Lecture 24

General Physics (PHY 2130)

http://www.physics.wayne.edu/~apetrov/PHY2130/

•  Solids and fluids   Fluids in motion

•  Oscillations   Simple harmonic motion

Lightning Review

Last lecture: 1.  Solids and fluids

  different states of matter; fluids   density, pressure, etc.

Review Problem: A piece of metal is released under water. The volume of the metal is 50.0 cm3 and its specific gravity is 5.0. What is its initial acceleration? (Note: when v = 0, there is no drag force.)

3

Example: A piece of metal is released under water. The volume of the metal is 50.0 cm3 and its specific gravity is 5.0. What is its initial acceleration? (Note: when v = 0, there is no drag force.)

FBD for the metal mawFF B =−=∑

VgFB waterρ=

The magnitude of the buoyant force equals the weight of the fluid displaced by the metal.

Solve for a: ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=−= 1

ρρ

ρρ

objectobject

water

objectobject

water

VVgg

VVgg

mFa B

Idea: apply Newton’s 2nd Law to the piece of metal:

x

y

w

FB

4

Since the object is completely submerged V=Vobject.

water

gravity specificρρ

=

where ρwater = 1000 kg/m3 is the density of water at 4 °C.

Given 0.5gravity specificwater

object ==ρ

ρ

2

objectobject

water m/s 8.710.5

11..

11ρρ

−=⎟⎠

⎞⎜⎝

⎛ −=⎟⎠

⎞⎜⎝

⎛ −=⎟⎟⎠

⎞⎜⎜⎝

⎛−= g

GSg

VVga

Example continued:

5

Fluid Flow

A moving fluid will exert forces parallel to the surface over which it moves, unlike a static fluid. This gives rise to a viscous force that impedes the forward motion of the fluid.

A steady flow is one where the velocity at a given point in a fluid is constant. Steady flow is laminar; the fluid flows in layers.

V1 = constant

V2 = constant

v1≠v2

An ideal fluid is incompressible, undergoes laminar flow, and has no viscosity.

Fluids in Motion: Streamline Flow • Streamline flow

•  every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier

•  also called laminar flow

• Streamline is the path •  different streamlines cannot cross each other •  the streamline at any point coincides with the direction of fluid

velocity at that point

Fluids in Motion: Turbulent Flow •  The flow becomes irregular

•  exceeds a certain velocity •  any condition that causes abrupt changes in velocity

7

The amount of mass that flows though the cross-sectional area A1 is the same as the mass that flows through cross-sectional area A2.

Equation of Continuity

AvtV=

Δ

Δis called the volume flow rate (units m3/s)

8

is the mass flow rate (units kg/s) Avtm

ρ=Δ

Δ

222111 vAvA ρρ =The continuity equation is

If the fluid is incompressible, then ρ1= ρ2.

9

Example: A garden hose of inner radius 1.0 cm carries water at 2.0 m/s. The nozzle at the end has radius 0.20 cm. How fast does the water move through the constriction?

( ) m/s 50m/s 0.2cm 0.20

cm 0.1 2

122

21

12

12

2211

=⎟⎠

⎞⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

=

vrrv

AAv

vAvA

ππ

Oscillations Simple Harmonic Motion

Recall: Hooke’s Law • Fs = - k x

•  Fs is the spring force •  k is the spring constant

•  It is a measure of the stiffness of the spring •  A large k indicates a stiff spring and a small k indicates a soft spring

•  x is the displacement of the object from its equilibrium position

•  The negative sign indicates that the force is always directed opposite to the displacement

•  The force always acts toward the equilibrium position •  It is called the restoring force

Hooke’s Law Applied to a Spring – Mass System

• When x is positive (to the right), F is negative (to the left)

• When x = 0 (at equilibrium), F is 0

• When x is negative (to the left), F is positive (to the right)

Example: two springs The springs 1 and 2 in Figure have spring constants of 40.0 N/cm and 25.0 N/cm, respectively. The object A remains at rest, and both springs are stretched equally. Determine the stretch.

