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Reference: Beginning Physics I

CHAPTER 13: FLUIDS AT REST (HYDROSTATICS)

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KEY WORD LIST

Hydrostatics, Density, Specific Gravity, Pressure, Hydrostatic Pressure, Gauge Pressure, Hydraulic Press, Open-Tube Manometer, Barometer, Archimedes’ Principle, Surface Tension, Capillarity

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GLOSSARY

For details on the following concepts, please consult CHAPTER 13.

HYDROSTATICS
Origin: “still water.” Hydrostatic is that branch of physics that deals with the static fluids, usually confined to the equilibrium and pressure of liquids.

DENSITY
The density d of any substance is defined as the mass per unit volume of the substance. If we have a uniform sample of materials (solid, liquid, or gas) of mass M and volume V, then

SPECIFIC GRAVITY
The specific gravity of a substance is defined as the ratio of the density of the substance to that of water.

PRESSURE
The pressure P on any surface is defined as the force per unit area acting perpendicular to that surface

HYDROSTATIC PRESSURE
The hydrostatic pressure is the pressure in a fluid at rest.

• For any point in a fluid at rest, the pressure on one side of a small surface is the same as the pressure on other side.
• The pressure at a given point in the fluid at rest has a definite value that represents the force per unit area on a small surface placed at that point, oriented in any arbitrary direction.
• The pressure in a fluid at rest is the same at all points on a horizontal plane.
• The pressure in a fluid at rest varies only with the depth in accord with the equation,

GAUGE PRESSURE
The gauge pressure is the difference between the actual, or absolute pressure P in a fluid, and the pressure exerted by the atmosphere ­P_A, which pervades the surface of the earth.

HYDRAULIC PRESS

OPEN-TUBE MANOMETER

BAROMETER

ARCHIMEDES’ PRINCIPLE
The fact that the buoyant force equals the weight of the displaced liquid is called the Archimedes’ Principle: the law of buoyancy.

SURFACE TENSION
Surface tension is caused by the pull of the molecules below the surface of the liquid on the molecules at the surface. This tends to pull the surface into a smooth and compact layer. The surface tension  is defined as force per unit length exerted by a liquid surface on an object, along its boundary of contact with the object. This force is parallel to the liquid surface and perpendicular to the boundary line of contact. For a straight boundary of length L and a total force F we have

CAPILLARITY
Because of adhesion, the water surface gets pulled toward the wall of the container and bends upward at the point of contact.

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Reference: Beginning Physics I

CHAPTER 12: SIMPLE HARMONIC MOTION (SHM)

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KEY WORD LIST

Simple Harmonic Motion, Periodic Motion, Period, Frequency, Reference Circle, Time Interval, SHM Time Equations, Spring Motion, Torsional Motion, Simple Pendulum

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GLOSSARY

For details on the following concepts, please consult CHAPTER 12.

SIMPLE HARMONIC MOTION (SHM)
It is the back-and-forth type vibratory motion of an object that is subjected to Hooke’s Law type force, which is a restoring force that is proportional to the displacement from the equilibrium position (F = -kx). Since the force is varying, the acceleration of the object, and its velocity is also varying.

The uniformity of circular motion and rotation does not occur in SHM.

PERIODIC MOTION
Periodic motion is motion repeated in equal intervals of time.

PERIOD (T)
The period (T) is defined as the time to make one complete repetition of the motion. Thus, T is the time interval from when the object traverses any position x moving in a given direction to the next time the object traverses position x moving in the same direction.

FREQUENCY (f)
The frequency f of the periodic motion is the number of repetitions per second. It is the reciprocal of the period: f = 1/T.

REFERENCE CIRCLE
Consider a particle undergoing uniform circular motion on a circle about the origin of a coordinate system.

As the particle moves around the circle, its shadow moves back and forth along the x-axis. Finding the time equations for the shadow’s motion is equivalent to finding the time equations for SHM. For this reason, this circle is called the reference circle for SHM.

TIME INTERVAL

SHM TIME EQUATIONS

In general,

SPRING MOTION

TORSIONAL MOTION

SIMPLE PENDULUM

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Reference: Beginning Physics I

CHAPTER 10: RIGID BODIES II: ROTATIONAL MOTION 

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KEY WORD LIST

Rotational Motion, Angular Displacement, Angular Velocity, Angular Acceleration, Period, Frequency, Torque, Moment of Inertia, Linear and Angular Relationships, Table of Analogs, Conservation of Angular Momentum, CM Frame

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GLOSSARY

For details on the following concepts, please consult CHAPTER 10.

ROTATIONAL MOTION
In the following sketch, a body in x-y plane is rotating around the z-axis. The orientation of the rigid body can be completely specified by giving the orientation angle  of a single chosen line segment etched in the body.

ANGULAR DISPLACEMENT
The angle  is called the angular displacement of the rigid body. By convention, the angle  is considered positive when it is measured counterclockwise from the x-axis.

ANGULAR VELOCITY
To get an idea of how fast the body is rotating, we define the average angular velocity in a given time interval as follows:

The instantaneous angular velocity is defined as the limit of average angular velocity as follows:

The angular velocity is positive for counterclockwise rotation. For constant angular velocity, we have

ANGULAR ACCELERATION
The average angular acceleration is the rate of change of the angular velocity.

The instantaneous angular acceleration is,

Thus, we have for constant acceleration,

PERIOD
The time to make one complete revolution is called the period of the motion. For constant angular velocity, the period stays the same from one revolution to the next.

FREQUENCY
The frequency is the number of revolutions per second.

TORQUE
We consider the axis of rotation fixed in the z-direction. Then the torque is along the z-axis, and the forces causing this torques and their displacements lie in the x-y plane. All the internal torques in a rigid body add up to zero. Thus, the only torque left is due to external forces,

MOMENT OF INERTIA
We define the moment of inertia of a body about the z-axis as,

LINEAR AND ANGULAR RELATIONSHIPS
At any instant, the angular and linear properties are related as follows:

DISPLACEMENT:                   s = R             and              s = R

VELOCITY:                              v = R            and              v = R

ACCELERATION:                    a_t = R          and              a_r = ² R

TABLE OF ANALOGS
Work done in rotation a rigid body, Kinetic energy in rotation, Work-energy theorem applied to a rotating object, the power of rotation, angular impulse, and angular momentum are all rotation analogs of the definitions for linear motion.

CONSERVATION OF ANGULAR MOMENTUM
If the resultant external vector torque (about the origin) for a system of particles is zero, then the vector sum of the angular momenta of all the particles stays constant in time.

For the special case of objects rotating about a fixed axis: If the total external torque about the axis is zero, then the total component of angular momentum along that axis does not change.

CM FRAME
The CM Frame is a coordinate system whose origin is fixed at the CM (Center of Mass) of the object. The CM Frame moves with the object, but its axes remain parallel to the axes of a coordinate system fixed in an inertial frame.

The translation of the object is the same as the translation of the CM. The rotation of the object is about an axis that passes through the CM. If the direction of this axis of rotation remains fixed, then all the laws of rotation hold.

The total kinetic Energy of an object in the inertial frame is given by,

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