Physics (Ch.10-14)

Right-Hand rule

1)Used to find the direction of the angular velocity; curl the fingers in the direction of rotation; thumb points to the direction of angular velocity 2) Used to find the direction of torque; curl fingers in the direction of rotation that will be caused by the tangential force F; the thumb points to the direction of torque The fingers point curl to the direction of rotation (either the initial or the one induced by F) => right thumb points in the direction of torque or angular acceleration

Orbital Maneuvers

Acceleration; motion on an elliptical orbit Large radius on higher orbits; higher period (=longer to complete the orbit) Deceleration; orbit of smaller radius and shorter period; can get somewhere faster

Translational Component of Rolling Motion

All parts of the wheel move in respect to the ground; tangential speed pointing to the direction of motion on the axle, the top and the bottom

Revolution

Arc length of displacement is the circumference of the circle 1 rev = 360o = 2π rad Revolutions derived from displacement

Damped Oscillation

Barely moves; amplitude is very small; critically damped system

Rolling Motion

Bike wheel -Moves both on the ground (translational motion) -Rotates around the axle (rotational motion) At one rotation of the wheel, the wheel has moved at a distance equal to the circumference on the ground Translational speed equal to the tangential speed; movement on the ground follows the tangential path K= 0.5 mv^2 + 0.5I w^2 Mass of pulley/disk used in the expression of the moment of inertia

Kinetic Energy of Rolling Object

Both translational and rotational kinetic energy Translational kinetic energy; usual (K=0.5mv^2) Rotational kinetic energy; involves moment of inertia; kinetic energy expensed for rotation around the center K=0.5Iω^2 Low rotational kinetic energy (small moment of inertia) => more kinetic energy of translation => higher velocity of translation Moment of inertia uses mass of the pulley/disk/object rotating; not of the hanging mass

Acceleration of Simple Harmonic Motion

Centripetal acceleration, as in uniform circular motion; points inwards and keeps object in orbit Only the x-component of acceleration considered; a= -Aω^2 cos(ωt) Acceleration and position vary in the same way over time, but with opposite signs Acceleration is maximum at the minimum position =>at the minimum position, acceleration needs to be positive to reach maximum position; needs to be positive to allow motion in the positive direction a(max) = A ω^2

Tangential Acceleration a(t)

Change in angular speed => change in tangential speed => tangential acceleration (when angular velocity changes) a(t) = r*α α is the change in angular velocity, which confers change in the tangential velocity, quantified by the tangential acceleration

Angular Acceleration (α)

Change in angular velocity (ω) over time Analogous to its linear acceleration counterpart Equation; ω=ω0 + αt

Angular Velocity (ω)

Deals with change of angle Change in the angular velocity requires an external net torque How fast the object is rotating around the center of rotation

Rotational Work

Derived from the original equation; W=F Δx Δx is the length of the arc corresponding to the change in theta Δx = R Δθ; W=F R Δθ However, torque = FR => W=τ Δθ Work depends on the displacement, as always, but the displacement is the length of the arc corresponding to the change in angular position theta (Arc length; r * θ)

Beat

Difference between frequencies that are very close; produce interference pattern; the sound produced by two frequencies that are very close together is not the same over time; varying interference pattern Frequency of fluctuations is quantified by beats; The two waves have the same amplitude, but varying frequencies when superimposed

Amplitude of Simple Harmonic Motion

Distance between midpoint and maximum or minimum One-half the total range of motion Range; two times the amplitude

Energy Conservation in Oscillatory Motion

E = K + U = 0.5mv^2 + 0.5kx^2 The kinetic and potential energy change over time since the motion is oscillatory. U=0.5kA^2cos^2(ωt) U=0.5kA^2 In simple harmonic motion, the total energy is proportional to the square of the amplitude of motion; both kinetic and potential energy dependent on A^2

Mechanical Energy in Astronomical Systems

E for an object away from the earth; E = K + U = 0.5mv^2 - GMm/r Kinetic energy encompasses the mass of the object, not the mass of the planet Potential energy includes both mass of planet and of object Problems; Asteroid starts at rest (Ki=0) and is at infinity (Ui=0); therefore the total initial mechanical energy is zero Ei= Ui + Ki = 0+0=0 The total mechanical energy of the asteroid remains constant since gravity is a conservative force; Ef = 0 => 0.5m(Vf)^2 - GmM/r = 0 The potential energy becomes increasing negative, and kinetic energy becomes increasingly positive; speed increases (//funnel; height decreases and speed increases) U is negative, because the asteroid is coming closer to the Earth (=origin); distance decreases

