PHY131H1F Summer Class 11. What term is used to describe an oscillator that runs down and eventually stops?

Transcription

1 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 1

2 Simple Harmonic Motion (SHM) 2

3 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 3

4 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 Condition for S.H.M. Angular frequency Period Mass on a Spring Small oscillations Pendulum Small angles 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 4

5 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 5

6 Driven Oscillations and Resonance Consider an oscillating system that, when left to itself, oscillates at a frequency f 0. We call this Suppose that this system is subjected This frequency is The amplitude of oscillations is generally not very high if f ext differs much from f 0. As f ext gets closer and closer to f 0, 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. 6

7 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/m 2, also defined as the pascal, where 1 pascal = Other units: 1 atm = Pa 1 mmhg = 133 Pa 1 kpa = 10 3 Pa 1 psi = 6890 Pa Atmospheric Pressure The global average sea-level pressure is Pa, or 1 atm. 7

8 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 p g. where 1 atm = Pa. ie 120 over 80 means the maximum your arteries is 120 mmhg or Pa. in The actual, or absolute pressure in your arteries has a maximum of p = p g + 1atm = 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 8

9 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. d A = d C > d B B. d A > d B > d C C. d A = d B = d C D. d A < d B < d C E. d A = d C < d B 9

10 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 F up is the pressure force on the, F down is the pressure force on the, and W F 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 F up is the pressure force on the bottom surface, F down is the pressure force on the top surface, and W F is the weight of the removed fluid F. Buoyancy: Archimedes Principle Step 2: Block B, with weight W B, is placed in the bubble. There is a net force on Block B: where W F is the weight of the removed fluid F, and W B 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, 10

11 Example A wooden sphere with a diameter of d = 10 cm and density ρ = 0.9 g/cm 3 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/cm 3. Example A sphere with a radius of r = 10 cm and density ρ = 2.0 g/cm 3 is suspended in water. What is the tension in the string? Note that the density of water in these units is 1.00 g/cm 3. 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. 11

12 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? A B C 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? A B C As the cart accelerates to the right, the heavier air molecules are left behind, to the left, creating a tilted pressure gradient Lower air Density, Pressure Higher air Density, Pressure Isobars (planes of equal pressure) 12

13 Another way of looking at it: Gravity acts like a pseudoforce, similar to the result of acceleration. This was noted by Einstein and lead to his theory of General Relativity in 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 pseudoforce experienced by an observer in a non-inertial (accelerated) frame of reference. 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 13