Modern Physics 2425

27 downloads 9830 Views 89KB Size Report
PHYSICS FORMULAS. 2425. Types of Errors: Personal errors due to bias or mistakes. Systematic errors due to miscalibration of instruments, personal bias, or  ...
PHYSICS FORMULAS 2425 Types of Errors: Personal errors due to bias or mistakes. Systematic errors due to miscalibration of instruments, personal bias, or reaction time.

Random errors are unknown or unpredictable, such as voltage or temperature fluctuations, vibration, etc.

Accuracy - how close measurement comes to accepted value Precision - how consistent or repeatable measurements are operation: A = L × W

2

g = 9.8 m/s = 32 f/s gmoon = 1.62 m/s2

(for constant a)

v =

(for constant a) average velocity

x = v 0t +

1

at 2 + 2ax

2

2

2

Mi Mf

− b ± b 2 − 4ac 2a

θ R = tan −1

y R

R=

A + B − 2 AB cos γ

sin θ =

2

γ

2

R

θ A

u = speed of the exhaust relative the to rocket [m/s]

r r r r Rr = Ar+ B +r C r Resultant = Sum of the vectors Rx = Ax + Bx + Cx x-component A x = A cos θ r r r r Ry = Ay + By + C y y-component A y = A sin θ R = Rx 2 + Ry 2

Law of Cosines:

(for constant a)

Addition of Multiple Vectors:

Quadratic Equation:

x=

2

v f − v i = u ln

 ∆M M × ∆V  ∆D = ±  +   V V2 

2

v 0 +v

burning of fuel.

M operation: D = V error:

v = v 0 + at

=v0 (for constant a) Rocket Science: The relationship between velocity and the

error: ∆A = ± ( L × ∆W + W × ∆L )

Division:

s]

v

Calculation of Errors: Multiplication:

Formulas for Velocity: [units: v: m/s; a: m/s2; x: m; t:

B

Ry Rx

Magnitude (length) of R or

tanθ R =

Ry Rx

Angle of the resultant

Unit Vectors: x

B sin γ

Cross Product or Vector Product:

i× j=k

j × i = −k i ×i = 0

Newton's Laws: First Law: Law of Inertia. An object at rest will remain at rest unless acted on by an external force. An object in motion will remain in motion unless . . . Second Law: ΣF = ma , Στ = Iα The sum of external forces on a object is equal to its mass (or inertia for rotational forces) times the acceleration. Third Law: Every action has an equal and opposite reaction. Law of Gravity: F = force of attraction exerted on each body G = gravitational constant 6.67 × 10-11 m1m2 F=G 2 [N · m 2/kg] or [m3/kg · s2] r r = distance between centers [m]

Positive direction:

i

j

k

Dot Product or Scalar Product:

i⋅ j = 0

i⋅i =1

j i

Velocity is the derivative of position with respect k to time:

v=

d dx dy dz ( xi + yj + zk) = i + j+ k dt dt dt dt

Acceleration is the derivative of velocity with respect to time:

a=

dv y dv dv d ( v x i + v y j + v zk ) = x i + j+ z k dt dt dt dt

Mass/Density:

Drag:

[kilograms]

M =V × D

mass = volume × density

vt =

Projectile Motion: v x0 = v0 cos θ 0 v y0 = v0 sin θ 0

horizontal component of velocity vertical component of velocity

x = v x 0t v y = v y 0 − gt

horizontal distance to find apex, let vy = 0

y = y 0 + v y 0 t − 21 gt 2 y = (tan θ0 ) x −

vertical distance

gx 2 2(v 0 cosθ0 ) 2

v y = v y 0 − 2 gy 2

2

Relative Motion:

vertical distance

v PA = v PB + v BA

v PA

2mg CρA

Force:

v PB + v BA = 1 + v PB v BA / c 2

D = Drag Force [N] C = Coeficient of drag [ ] ρ = density [kg/m3] (air: 1.2, water: 1000) A = effective cross-sectional area [m2] v = speed [m/s] vt = Terminal Velocity [m/s] g = acceleration due to gravity [9.8 m/s2]

