Physics 1 for KMA

Август 2, 2022

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Physics 1 for KMA

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2. Lecture 4 Rotation of rigid bodies. Angular momentum and torque. Properties of fluids. Flotation. Bernulli equation.
3. Rotation of Rigid Bodies When a rigid object is rotating about a fixed axis, every particle
4. Radians
5. Angular kinematics Angular displacement: Instantaneous angular speed: Instantaneous angular acceleration:
6. Angular and linear quantities Every particle of the object moves in a circle whose center is
7. Total linear acceleration Tangential acceleration is perpendicular to the centripetal one, so the magnitude of total
8. Angular velocity Angular velocity is a vector. The right hand rule is applied: If the fingers
9. Rotational Kinetic Energy Moment of rotational inertia Rotational kinetic energy
10. Calculations of Moments of Inertia
11. Uniform Thin Hoop
12. Uniform Rigid Rod
13. Uniform Solid Cylinder
14. Moments of Inertia of Homogeneous Rigid Objects with Different Geometries
16. Parallel-axis theorem Suppose the moment of inertia about an axis through the center of mass of
18. Torque When a force is exerted on a rigid object pivoted about an axis, the object
19. The force F has a greater rotating tendency about axis O as F increases and as
20. The force F1 tends to rotate the object counterclockwise about O, and F2 tends to rotate
21. Torque is not Force Torque is not Work Torque should not be confused with force. Forces
22. Rotational Dynamics Let’s add which equals zero, as and are parallel. Then: So we get
23. Rotational analogue of Newton’s second law Quantity L is an instantaneous angular momentum. The torque acting
24. Net External Torque The net external torque acting on a system about some axis passing through
25. Angular Momentum of a Rotating Rigid Object Angular momentum for each particle of an object: Angular
26. Angular acceleration
27. The Law of Angular Momentum Conservation The total angular momentum of a system is constant if
28. Change in internal structure of a rotating body can result in change of its angular velocity.
29. When a rotating skater pulls his hands towards his body he spins faster.
30. Three Laws of Conservation for an Isolated System Full mechanical energy, linear momentum and angular momentum
31. Work-Kinetic Theory for Rotations Similarly to linear motion:
32. The net work done by external forces in rotating a symmetric rigid object about a fixed
33. Equations for Rotational and Linear Motions
34. Gyroscope One typical type of gyroscope is made by suspending a relatively massive rotor inside three
35. At high speeds, the gyroscope exhibits extraordinary stability of balance and maintains the direction of the
36. If a gyroscope is tipped, the gimbals will try to reorient to keep the spin axis
37. Precession of Spinning Wheel
38. Fluids and liquids
39. Relative density Relative density or specific gravity is the ratio of the density of a substance
41. Specific volume of a substance is the ratio of the substance's volume to its mass. It
42. Pressure
43. Manometer The difference in fluid height in a liquid column manometer is proportional to the pressure
44. Static Fluid Pressure Pstatic fluid = ρgh where ρ = m/V = fluid density g =
45. Pressure Thrust Thrust is a total force in a particular direction. The unit of thrust, therefore
46. Atmospheric Pressure The surface of the earth is at the bottom of an atmospheric sea. The
47. Atmospheric constituents
48. Barometer
49. Aneroid barometer An aneroid barometeru ses a small, flexible metal box called an aneroid cell (capsule),
51. The Barometric Formula μair=28.9644 g/mol mair= μair/Na
52. Pascal's Principle Pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions
54. Hydraulic Press
55. Lift pump The lift pump, also known as a suction pump, operates as follows: on the
56. Force pump The force pump, also known as a pressure pump, operates as follows: on the
57. Rotary Pumps Rotary vane pump Scroll pump
58. Height limitation Total Dynamic Head (TDH) is the total equivalent height that a fluid is to
59. Total Dynamic Height TDH = Static Height + Static Lift + Friction Loss Static Height is
60. Viscosity The resistance to flow of a fluid and the resistance to the movement of an
61. Experimentally, under conditions of laminar flow, the force required to move a plate at constant speed
63. Drag force due viscosity In a viscous fluid, a boundary layer is formed. This causes a
64. Effect of Temperature on Viscosity The temperature dependence of liquid viscosity is the phenomenon by which
65. Liquid Damping Damping is an effect that reduces the amplitude of oscillations in an oscillatory system
68. Buoyancy Buoyancy arises from the fact that fluid pressure increases with depth and from the fact
69. Archimedes' Principle The buoyant force on a submerged object is equal to the weight of the
70. Hydrometer A hydrometer is an instrument used to measure the specific gravity (or relative density) of
72. Determine, what liquid is denser?
73. This liquid is lighter. This liquid is denser. This liquid is lighter.
74. Fluid Kinetic Energy The kinetic energy of a moving fluid is more useful in applications like
75. Fluid Potential Energy The potential energy of a moving fluid is more useful in applications like
76. Bernoulli Equation
77. Venturi meter The Venturi effect is the reduction in fluid pressure that results when a fluid
78. Venturi effect Q is volumetric flow rate So Venturi meter can be used to measure the
79. Water Eductor Liquid Jet Eductors use the kinetic energy of a motive liquid to entrain another
82. Скачать презентацию

