Презентация Rotation of rigid bodies. Angular momentum and torque. Properties of fluids онлайн
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Слайды и текст к этой презентации:
№1 слайд
Содержание слайда: Physics 1
Voronkov Vladimir Vasilyevich
№2 слайд
Содержание слайда: Lecture 4
Rotation of rigid bodies.
Angular momentum and torque.
Properties of fluids.
№3 слайд
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
Содержание слайда: Radians
Angle in radians equals the ratio of the arc length s and the radius r:
№5 слайд
Содержание слайда: Angular kinematics
Angular displacement:
Instantaneous angular speed:
Instantaneous angular acceleration:
№6 слайд
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
Содержание слайда: Total linear acceleration
Tangential acceleration is perpendicular to the centripetal one, so the magnitude of total linear acceleration is
№8 слайд
Содержание слайда: Angular velocity
Angular velocity is a vector.
№9 слайд
Содержание слайда: Rotational Kinetic Energy
№10 слайд
Содержание слайда: Calculations of Moments of Inertia
№11 слайд
Содержание слайда: Uniform Thin Hoop
№12 слайд
Содержание слайда: Uniform Rigid Rod
№13 слайд
Содержание слайда: Uniform Solid Cylinder
№14 слайд
Содержание слайда: Moments of Inertia of Homogeneous Rigid Objects
with Different Geometries
№15 слайд
Содержание слайда:
№16 слайд
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
Содержание слайда:
№18 слайд
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
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
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
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
Содержание слайда: Rotational Dynamics
Let’s add which equals zero, as
and are parallel.
Then: So we get
№23 слайд
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
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
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
Содержание слайда: Angular acceleration
№27 слайд
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
Содержание слайда: Change in internal structure of a rotating body can result in change of its angular velocity.
№29 слайд
Содержание слайда: When a rotating skater pulls his hands towards his body he spins faster.
№30 слайд
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
Содержание слайда: Work-Kinetic Theory for Rotations
Similarly to linear motion:
№32 слайд
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
Содержание слайда: Equations for Rotational and Linear Motions
№34 слайд
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
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
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
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
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