Column of air open at one end and closed at the other

Each successive harmonic differs by half a wavelength Each harmonic has different frequency; because different wavelength The first harmonic is λ/4 All integer harmonics present when column is open at one end (//string) Standing waves; column open at one end f1=v/2L f(n)= nf(1) = n v/2L n=1,2,3 λ (n) = λ(1) /n = 2L / n Each harmonic has a frequency that is a multiple of the fundamental frequency and has an additional λ/2 => harmonics have both different frequencies and wavelengths

Angular Frequency of Simple Harmonic Motion

Equivalent to the angular speed (ω) for periodic motion Frequency; number of cycles per time ω= 2π f = 2π/T (measured in s^-1)

Find the mass of a rope

Find velocity; depends on the mass per unit length Mass per unit length is mass per length Find length

Kepler's Third Law

Focused on the relationship between the radius, the mean distance of the planet from the sun, and the period (T), the time required for one full revolution T = (constant) r^1.5 As the planet moves in a circular ellipse, the gravitational force from the central planet generates a centripetal acceleration pointing toward the central planet and keeps the second planet in orbit **The period does not depend on the mass of the planet being orbited; only on the mass of the planet exerting the gravitational force, ex. the sun

Balance

For mobiles; m1x1=m2x2 Center of mass for an object hanged lies on the same axis as the suspension point so that the torque caused by gravity is zero

Wavelength

From one extreme to the other and then back One wavelength crossed within one period Speed of a wave dependent on the wavelength For standing waves, wavelength quantified relative to the length of the string For the first harmonic, l=2L => it takes twice the length to go from one extreme to the other and back

Torque (τ) and Moment of Inertia

High Torque (τ); because of high moment of inertia More mass distributed away from the center of rotation; greater torque => greater rotational force Net torque causes a change in the rotational motion of an object ***Torque => changes the angular velocity; causes angular acceleration α High torque; lower angular acceleration

Intensity and Power

I= P/A = P/4pi r^2 The power does not change--intensity is measured relative to the position

Interference

Identical, in phase sources Constructive or destructive according to the differences in the path length of two sources in phase; -If path lengths differ by an integer multiple of the wavelength; constructive interference -If the path lengths differ by a non-integer multiple of wavelength, destructive interference Calculate distance from each source to detector; calculate difference between those distances Divide that distance by the wavelength to chack if it is an integer multiple of the wavelength

Reflection of waves

If rope is mounted on the wall; wave is inverted If rope mounted on a ring; wave returns without inversion

Black Holes

Immense escape speed (= equal to the speed of light); not even light can escape Very concentrated; high mass yet small radius

Escape Speed

Initial Mechanical Energy for an object on the surface of earth has non-zero individual initial kinetic and potential energies; *The sum of initial kinetic and potential energy is zero, because their non-zero values cancel out **The final kinetic and potential energies are zero; both at rest and at infinite distance Ve=(2GM/R)^0.5 Mass of planet used in the expression Low escape speed of the moon, since G is lower; easier to launch a rocket from there Moon does not have an atmosphere because of the low escape speed; particles can easily escape--not attracted strongly by gravity since small G When the initial speed of a rocket is less than the escape speed, the rocket momentarily comes to rest at a finite distance r For black holes, escape speed is the speed of light (=3e8 m/s)

Intensity and Intensity Level

Intensity (I); energy per time per area; Intensity Level (β); measure of the relative loudness relative to the threshold of hearing β = 10 log (I/I0) I0 = 10^-12 W/m^2 Each increase of 10 dB in the intensity level leads to doubling of the intensity