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Lecture 4
Rotation of rigid bodies.
Angular momentum and torque.
Properties of fluids.
Flotation.
Bernulli

equation.

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Rotation of Rigid Bodies
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.

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Radians

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Angular kinematics
Angular displacement:
Instantaneous angular speed:
Instantaneous angular acceleration:

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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:

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Total linear acceleration
Tangential acceleration is perpendicular to the centripetal one, so

the magnitude of total linear acceleration is

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Angular velocity
Angular velocity is a vector.
The right hand rule

is applied: If the fingers of your right hand curl along with the rotation your thumb will give the direction of the angular velocity.

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Rotational Kinetic Energy

Moment of rotational inertia
Rotational kinetic energy

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Calculations of Moments of Inertia

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Uniform Thin Hoop

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Uniform Rigid Rod

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Uniform Solid Cylinder

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Moments of Inertia of Homogeneous Rigid Objects with Different Geometries

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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

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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 τ (Greek tau).

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The force F has a greater rotating tendency about axis O

as F increases and as the moment arm d increases. The component F sinφ tends to rotate the wrench about axis O.

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The force F1 tends to rotate the object counterclockwise about O,

and F2 tends to rotate it clockwise.

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

The force F1 tends to rotate the object counterclockwise about O, and F2 tends to rotate it clockwise.

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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.

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Rotational Dynamics
Let’s add which equals zero, as
and are parallel.

Then: So we get

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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.

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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:

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Angular Momentum of a Rotating Rigid Object
Angular momentum for each particle

of an object:
Angular momentum for the whole object:
Thus:

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Angular acceleration

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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.

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Change in internal structure of a rotating body can result in

change of its angular velocity.

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When a rotating skater pulls his hands towards his body he

spins faster.

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Three Laws of Conservation for an Isolated System
Full mechanical energy, linear

momentum and angular momentum of an isolated system remain constant.

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Work-Kinetic Theory for Rotations
Similarly to linear motion:

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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.

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Equations for Rotational and Linear Motions

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Gyroscope
One typical type of gyroscope is made by suspending a relatively

massive rotor inside three rings called gimbals. Mounting each of these rotors on high quality bearing surfaces insures that very little torque can be exerted on the inside rotor.

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At high speeds, the gyroscope exhibits extraordinary stability of balance and

maintains the direction of the high speed rotation axis of its central rotor. The implication of the conservation of angular momentum is that the angular momentum of the rotor maintains not only its magnitude, but also its direction in space in the absence of external torque. The classic type gyroscope finds application in gyro-compasses.