[F is in Newtons; m is kilograms]

Newtons = kg × m / s = 2

dynes = grams × cm / s 2

grams × g

= dynes × 10, 000 1000 1 lb = 4. 448 N

F = ma

force = mass X acceleration

F=

force =

∆ρ ∆t J F= ∆t

vertical velocity

The relative velocity of object P with respect to A is equal to the velocity of P with respect to B plus the velocity of B with respect to A. For velocities approaching the speed of light, the formula changes to:

D = 21 CρAv 2

force =

change in momentum time interval impulse time interval

conservative force - work done is independent of the path taken non-conservative force - depends on the path taken F =G

m 1m 2 r2

force of gravitational attraction, where G is the constant of universal gravitation 6. 673 × 10−11

N ⋅m 2 kg 2

c = the speed of light = 299,792,458 m/s

Atwood's Machine: Inclined Plane:

F = mg sinθ W = mg Fn = mg cosθ (the normal force)

[F and W are in Newtons; m is kilograms] Fk

a=

 m2 − m1  a= g  m1 + m2  m1

F θ

W

Fn

Fk = µk Fn tanθ

Acceleration in m/s2:

(force of friction, opposite the direction of movement) = µk The coefficient of friction µk is found when the

angle θ is adjusted for zero acceleration of the sliding object. g sinθ (acceleration)

Tension in Newtons:

 2m1 m2  T = g  m1 + m2 

m2

Tension: [Newtons]

T m θ T = m (g + a ) T = m (g − a )

W

Fn

(where m is accelerating upward) (where m is accelerating downward)

PE s =

Work:

[joules or Newton-meters]

T=

W = F•d

2

F cos θ

1 f

T = 2π W = ( F cos θ )s

kx

(elastic potential energy)

Simple Harmonic Motion:

F θ

1 2

s (work done on the object by F)

a=−

work = force × displacement = PE i − PE f (work done by gravity, y is

(T is period in seconds; f is frequency in Hz)

m k

k x m

T = period (s) m = mass (kg) k = spring constant (N/m) (acceleration) x is the location in meters

W = KE f − KEi

k 2 ( A − x2) m x = A cos( 2π f t )

The work done by a conservative force on a particle is independent of the path taken.

Pendulum:

see also: Energy, Spring

T = 2π

W g = mgy

i

− mgy

f

vertical distance in meters)

Power:

[watts]

P=

Power is the rate of work. P =

Energy: KE =

1 2

dW dt W

∆t

= Fv

watts =

mv

v = fλ

second

v=

A falling object loses potential energy as it gains kinetic energy. In an isolated system, energy can be transferred from one type to another but total energy remains the same. W net = ∆KE = − ∆PE E total = KE + PE (mechanical energy) PE i + KE i = PE f + KE f

[i = initial;

f = final, energy is

conserved]

+ 12 mv i = mgy f + 12 mv f 2

2

[F is in Newtons; W is in Joules; x is in meters; k is in Newtons per meter: N/m]

F =

spring with a spring constant k a distance x) (average force required to compress a spring--or

kx

average force output from a decompression over a distance x) kx

2

W = − kx 1 2

center of mass is unaffected. In an elastic collision, kinetic energy KE = 12 mv 2 is conserved.

m1v1i + m2 v 2 i = m1v1 f + m2 v 2 f

m1 − m2 2m2 v1i + v m1 + m2 m1 + m2 2i m − m1 2m1 v1i + 2 v = m1 + m2 m1 + m2 2i

spring

(work done on a spring by an applied force) 2

(work done by a spring)

(momentum)

v1 f =

(elastic only)

v2 f

(elastic only)

p = mv

∑F =

dp dt

Linear Momentum in a system of particles: M = total mass of the system [m] P = Mv cm vcm = velocity of the center of mass [m/s]