Moment of Inertia

Kinetic energy depends on the distribution of mass around the axis of rotation Mass near the axis of rotation does not contribute to rotation More mass away from the center of rotation; higher moment of inertia; more rotation Takes mass far from the center to drive rotation around the axis of rotation Pronounced rotation with high moment of Inertia **hoop has a high moment of inertia, because all of its mass is distributed away from the center Hollow objects have most of their mass away from the center; high rotational kinetic energy, because high moment of inertia I = m*r^2 [Different per shape of object] Rotating person holding weights with stretched arms; moment of inertia for body/shoulder width and of stretched arms with weights High moment of inertia; lower angular acceleration (view torque definition τ= Iα)

Newton's Second Law for Rotational Motion

L= I ω Change in angular momentum (ΔL) is equal to the change in angular velocity, provided that the shape and mass of the body are constant ΔL=I(Δω)/Δt However, (Δω)/Δt = α, therefore ΔL=Iα=Στ Momentum is conserved, when no net torque is applied to the system L(1) = L(2) => I(1)ω(1) = I(2)ω(2) => mr^2 ω(1) = mr^2 ω(2) High r (=arms extended); low angular velocity Low r (=arms flexed); high angular velocity

Damped Oscillation

Loss of mechanical energy, because of the action of a non-conservative force (friction or air resistance) Damped oscillation; nonconservative force opposes the harmonic motion; force is proportional to the speed of the mass and opposite in direction =>force can be expressed as the opposite velocity of the oscillating mass multiplied by a damping constant F = -bv b; damping constant--measures strength of damping force With small damping constant, oscillation continues, but with an exponentially smaller amplitude (=underdamping) With large damping constant, the system relaxes at the equilibrium position (=critically dumped) Oscillation taking longer; overdamped oscillation

Physical Pendulum

Mass in not concentrated at a point, but distributed over a finite volume. Period depends on moment of inertia from the axis of rotation, distance from the center of mass (l) T = 2π (l/g)^0.5 (I/ml^2)^0.5 The first part is the period of a simple pendulum with all the mass concentrated at one point; the second component takes into account the shape and size of the physical pendulum Small moment of inertia (=mass more concentrated around the origin of rotation) => smaller period Completion of one cycle is faster when the mass is concentrated around the origin of rotation(=>low I)

Pendulum

Mass suspended by string/rod of length L Oscillation from equilibrium (angle is zero) to two extremes U=mgL(1-cosθ) Minimum potential energy at equilibrium Forces acting on mass; tension and weight =>tension acts in a radial direction; keeps mass along its circular path Net tangential force; x component of weight (F=mgsinθ) Arc displacement; length of arc; s=Lθ => θ=s/L F=mgsinθ ~ F=mgθ~ mg(s/L) The restoring force (^^) is proportional to displacement For small angles sinθ = θ Period of pendulum = 2π(L/g)^0.5 **Period is independent of the mass and the amplitude; only depends on the length of the string and g Increasing the hanging mass (essentially the weight) decreases the period and makes the pendulum/clock run faster because higher gravitational force is acting on it (in mass hanging from the spring, period decreases with increasing mass)

Periodic Motion

Motion that repeats itself over and over; the time required to complete one cycle is characteristic of that motion

Centripetal Acceleration

Movement around a circular path has centripetal acceleration moving inwards--for stable angular velocity; Angular speed is constant, but its direction changes a(cp) = r(ω^2) Always perpendicular to the path

Frequency

Number of oscillations per time; how many oscillations/cycles within a time interval Measured in hertz (= cycles per time) Different length of column; different frequency for standing waves

Angular Momentum

Object moving with an angular speed along the circumference of a circle of radius r => has angular momentum of L L= I ω = mr^2 (v/r) = rmv = rp L = rmv sinθ **When a particle is moving on a circular path, only the tangential component of its velocity will contribute to the angular momentum; hence sinθ

Strings

One fundamental frequency added to each successive harmonic--half a wavelength added to the standing wave

Power produced by torque

P=W/Δt => τΔθ/Δt = τω

Period of a Mass on a Spring

Period of a mass on a spring; time required for the mass to move from the maximum to the minimum position and then back to the maximum T=2π (m/k)^0.5 The period increases with mass; more time required to move a heavy mass Period decreases with spring constant Period independent of the amplitude;

Kepler's First Law

Planets follow elliptical orbits, with the sun at one focus of the ellipse

Kepler's Second Law

Planets sweep the same area within the same time interval Angular momentum (L=Iω = Imr^2)

