College Physics I Final Exam Review
College Physics I Final Exam Review Phys 1301
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This 10 page Study Guide was uploaded by Aneeqa Akhtar on Saturday April 30, 2016. The Study Guide belongs to Phys 1301 at University of Texas at Dallas taught by Dr. Rodrigues in Spring 2016. Since its upload, it has received 60 views. For similar materials see College Physics I in Physics 2 at University of Texas at Dallas.
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Date Created: 04/30/16
PHYS 1301 Final Exam Review (Chapters 10-13) Chapter 10: Equilibrium of a rigid body Translational Equilibrium: perfect balances of the forces acting on the object/particle ΣF = ma = 0 Rotational Equilibrium: perfect balance of the torques acting on the object (rigid body) Στ = Iα = 0 Examples: 1. With F₁ = 50N and F₂ = 120N, what magnitude of force F₃ is needed to put the object in rotational equilibrium? Recall: τ = Fℓ F: perpendicular force Στ = Iα = 0 ℓ: distance from F to axis of rotation τ₁ + τ₂ + τ₃ = 0 F₁ (0.15m + 0.15m) - F₂ (0.15m) + F₃ (0.10m) = 0 I-----------I---------I---------3 + 0.10F₃ = 0 15 cm 15 cm 10 cm F₃ = 30N 2. The horizontal beam in the figure below weighs 120N, and its center of gravity is at its center. a. Find the tension in the cable. Στ = 0 τTy+ τWb+ τ300= 0 3 m T y T T (4m) – Wb (2m) – (300N) (4m) = 0 4 m [Tsin(α)](4m) – (120N) (2m) – (300N)(4m) = 0 [Tsin(36.9⁰)](4m) – (240Nm) – (1200Nm) = 0 W b α = arctan( ¾ ) = 36.9⁰ T = 1440Nm / [4sin(36.9⁰)] = 600 N T = 600 N 300 N b. Find the horizontal component of the force exerted on the beam at the wall. Fw-b – T//= 0 Fw-b – 600cos(36.9⁰) = 0 Fw-b = 480 N Chapter 11: Elasticity and Periodic Motion Stress: force per unit area causing the deformation (stretching, squeezing, twisting) of an object Four different types of stress: 1. Tensile Stress 2. Compressive Stress 3. Sheer Stress 4. Volume Stress Strain: the resulting (normalized) deformation caused by stress For small deformations, the following relationship holds: stress strain= constant = Young’s Modulus = Y (General form of Hooke’s Law) Tensile Stress and Strain Tensile stress = units: N/m^2 = Pa Tensile strain = units: none acts as normal to cross-section A Compressive Stress and Strain Compressive stress = units: N/m^2 = Pa Compressive strain = units: non It has been shown experimentally, that for small values of stress and strain, one can write: Y = tensile stress OR compressive stress tensile strain compressive strain Therefore: Y depends on the type of material of the object under stress is made of. The higher Y, the harder the object is to stress. Particular case of a spring Volume Stress and Strain When someone is underwater, stress is now caused by the uniform pressure. The resulting deformation (strain) is a change in volume caused by the change in pressure. Volume Stress: ∆p = p - pₒ Volume Strain: ∆v / vₒ If Hooke’s law is obeyed: stress = constant = Bulk Modulus = B strain Negative sign, because a higher p causes a low V Shear Stress and Strain We talk about shear stress and strain when we are dealing with forces that are tangent to a surface Sheer stress = F/// A Sheer strain = x/h = tan(Ø) Periodic Motion: oscillatory (repetitive) motion with a well-defined period Examples: rocking chair, bouncing ball, swing in motio, planet orbiting Simple Harmonic Motion (SHM): 1. Motion is “simple” and “harmonic”: It is described by a sin/cos function with a unique frequency. 2. Periodicity is kept by a “restoring force” Quantities describing periodic motion Amplitude: maximum displacement from equilibrium position Cycle: one complete cycle of motion Period: the time it takes for a complete cycle of motion Frequency (f): number of cycles within one second F = 1/T Angular frequency: Ѡ = (2π)/T = 2πf Periodic Motion Graphs period x amplitude t Displacement, velocity, and acceleration graphs: Energy in Simple Harmonic Motion Mechanical energy of a spring-mass system: kinetic energy and elastic energy We can find total energy when x = A This value has to remain constant throughout the motion Solving for vₓ: Acceleration For the specific case of a mass-spring system, the restoring force is:r= -kx The acceleration, as a function of x, is then given by: maₓ = -kx The Simple Pendulum: an idealized model of a: a. Point mass, suspended by a b. Weightless string, in a c. Uniform gravitational field L L Ø x What happens when you release the mass? PERIODIC MOTION But therestoringforceisnot proportional to the displacement F t= - (mg) sinØ For small values of Ø: sinØ ≈ Ø Ø = x/L Therefore, F t= - (mg) x/L And restoring force is now proportional to displacement! Now: F = max = - (mg) x/L Before: aₓ = - Ѡ^2 x -Long strings -> slow oscillations Therefore: -mѠ^2 x = - (mg) x/L -The frequency (and period) is Ѡ = 2πf = 2π/T = √g/L independent of mass Chapter 12: Mechanical Waves and Sound Waves: occur whenever a distinctive “pattern” travels (propagates) from one region to another Example: radio waves, strings of a guitar, ocean waves Mechanical Waves: requires a medium (particles) to propagate. The characteristics of that wave (e.g. speed) will depend on the properties of the medium. Sound waves, for instance, cannot propagate in a vacuum. It requires a medium (air) to probate through -> mechanical wave Mechanical Waves fall within three categories: 1. Longitudinal waves 2. Transverse waves 3. Transverse AND longitudinal