Motion of the Spring-Mass System • Assume the object is initially pulled to x = A and released from rest

• As the object moves toward the equilibrium position, F and a decrease, but v increases

• At x = 0, F and a are zero, but v is a maximum • The object’s momentum causes it to overshoot the equilibrium position

• The force and acceleration start to increase in the opposite direction and velocity decreases

• The motion continues indefinitely

Simple Harmonic Motion • Motion that occurs when the net force along the direction of motion is a Hooke’s Law type of force •  The force is proportional to the displacement and in the

opposite direction • The motion of a spring mass system is an example of Simple Harmonic Motion

• Not all periodic motion over the same path can be considered Simple Harmonic motion

• To be Simple Harmonic motion, the force needs to obey Hooke’s Law

Amplitude, Period and Frequency • Amplitude, A

•  The amplitude is the maximum position of the object relative to the equilibrium position

•  In the absence of friction, an object in simple harmonic motion will oscillate between ±A on each side of the equilibrium position

•  The period, T, is the time that it takes for the object to complete one complete cycle of motion •  From x = A to x = - A and back to x = A

•  The frequency, ƒ, is the number of complete cycles or vibrations per unit time

Acceleration of an Object in Simple Harmonic Motion

• Newton’s second law will relate force and acceleration

• The force is given by Hooke’s Law

F = - k x = m a or a = -kx / m

• The acceleration is a function of position • Acceleration is not constant and therefore the uniformly

accelerated motion equation cannot be applied

Elastic Potential Energy • The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy

PEs = ½kx2

• The energy is stored only when the spring is stretched or compressed

• Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy

Energy in a Spring Mass System

•  Consider a situation:

1.  A block sliding on a frictionless system collides with a light spring

2.  The block attaches to the spring

Energy Transformations

• The block is moving on a frictionless surface • The total mechanical energy of the system is the kinetic energy of the block

Energy Transformations, 2

•  The spring is partially compressed •  The energy is shared between kinetic energy and elastic

potential energy •  The total mechanical energy is the sum of the kinetic energy

and the elastic potential energy

Energy Transformations, 3

• The spring is now fully compressed • The block momentarily stops • The total mechanical energy is stored as elastic potential energy of the spring

Energy Transformations, 4

• When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block

•  The spring force is conservative and the total energy of the system remains constant

Velocity as a Function of Position • Conservation of Energy allows a calculation of the velocity of the object at any position in its motion

• Speed is a maximum at x = 0 • Speed is zero at x = ±A •  The ± indicates the object can be traveling in either

direction

( )22 xAmkv −±=

25

When a mass-spring system is oriented vertically, it will exhibit SHM with the same period and frequency as a horizontally placed system.

A Vertical Mass and Spring

At equilibrium position (b)

0=−+=∑ mgkdFy

)(, ydkF yspring −=

kymgkykdmgydkFy −=−−=−−=∑ )(

At (c), displaced the equilibrium by y

kyFy −=∑

Example: oscillator A block of mass 1.00 kg is attached to a spring with a spring constant of 30.0 N/m, which is stretched 0.200 m from its equilibrium position. How much work must be done to stretch it an additional 0.100 m? What maximum speed will the block attain if the system is then let go?

Simple Harmonic Motion and Uniform Circular Motion • A ball is attached to the rim of a turntable of radius A

• The focus is on the shadow that the ball casts on the screen

• When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion

Period and Frequency from Circular Motion

• Period

•  This gives the time required for an object of mass m attached to a spring of constant k to complete one cycle of its motion

• Frequency

• Units are cycles/second or Hertz, Hz

• The angular frequency is related to the frequency

km2T π=

mk

21

T1ƒ

π==

mkƒ2 =π=ω

Motion as a Function of Time

• Use of a reference circle allows a description of the motion

• x = A cos (2πƒt) •  x is the position at time t •  x varies between +A and -A

Graphical Representation of Motion

• When x is a maximum or minimum, velocity is zero

• When x is zero, the velocity is a maximum

• When x is a maximum in the positive direction, a is a maximum in the negative direction

Verification of Sinusoidal Nature

• This experiment shows the sinusoidal nature of simple harmonic motion

• The spring mass system oscillates in simple harmonic motion

• The attached pen traces out the sinusoidal motion

32

Example: The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point?

Idea: let’s use energy conservation: at equilibrium x = 0:

222

21

21

21 mvkxmvUKE =+=+=

Since E=constant, at equilibrium (x = 0) the KE must be a maximum.

Thus, v = vmax = Aω.

( )( ) cm/sec 8.62rads/sec 6.12cm 5.0 and

rads/sec 6.12s 50.0

22

===

===

Aωv

Tππ

ω