Hooke's Law (force required to compress a

1 2

Collisions: In all collisions, momentum is conserved and the

8

F = kx

W =

F µ

Momentum: [kg · m/s]

speed light 3.00 × 10 m/s See also: Rotation and Torque

1 2

(f is frequency in Hz; λ is wavelength in m)

[y = vertical distance]

E is the mass energy, m is mass, c is the

2

Spring:

Third Order Approximation

(kinetic energy)

∆KE = KE f − KE i = 21 mv 2 − 21 mv 0 2 = Work

E = mc

First Order Approximation for small angles L is length in m; g is gravity

(F is tension in N; µ is mass per unit length of string in kg/m) see also: Oscillation

(gravitational potential energy, y is vertical distance in meters)

i

(x is position in m; f is frequency Hz)

Waves:

joules

PE = mgy

mgy

L g

(A is amplitude in m; x is position)

L 1 θ 9 θ sin 4  1 + sin 2 + g 4 2 64 2

T = 2π

[joules] 2

v=±

in

Impulse: [kg · m/s]

J = ∆p = F∆ t = mv f − mv i

impulse = force × duration or the change in momentum see also Force

Center of Mass: The center of mass of a body or a system of bodies is the point that moves as though all of the mass were concentrated there and all external forces were applied there. xcm = distance from origin [m] 1 n x cm = mi x i M = total mass [m] M i =1 m = mass of object [m] This can be applied to x = distance of object from origin y and z axis as well. [m]



Rotation and Torque: [θ is in radians]

∆θ ω= ∆t ω = ω 0 + αt θ = ω 0 t + 12 αt 2

τ = torque (vector) (positive is in the counterclockwise direction) [N · m]

τ = rFt = r⊥ F = rF sin φ

τ = magnitude of the torque r = radius [m] F = force [N] r⊥ = perpendicular distance between axis and an extended line running through F. φ = the angle between r and F [° or rad]

∑ τ = Iα

∑τ = the net torque acting on a body [N · m]

I = Inertia [kg · m2] α = angular acceleration [rad/s2]

(if constant acceleration) [rad/s] (if constant acceleration) [radians] (if constant acceleration)

I =

[kg · m2]

Inertia:

orbiting object:

I = mr

sphere:

Is =

2 5

2

mr

∑ mr

(inertia)

Ir =

ring: 2

2

1 2

m (r 1 + r 2 ) 2

Id =

disk or cyl.:

2

1 2

mr

2

average angular acceleration [rad/s2]

thin rod (on side): I =

tangential speed [m/s]

cylinder on its side (axis ctr): I = m

velocity of the center of mass [m/s]

Parallel Axis Theorem: If you know the rotational inertia

a t = rα

at = tangential acceleration [m/s2] r = radius [m] α = angular acceleration [rad/s2] 2

vt = rω 2 r

ar = radial acceleration or centripetal acceleration [m/s2]

ar

1 12

ml

2

rod (axis end): I = 13 ml

(

r2 4

l + 12 2

2

)

of a body about any axis that passes through its center of mass, you can find its rotational intertia about any other axis parallel to that axis with the parallel-axis theorem:

I = I cm + Mh 2

(directed inward to center)

v = speed [m/s] r = radius [m] ω = angular speed [rad/s]

at

I = Inertia [kg · m2] Icm = Inertia with axis at the center of mass [kg · m2] M = mass [kg] h = distance from the center of mass to the axis [m]

Kinetic Energy: [Joules]

KE r = 12 Iω 2 rotational kinetic energy KEt = 12 mv 2 translational kinetic energy KE = 12 I cmω 2 + 12 mv cm 2 rolling kinetic energy

total acceleration [m/s2]

a = at 2 + a r 2 Fc = ma r = m

τ =r×F

average angular speed [rad/s]

ω 2 = ω 0 2 + 2αθ ∆ω α = ∆t v t = rω v cm = rω

ar =

Torque:

vt 2 r

Fc = centripetal force [N] ar = radial acceleration or centripetal acceleration [m/s2] (directed inward to center)