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If a gyroscope is tipped, the gimbals will try to reorient

to keep the spin axis of the rotor in the same direction. If released in this orientation, the gyroscope will precess in the direction shown because of the torque exerted by gravity on the gyroscope.

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Precession of Spinning Wheel

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Fluids and liquids

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Relative density
Relative density or specific gravity is the ratio of the

density of a substance to the density of a given reference material. Specific gravity usually means relative density with respect to water.
If the reference material is water then a substance with a relative density (or specific gravity) less than 1 will float in water. For example, an ice cube, with a relative density of about 0.91, will float. A substance with a relative density greater than 1 will sink.

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Specific volume of a substance is the ratio of the substance's

volume to its mass. It is the reciprocal of density and is an intrinsic property of matter:

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Pressure

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Manometer
The difference in fluid height in a liquid column manometer is

proportional to the pressure difference.
P1-P2=ρgh

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Static Fluid Pressure
Pstatic fluid = ρgh where ρ = m/V = fluid

density
g = gravitational acceleration
h = depth of fluid
The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the acceleration of gravity

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Pressure Thrust
Thrust is a total force in a particular direction. The

unit of thrust, therefore is the same as that of force: Newtons (N). Pressure is the force or thrust applied per unit area.
F=P·A

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Atmospheric Pressure
The surface of the earth is at the bottom of

an atmospheric sea. The standard atmospheric pressure is measured in various units:
1 atmosphere = 760 mmHg = 101.3 KPa
The bar is a unit of pressure defined as 100 kilopascals. It is about equal to the atmospheric pressure on Earth at sea level.
The unit mmHg is often called torr, particularly in vacuum applications: 760 mmHg = 760 torr

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Atmospheric constituents

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Barometer

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Aneroid barometer
An aneroid barometeru ses a small, flexible metal box called

an aneroid cell (capsule), which is made from an alloy of beryllium and copper. The evacuated capsule (or usually more capsules) is prevented from collapsing by a strong spring. Small changes in external air pressure cause the cell to expand or contract. This expansion and contraction drives mechanical levers such that the tiny movements of the capsule are amplified and displayed on the face of the aneroid barometer.

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The Barometric Formula
μair=28.9644 g/mol
mair= μair/Na

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Pascal's Principle
Pressure exerted anywhere in a confined incompressible fluid is transmitted

equally in all directions throughout the fluid such that the pressure ratio (initial difference) remains the same.

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Hydraulic Press

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Lift pump
The lift pump, also known as a suction pump, operates

as follows:
on the upstroke of the plunger, the lower valve opens, the upper valve (situated on or in the plunger itself) is closed, and the low air pressure produced in the barrel allows atmospheric pressure on the surface of the water source, down below, to make the water move up the downpipe and eventually fill the barrel below the plunger.
On the downstroke, the lower valve closes, the upper one opens, and water is forced into the barrel above the upper valve. On the next upstroke, the water above the plunger is forced out of the spout, located at the top of the barrel, at the same time as the volume below the barrel fills up with water again.

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Force pump
The force pump, also known as a pressure pump, operates

as follows:
on the upstroke of the plunger, the outlet or delivery valve is closed and the inlet valve opens. The low air pressure produced in the barrel causes the water below to move up the downpipe and eventually fill the barrel.
On the downstroke, the inlet valve closes, the outlet valve opens, and the water is forced out via the outlet pipe, which is located at the bottom of the barrel.

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Rotary Pumps
Rotary vane pump
Scroll pump

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Height limitation
Total Dynamic Head (TDH) is the total equivalent height that

a fluid is to be pumped, taking into account friction losses in the pipe.

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Total Dynamic Height
TDH = Static Height + Static Lift + Friction

Loss
Static Height is the maximum height reached by the pipe after the pump (also known as the 'discharge head').
Static Lift is the height the water will rise before arriving at the pump (also known as the suction head).
Friction Loss - in any real moving fluid, energy is dissipated due to friction; turbulence dissipates even more energy for high Reynolds number flows. Friction loss is divided into two main categories, "major losses" associated with energy loss per length of pipe, and "minor losses" associated with bends, fittings, valves, etc.

