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№1 слайд![Physics Voronkov Vladimir](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img0.jpg)
Содержание слайда: Physics 1
Voronkov Vladimir Vasilyevich
№2 слайд![Lecture Rotation of rigid](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img1.jpg)
Содержание слайда: Lecture 4
Rotation of rigid bodies.
Angular momentum and torque.
Properties of fluids.
№3 слайд![Rotation of Rigid Bodies in](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img2.jpg)
Содержание слайда: Rotation of Rigid Bodies in General case
When a rigid object is rotating about a fixed axis, every particle of the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration. So the rotational motion of the entire rigid object as well as individual particles in the object can be described by three angles. Using these three angles we can greatly simplify the analysis of rigid-object rotation.
№4 слайд![Radians Angle in radians](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img3.jpg)
Содержание слайда: Radians
Angle in radians equals the ratio of the arc length s and the radius r:
№5 слайд![Angular kinematics Angular](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img4.jpg)
Содержание слайда: Angular kinematics
Angular displacement:
Instantaneous angular speed:
Instantaneous angular acceleration:
№6 слайд![Angular and linear quantities](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img5.jpg)
Содержание слайда: Angular and linear quantities
Every particle of the object moves in a circle whose center is the axis of rotation.
Linear velocity:
Tangential acceleration:
Centripetal acceleration:
№7 слайд![Total linear acceleration](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img6.jpg)
Содержание слайда: Total linear acceleration
Tangential acceleration is perpendicular to the centripetal one, so the magnitude of total linear acceleration is
№8 слайд![Angular velocity Angular](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img7.jpg)
Содержание слайда: Angular velocity
Angular velocity is a vector.
№9 слайд![Rotational Kinetic Energy](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img8.jpg)
Содержание слайда: Rotational Kinetic Energy
№10 слайд![Calculations of Moments of](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img9.jpg)
Содержание слайда: Calculations of Moments of Inertia
№11 слайд![Uniform Thin Hoop](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img10.jpg)
Содержание слайда: Uniform Thin Hoop
№12 слайд![Uniform Rigid Rod](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img11.jpg)
Содержание слайда: Uniform Rigid Rod
№13 слайд![Uniform Solid Cylinder](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img12.jpg)
Содержание слайда: Uniform Solid Cylinder
№14 слайд![Moments of Inertia of](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img13.jpg)
Содержание слайда: Moments of Inertia of Homogeneous Rigid Objects
with Different Geometries
№15 слайд![](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img14.jpg)
№16 слайд![Parallel-axis theorem Suppose](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img15.jpg)
Содержание слайда: Parallel-axis theorem
Suppose the moment of inertia about an axis through the center of mass of an object is ICM. Then the moment of inertia about any axis parallel to and a distance D away from this axis is
№17 слайд![](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img16.jpg)
№18 слайд![Torque When a force is](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img17.jpg)
Содержание слайда: Torque
When a force is exerted on a rigid object pivoted about an axis, the object tends to rotate about that axis. The tendency of a force to rotate an object about some axis is measured by a vector quantity called torque (Greek tau).
№19 слайд![The force F has a greater](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img18.jpg)
Содержание слайда: The force F has a greater rotating tendency about axis O as F increases and as the moment arm d increases. The component F sin tends to rotate the wrench about axis O.
№20 слайд![We use the convention that](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img19.jpg)
Содержание слайда: We use the convention that the sign of the torque resulting from a force is positive if the turning tendency of the force is counterclockwise and is negative if the turning tendency is clockwise. Then
№21 слайд![Torque is not Force Torque is](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img20.jpg)
Содержание слайда: Torque is not Force
Torque is not Work
Torque should not be confused with force. Forces can cause a change in linear motion, as described by Newton’s second law. Forces can also cause a change in rotational motion, but the effectiveness of the forces in causing this change depends on both the forces and the moment arms of the forces, in the combination that we call torque. Torque has units of force times length: newton · meters in SI units, and should be reported in these units.