waves Transverse Waves: the particles of the medium move up and down, perpendicular to the direction of the wave propagation -> transverse motion Longitudinal Waves: particles move back and forth, along the direction of the wave propagation -> longitudinal motion Transverse & Longitudinal Waves: particles move along AND transverse to the direction of wave propagation. Energy: 1. Particles forming the medium wave move around their equilibrium position 2. Only the “wave pattern” moves from one region to another (e.g. from one extreme of the rope to another) 3. Energy is moving through the medium, going from one place to another -Waves transport energy from one point to another, but they do not transport matter -The energy propagates with what we call wave speed, the speed of the disturbance. Periodic Transverse Waves: Wavelength (λ) = distance between 2 consecutive wave patterns Wave Speed = speed at which wave pattern moves/propagates V = λ/T = λf wave frequency wave period Wave Speeds: speed of a transverse mechanical wave depends on the mechanical properties of the propagating medium Speed of a longitudinal mechanical wave depends on the mechanical properties of the medium Speed is: -proportional to the square root of an elastic factor -inversely proportional to the square root of the density of the material Mathematical Descriptions of Waves For a stationary sinusoidal wave: y(x) = Asin (2π/λ)x For a propagating sinusoidal wave: y(x) = Asin[(2π/λ)x – Ѡt] For a wave moving to the right (+x) / AKA mathematical description of a wave Reflection: the return of a mechanical wave (or pulse) after reaching a boundary or interface Echo: the reflection of sound (longitudinal) wave in some boundary away from where the sound was created Principle of Superposition: Whenever two waves overlap, the final displacement of any point on the rope is given by the vector addition of the displacements caused, individually, by the interacting waves Standing Waves: perturbations (displacements in a rope) that do not travel from one region to another, unlike regular “traveling” waves The perturbation simple “stands” there, independently of time It is the result of the superposition of two transverse waves of the same wavelength, but traveling in opposite directions. Wavelengths for standing waves o For a string fixed at both ends, only specific types of standing waves can exist. This is because there must be nodes in both ends of the string. Therefore, the wavelength of the standing wave depends on the length (L) of the string. o Only standing waves with the following wavelengths are possible: o These are called the normal modes of the string of length L. Frequencies for standing waves o F = v/λ. Since the v is the same for all waves, these are the frequencies allowed: Fundamental Frequency o Other frequencies are harmonics (or overtones) of the fundamental frequency f₁ Interference: the interaction (principle of superposition) between two or more waves overlapping in the same region of space a standing wave is an example of the interference of two waves with the same frequency, wavelength, and speed but opposite directions two types of interactions are particularly interesting: o constructive interference o destructive interference Sound and Hearing Points to remember: o Sound waves are mechanical longitudinal waves. Therefore, they require matter to propagate. o Sounds waves can propagate in solids, liquids, and gases. (There is sound underwater!) o The human ear can hear sound waves with frequencies between 20 Hz and 20 kHz. Infrasonic waves: under 20 Hz Ultrasonic waves: above 20 kHz o The amplitude of sound waves can be describe in terms of pressure (force/area) o Microphones and our ears sense pressure variations Waves reach the eardrum Sound waves make the eardrum, and 3 tiny bones vibrate, which then set the fluid in the inner ear to vibrate Sensory cells in the inner ear convert the vibrations into nerve impulses that are transmitted to the brain via the auditory/cochlear nerve. Sound Intensity Intensity: intensity of a wave is the average power (P in watts) transported by a wave, per unit area (given in m^2), across a surface perpendicular to the direction or propagation. o We often assume an isotropic source. It transmits the P in all directions equally. In that case: A = A spher= 4πr^2 (r = distance from source) Therefore: Units: W/m^2 o Now, for a point source transmitting sound uniformly in all directions: Instead of W/m^2, the intensity of sound is also expressed in decibel (dB), which is defined in respect to a reference intensity (Iₒ) ( which is in general, Iₒ = 10 ^ -12 W/m^2 Frequency Beat: the result of two sound waves of slightly different frequencies interfering, alternating constructive and destructive interference, to produce a sound that is alternatively soft and loud. The frequency of the soft/loud cycle is given by: The Doppler Effect: the apparent change in frequency (or wavelength) of a wave to the relative motion between the sound source and the observer (listener) Source or listener is moving Listener moving: o Listener moving towards the source, +VL o Listener moving away from source, -V L Source moving; o Source moving away from listener, +V S o Source moving towards listener, -VS Doppler Effect: Apparent Wavelength Source is moving, listener is stationary V = sound speed = 344 m/s Vs= source speed
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