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Viscosity
The resistance to flow of a fluid and the resistance to

the movement of an object through a fluid are usually stated in terms of the viscosity of the fluid.

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Experimentally, under conditions of laminar flow, the force required to move

a plate at constant speed against the resistance of a fluid is proportional to the area of the plate and to the velocity gradient perpendicular to the plate. The constant of proportionality is called the viscosity .

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Drag force due viscosity
In a viscous fluid, a boundary layer is

formed. This causes a net drag due to skin friction. Further, because the ideal pressure now acts on the boundary layer, as opposed to the ship, and the boundary layer grows along the length of the ship, the net opposing forces are greater than the net supporting forces. This further adds to the resistance.

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Effect of Temperature on Viscosity
The temperature dependence of liquid viscosity

is the phenomenon by which liquid viscosity tends to decrease (or, alternatively, its fluidity tends to increase) as its temperature increases.

here η0 and b are constants.
This is an empirical model that usually works for a limited range of temperatures.

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Liquid Damping
Damping is an effect that reduces the amplitude of oscillations

in an oscillatory system
Fluid viscous damping is a way to add energy dissipation to the lateral system of a building structure. A fluid viscous damper dissipates energy by pushing fluid through an orifice, producing a damping pressure which creates a force. These damping forces are 90 degrees out of phase with the displacement driven forces in the structure. This means that the damping force does not significantly increase the seismic loads for a comparable degree of structural deformation.

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Buoyancy
Buoyancy arises from the fact that fluid pressure increases with depth

and from the fact that the increased pressure is exerted in all directions (Pascal's principle) so that there is an unbalanced upward force on the bottom of a submerged object.

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Archimedes' Principle
The buoyant force on a submerged object is equal to

the weight of the fluid displaced.
The upward thrust which the surrounding fluid exerts on an object is referred to as the force of buoyancy.

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Hydrometer
A hydrometer is an instrument used to measure the specific

gravity (or relative density) of liquids; that is, the ratio of the density of the liquid to the density of water.
A hydrometer is usually made of glass and consists of a cylindrical stem and a bulb weighted with mercury or lead shot to make it float upright. The liquid to be tested is poured into a tall container, often a graduated cylinder, and the hydrometer is gently lowered into the liquid until it floats freely. The point at which the surface of the liquid touches the stem of the hydrometer is noted. Hydrometers usually contain a scale inside the stem, so that the specific gravity can be read directly. A variety of scales exist, and are used depending on the context.
Hydrometers may be calibrated for different uses, such as a lactometer for measuring the density (creaminess) of milk, a saccharometer for measuring the density of sugar in a liquid, or an alcoholometer for measuring higher levels of alcohol in spirits.

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Determine, what liquid is denser?

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This liquid is lighter.
This liquid is denser.
This liquid is lighter.

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Fluid Kinetic Energy
The kinetic energy of a moving fluid is more

useful in applications like the Bernoulli equation when it is expressed as kinetic energy per unit volume

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Fluid Potential Energy
The potential energy of a moving fluid is more

useful in applications like the Bernoulli equation when is expressed as potential energy per unit volume

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Bernoulli Equation

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Venturi meter
The Venturi effect is the reduction in fluid pressure that

results when a fluid flows through a constricted section of pipe.
The Venturi effect is named after Giovanni Battista Venturi (1746–1822), an Italian physicist.

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Venturi effect
Q is volumetric flow rate
So Venturi meter can be used

to measure the flow rate.

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Water Eductor
Liquid Jet Eductors use the kinetic energy of a motive

liquid to entrain another liquid, completely mix the two, and then discharge the mixture against a counter pressure and are used for pumping and mixing operations.

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