • t = Ia

    • The torque is directly proportional to the angular acceleration and the constant of proportionality is the moment of inertia


Torque and angular acceleration extended l.jpg
Torque and Angular Acceleration, Extended Angular Acceleration

  • Consider the object consists of an infinite number of mass elements dm of infinitesimal size

  • Each mass element rotates in a circle about the origin, O

  • Each mass element has a tangential acceleration


Torque and angular acceleration extended cont l.jpg
Torque and Angular Acceleration, Extended cont. Angular Acceleration

  • From Newton’s Second Law

    • dFt= (dm) at

  • The torque associated with the force and using the angular acceleration gives

    • dt = r dFt = atr dm = ar 2dm

  • Finding the net torque

    • This becomes St = Ia


Torque and angular acceleration extended final l.jpg
Torque and Angular Acceleration, Extended final Angular Acceleration

  • This is the same relationship that applied to a particle

  • The result also applies when the forces have radial components

    • The line of action of the radial component must pass through the axis of rotation

    • These components will produce zero torque about the axis


Torque and angular acceleration wheel example l.jpg
Torque and Angular Acceleration, Wheel Example Angular Acceleration

  • The wheel is rotating and so we apply St = Ia

    • The tension supplies the tangential force

  • The mass is moving in a straight line, so apply Newton’s Second Law

    • SFy = may = mg - T


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Torque and Angular Acceleration, Multi-body Ex., 1 Angular Acceleration

  • Both masses move in linear directions, so apply Newton’s Second Law

  • Both pulleys rotate, so apply the torque equation


Torque and angular acceleration multi body ex 2 l.jpg
Torque and Angular Acceleration, Multi-body Ex., 2 Angular Acceleration

  • The mg and n forces on each pulley act at the axis of rotation and so supply no torque

  • Apply the appropriate signs for clockwise and counterclockwise rotations in the torque equations


Work in rotational motion l.jpg
Work in Rotational Motion Angular Acceleration

  • Find the work done by F on the object as it rotates through an infinitesimal distance ds = r dq

  • dW = F.d s

    = (F sin f) r dq

    dW = tdq

    The radial component of F

    does no work because it is

    perpendicular to the

    displacement


Power in rotational motion l.jpg
Power in Rotational Motion Angular Acceleration

  • The rate at which work is being done in a time interval dt is

  • This is analogous to P = Fv in a linear system


Work kinetic energy theorem in rotational motion l.jpg
Work-Kinetic Energy Theorem in Rotational Motion Angular Acceleration

  • The work-kinetic energy theorem for rotational motion states that the net work done by external forces in rotating a symmetrical rigid object about a fixed axis equals the change in the object’s rotational kinetic energy


Work kinetic energy theorem general l.jpg
Work-Kinetic Energy Theorem, General Angular Acceleration

  • The rotational form can be combined with the linear form which indicates the net work done by external forces on an object is the change in its total kinetic energy, which is the sum of the translational and rotational kinetic energies


Energy in an atwood machine example l.jpg
Energy in an Atwood Machine, Example Angular Acceleration

  • The blocks undergo changes in translational kinetic energy and gravitational potential energy

  • The pulley undergoes a change in rotational kinetic energy


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Summary of Useful Equations Angular Acceleration


Rolling object l.jpg
Rolling Object Angular Acceleration

  • The red curve shows the path moved by a point on the rim of the object

    • This path is called a cycloid

  • The green line shows the path of the center of mass of the object


Pure rolling motion l.jpg
Pure Rolling Motion Angular Acceleration

  • In pure rolling motion, an object rolls without slipping

  • In such a case, there is a simple relationship between its rotational and translational motions


Rolling object center of mass l.jpg
Rolling Object, Center of Mass Angular Acceleration

  • The velocity of the center of mass is

  • The acceleration of the center of mass is


Rolling object other points l.jpg
Rolling Object, Other Points Angular Acceleration

  • A point on the rim, P, rotates to various positions such as Q and P ’

  • At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs


Rolling motion cont l.jpg
Rolling Motion Cont. Angular Acceleration

  • Rolling motion can be modeled as a combination of pure translational motion and pure rotational motion


Total kinetic energy of a rolling object l.jpg
Total Kinetic Energy of a Rolling Object Angular Acceleration

  • The total kinetic energy of a rolling object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass

    • K = ½ ICMw2 + ½ MvCM2


Total kinetic energy example l.jpg
Total Kinetic Energy, Example Angular Acceleration

  • Accelerated rolling motion is possible only if friction is present between the sphere and the incline

    • The friction produces the net torque required for rotation


Total kinetic energy example cont l.jpg
Total Kinetic Energy, Example cont Angular Acceleration

  • Despite the friction, no loss of mechanical energy occurs because the contact point is at rest relative to the surface at any instant

  • Let U = 0 at the bottom of the plane

  • Kf + U f = Ki + Ui

  • Kf = ½ (ICM / R 2) vCM2 + ½ MvCM2

  • Ui = Mgh

  • Uf = Ki = 0


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