Do not confuse torque and work, which have the same units but are very different concepts.
№22 слайд![Rotational Dynamics Let s add](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img21.jpg)
Содержание слайда: Rotational Dynamics
Let’s add which equals zero, as
and are parallel.
Then: So we get
№23 слайд![Rotational analogue of Newton](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img22.jpg)
Содержание слайда: Rotational analogue of Newton’s second law
Quantity L is an instantaneous angular momentum.
The torque acting on a particle is equal to the time rate of change of the particle’s angular momentum.
№24 слайд![Net External Torque The net](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img23.jpg)
Содержание слайда: Net External Torque
The net external torque acting on a system about some axis passing through an origin in an inertial frame equals the time rate of change of the total angular momentum of the system about that origin:
№25 слайд![Angular Momentum of a](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img24.jpg)
Содержание слайда: Angular Momentum of a Rotating Rigid Object
Angular momentum for each particle of an object:
Angular momentum for the whole object:
Thus:
№26 слайд![Angular acceleration](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img25.jpg)
Содержание слайда: Angular acceleration
№27 слайд![The Law of Angular Momentum](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img26.jpg)
Содержание слайда: The Law of Angular Momentum Conservation
The total angular momentum of a system is constant if the resultant external torque acting on the system is zero, that is, if the system is isolated.
№28 слайд![Change in internal structure](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img27.jpg)
Содержание слайда: Change in internal structure of a rotating body can result in change of its angular velocity.
№29 слайд![When a rotating skater pulls](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img28.jpg)
Содержание слайда: When a rotating skater pulls his hands towards his body he spins faster.
№30 слайд![Three Laws of Conservation](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img29.jpg)
Содержание слайда: Three Laws of Conservation for an Isolated System
Full mechanical energy, linear momentum and angular momentum of an isolated system remain constant.
№31 слайд![Work-Kinetic Theory for](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img30.jpg)
Содержание слайда: Work-Kinetic Theory for Rotations
Similarly to linear motion:
№32 слайд![The net work done by external](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img31.jpg)
Содержание слайда: The net work done by external forces in rotating a symmetric rigid object about a fixed axis equals the change in the object’s rotational energy.
№33 слайд![Equations for Rotational and](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img32.jpg)
Содержание слайда: Equations for Rotational and Linear Motions
№34 слайд![Independent Study for IHW](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img33.jpg)
Содержание слайда: Independent Study for IHW2
Vector multiplication (through their components i,j,k).Right-hand rule of Vector multiplication.
Elasticity
Demonstrate by example and discussion your understanding of elasticity, elastic limit, stress, strain, and ultimate strength.
Write and apply formulas for calculating Young’s modulus, shear modulus, and bulk modulus. Units of stress.
№35 слайд![Fluids Fluids Define absolute](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img34.jpg)
Содержание слайда: Fluids
Fluids
Define absolute pressure, gauge pressure, and atmospheric pressure, and demonstrate by examples your understanding of the relationships between these terms.
Pascal’s law.
Archimedes’s law.
Rate of flow of a fluid.
Bernoulli’s equation.
Torricelli’s theorem.
№36 слайд![Literature to Independent](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img35.jpg)
Содержание слайда: Literature to Independent Study
Lecture on Physics Summary by Umarov. (Intranet)
Fishbane Physics for Scientists… (Intranet)
Serway Physics for Scientists… (Intranet)
№37 слайд![Problems A solid sphere and a](/documents_6/29a7d57069a3a0b8df02ca325d63e75f/img36.jpg)
Содержание слайда: Problems
A solid sphere and a hollow sphere have the same mass and radius. Which momentum of rotational inertia is higher if it is? Prove your answer with formulae.
What are the units for, are these quantities vectors or scalars:
Angular momentum
Angular kinetic energy
Angular displacement
Tangential acceleration
Angular acceleration
Torque