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PHY131H1F Summer – Class 11 Today: •  Hanging Springs •  The Pendulum •  Damped Oscillations;

Shock Absorbers •  Driven Oscillations;

Resonance •  Fluids •  Pressure •  Pascal’s Law •  Gauge Pressure

Italian opera singer Luigi Infantino tries to break a wine glass by singing top 'C' at a rehearsal.

What term is used to describe an oscillator that “runs down” and eventually stops?

A.  Tired oscillator B.  Out of shape oscillator C.  Damped oscillator D.  Resonant oscillator E.  Driven oscillator

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Simple Harmonic Motion (SHM)

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End of chapter problem P14.17

•  A spring is hung from the ceiling. When a block is attached to its end, it stretches 2.0 cm before reaching its new equilibrium length. The block is then pulled down slightly and released. What is the frequency of oscillation?

The Pendulum Suppose we restrict the pendulum’s oscillations to small angles (< 10°). Then we may use the small angle approximation sin θ ≈ θ, where θ is measured in radians. Since θ = s/L, the net force on the mass is

and the angular frequency of the motion is found to be

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Two pendula have the same length, but different mass. The force of gravity, F=mg, is larger for the larger mass. Which will have the longer period?

A. the larger mass B. the smaller mass C. neither

Mass on Spring versus Pendulum

Mass on a Spring

Pendulum

Condition for S.H.M.

Small oscillations

Small angles

Angular frequency Period

A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, two people sit on the swing, the natural frequency of the swing is

A. greater B. the same C. smaller

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A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, the person stands on the swing, the natural frequency of the swing is

A. greater B. the same C. smaller

Damped Oscillations When a mass on a spring experiences the force of the spring as given by Hooke’s Law, as well as a drag force of magnitude , the solution is

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Driven Oscillations and Resonance

•  Consider an oscillating system that, when left to itself, oscillates at a frequency f0. We call this

•  Suppose that this system is subjected This frequency is

•  The amplitude of oscillations is generally not very high if fext differs much from f0.

•  As fext gets closer and closer to f0, the

14.8 Externally Driven Oscillations

Chapter 15. Definition: Density

The ratio of a fluid’s or object’s mass to its volume is called the mass density, or sometimes simply “the density.”

The SI units of mass density are

The density of water is

Your body is composed of about 60% water.

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Pressure is due to the net force of the molecules in a fluid colliding with the walls.

Each collision exerts

Definition: Pressure A fluid in a container presses with an outward force against the walls of that container. The pressure is defined as the ratio of the force to the area on which the force is exerted.

The SI units of pressure are N/m2, also defined as the pascal, where 1 pascal = Other units:

1 atm = 1.01×105 Pa 1 mmHg = 133 Pa 1 kPa = 103 Pa 1 psi = 6890 Pa

Atmospheric Pressure

The global average sea-level pressure is 1.01×105 Pa, or 1 atm.

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Gauge Pressure Pressure gauges, such as tire gauges and blood pressure monitors, measure not the actual or absolute pressure p but what is called gauge pressure pg.

where 1 atm = 1.01×105 Pa.

•  ie “120 over 80” means the maximum in your arteries is 120 mmHg or 1.6×104 Pa.

•  The actual, or “absolute” pressure in your arteries has a maximum of p = pg + 1atm = 1.6×104 + 1.01×105 Pa = Pa

Is gauge pressure larger, smaller, or equal to true pressure?

A.  Larger B.  Smaller C.  equal to

Fluids

•  Fluids include both Liquids and Gases: what’s the difference?

•  Gas: Pressure and Volume are related by the ideal gas law:

At constant temperature, if the Pressure of a gas is increased, its

•  Liquid: Pressure “Incompressible”

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Pascal’s Law for liquids

Pascal’s Law for liquids •  Consider a small element of fluid in a beaker. •  Pressure acts

•  Gravity pulls it •  To balance the force of gravity, the upward

pressure on the bottom surface must be

“buoyancy”

•  This is the equation for the pressure of an incompressible fluid in hydrodynamic equilibrium in a gravitational field.

•  Pressure increases with depth! Scuba divers know this!

Water is slowly poured into the container until the water level has risen into tubes A, B, and C. The water doesn’t overflow from any of the tubes. How do the water depths in the three columns compare to each other?

A.  dA = dC > dB B.  dA > dB > dC C.  dA = dB = dC D.  dA < dB < dC E.  dA = dC < dB

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Buoyancy: Archimedes Principle

•  Let’s do a “thought experiment” (Gedanken).

•  Imagine a beaker with a fluid and a block, B, hanging near it.

•  There is a fluid element

•  The fluid element F is in mechanical equilibrium:

where Fup is the pressure force on the , Fdown is the pressure force on the , and WF is

Buoyancy: Archimedes Principle

•  Step 1: Remove F from the beaker and place it in a small container, leaving an empty bubble of the

•  The bubble is , since its weight is

much less than that of the removed fluid, but the pressure forces are the same.:

where Fup is the pressure force on the bottom surface, Fdown is the pressure force on the top surface, and WF is the weight of the removed fluid F.

Buoyancy: Archimedes Principle

•  Step 2: Block B, with weight WB, is placed in the bubble.

•  There is a net force on Block B:

where WF is the weight of the removed fluid F, and WB is the weight of the block B.

•  This is equal to the force of gravity, , plus a new force called ,which is due to the pressure gradient in the fluid.

Archimedes’ principle: When an object is immersed in a fluid,

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Example

•  A wooden sphere with a diameter of d = 10 cm and density ρ = 0.9 g/cm3 is held under water by a string. What is the tension in the string?

•  Note that the density of water in these units is 1.00 g/cm3.

Example

•  A sphere with a radius of r = 10 cm and density ρ = 2.0 g/cm3 is suspended in water. What is the tension in the string?

•  Note that the density of water in these units is 1.00 g/cm3.

The buoyant force on an object submerged in a liquid depends on

A.  the object’s mass. B.  the object’s volume. C.  the density of the liquid. D.  both A and B. E.  both B and C.

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•  In Class Discussion Question & Demonstration •  A cart is covered by an enclosed transparent box.

A ball is attached to the top of the box by a string. Predict: As the box is accelerating toward the right, which will be the best sketch of the situation?

B C A

•  In Class Discussion Question & Demonstration •  A cart is covered by an enclosed transparent box. A

helium-balloon is attached to the bottom of the box by a string. Predict: As the box is accelerating toward the right, which will be the best sketch of the situation?

B C A

As the cart accelerates to the right, the heavier air molecules are left behind, to the left, creating a tilted pressure gradient

Higher air Density, Pressure

Lower air Density, Pressure

Isobars (planes of equal pressure)

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•  Another way of looking at it: Gravity acts like a pseudo-force, similar to the result of acceleration. This was noted by Einstein and lead to his theory of General Relativity in 1915.

•  Einstein’s Equivalence Principle states that “the gravitational ‘force’ as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.” http://en.wikipedia.org/wiki/Equivalence_principle

Before Next Class (the last class!):

•  Read Chapter 15 of Knight. •  Complete MasteringPhysics.com Problem Set 9

due by June 16 at 11:59pm •  Do Suggested End-of-chapter Exercises and

Problems from Knight: Ch.14: 13, 17, 23, 27, 51, 71 and 77