Driven Oscillations and Resonance

Possible to increase energy of the system and counteract the loss of mechanical energy =>requires external forces with positive work Driven oscillation; moving the support point back and forth to "drive" the weight; the response depends on the frequency of movement of the support point Hand should drive the support point with an intermediate frequency--at natural frequency, amplitude becomes larger Large amplitudes at a driving frequency equal to the natural frequency => frequency matching produces large response (=resonance) Low and broad resonance peak for damped oscillations, since the amplitude becomes exponentially smaller

Torque (τ)

Rotational force; tendency of a force to cause rotation increases with the distance r from the axis of rotation Rotation requires force to be tangential to the axis of rotation Higher r increases the magnitude of rotational force Radial force produces zero torque; force needs to be tangential to the circular path to cause rotational force τ= rFsinθ τ= Iα τ= ΔL/Δt

How to find the angular acceleration (α)

Since τ= Iα, calculate torque (τ) from τ=rF and substitute it into the first equation => use torque to calculate the angular acceleration, since torque causes a change in angular velocity rF = Iα

Rotational Component of Rolling Motion

Tangential velocity on top and on the bottom of the wheel Tangential velocity on the bottom points to the direction opposite to the direction of motion

Fundamental Frequency

The first harmonic; each successive harmonic differs by one fundamental frequency Each harmonic results from addition to half a wavelength to the previous standing wave => For strings successive harmonics differ by the frequency of waves; therefore also by half a wavelength, since one fundamental frequency consists of half a wavelength along one L

Simple Harmonic Motion

The restoring force of a spring (=points in a direction opposite to the displacement) causes oscillations following simple harmonic motion; -Mass oscillated from a maximum to a lower height -Since motion is repetitive, the position and velocity of an object can be predicted as a function of sine and cosine

Standing Waves

Three types -Strings; close at both ends -Closed only at one end -Open at both ends

Ropes tied together

Tied together = Same Tension v=L/t Calculate v from v=(F/μ)^0.5

Gravitational Potential Energy

U=mgh is valid only near the surface of the earth, where the acceleration of gravity is zero U approaches zero as r approaches infinity; when origin is at ground level

Find Mass of Planet when dropping an object

Weight = Gravitational Force mg = G(Mm)/r^2 On the surface of earth; use diameter of earth as R Only mass of earth when calculating (g); the other mass is canceled out

In Phase

When one source emits a crest, the other also emits a crest; constructive interference

Tangential velocity (v_t)

Why? For every object moving in a circular path, its velocity (direction of movement) is tangential to the path at any point => The tangential velocity is the tangential velocity at any point; linear velocity is derived from the angular velocity Tangential velocity depends on the radius; object further from the center crosses a larger distance to move around the center, while object closer to the center moves crosses a smaller distance to move around Linear velocity of some particle on the circular path v(t) = r ω

Harmonics

family of resonant frequencies that are multiples of the natural frequency

Velocity of Simple Harmonic Motion

v= -Aω sin(ωt) The velocity is tangential to the object's circular path, hence the x component of angular velocity is considered When the displacement from equilibrium is maximum, the object is momentarily at rest before turning around => turning points of the motion Velocity is maximum when the object is in equilibrium; when it passes at x=0 Vmax= Aω Speed is maximum at equilibrium, because as the mass passes x=0 to compress or extend the spring, it slows down; reaches zero at +A or -A

Position of Simple Harmonic Motion

x=A cos(ωt) = A cos(2πt/T) A is the amplitude/radius Position depends on the amplitude, the period (characteristic of simple harmonic motion) and on the time

Torque (τ) and Angular Acceleration (α)

τ= Iα Angular acceleration (α) is therefore proportional to the torque (= the force that causes rotation) -Higher torque; higher change in angular velocity; greater angular acceleration -Angular acceleration (α) negatively proportional to the moment of inertia => high moment of inertia; low acceleration ex. the hoop, which has a great moment of inertia since most of the mass is away from the center of rotation, will experience a smaller angular acceleration (α) Large moment of inertia => small angular acceleration (α) Expression is the rotational equivalent to Newton's second law

Angular velocity equation

ω=ω0+ αt