2π r 2π T= = v ω

 4π 2  3 2  r = K sr 3 T =  GM s 

T

∑ F = T − Mg = Ma ∑ τ = TR = Iα

T = period [s]

Kepler's Third Law (planetary motion)

Yo-yo:

0

where T = the period 2 G = 6. 673 × 10−11 N ⋅m2

kg

K s = 2. 97 × 10

−19 s 2 m3

a = −α R0 a=

T = tension [N] M = mass [kg] R0 = radius of axle [m]

−g 1 + I / MR0 2

R0

Mg

Angular Momentum:

Oscillation:

l = Iω rigid body on fixed axis [kg · m2/s or J · s] l = r × p = m(r × v ) l = angular momentum of a particle [J · s] r = a position vector p = linear momentum [kg · m2/s or J · s] m = mass [kg] v = linear velocity [m/s] I i ωi = I f ωf

tan θ =

v

2

F = force [N] A = crossectional

Fl0 A∆L

area [m2]

V ∆P

V= original vol. [m3] ∆V = chg. in vol.

∆V

[m3]

l0 = initial length [m] ∆L = chg. in length [m]

∆P = change in pressure [Pa or N/m2]

f

F = ma = −( mω 2 ) x

U ( t ) = 12 kx m 2 cos 2 (ω t + φ )

Potential Energy

K ( t ) = 12 kx m 2 sin 2 (ω t + φ )

Kinetic Energy

y = mx + b Ax + By + C = 0

d (sin u ) = u' cos u dx

(m = − A / B )

y − y1 = m ( x − x1 )

A

Ax + By = Ax

1

R = rate of flow [m3/s] A = crossectional area [m2] v = velocity [m/s]

R = A1v1 = A2 v 2

+ By 1

point-slope, alt.

a

+

y b

=1

(m = − b / a )

2-point

intercept a = x-intercept b = y-intercept

Tom Penick [email protected] www.teicontrols/notes December 6, 1997

Bernoulli's Equation:

P1 + 12 ρ v12 + ρ gy1 = P2 + 12 ρ v 2 2 + ρ gy 2 P1 = pressure [Pa or N/m2] v = velocity [m/s] y = height [m]

x

slope-intercept first degree point-slope

 y 2 − y1  y − y1 =   ( x − x1 )  x 2 − x1 

Rate of Flow:

For a horizontal pipe:

Total Mechanical Energy

Equations of a Line:

F a

[rad] k = spring constant [N/m] T = period [s] f = frequency [Hz] F = force [N] m = mass [kg] I = moment of inertia [kg · m2] h = distance between axis and center of mass [m]

d (cos u) = − u'sin u dx

P0 = atmospheric pressure if applicable [Pa or N/m2] ρ = density [kg/m3] g = gravity [m/s2] h = height [m]

f = force [N] a = area [m2] F = force [N] A = area [m2]

1 f

x' = velocity of the oscillating object [m/s] x'' = acceleration of the oscillating object m/s2]

1 atm = 1.01 × 105 Pa = 760 torr = 14.7 in2

f F = a A

k (spring) m

ω=

E = U + K = 12 kx m 2

Pressure in a liquid: (due to gravity) [Pa or N/m2]

P = P0 + ρ gh

x = position [m] xm = amplitude [m] ω = angular frequency [rad/s] t = time [s] φ = phase angle [rad] (ω t + φ) = phase of the motion

m (spring) k I T = 2π mgh

Volume Elasticity: Bulk Modulus [Pa or N/m2] B =−

2π = 2π f T

T = 2π

rg

Elasticity in Length: Young's Modulus [Pa or N/m2]

Y=

ω=

T=

Angular momentum is conserved when torque is zero.

An Optimally Banked Curve:

x = x m cos(ω t + φ )

The Position Function for oscillating motion:

ρ = density [kg/m3] g = gravity [m/s2]

P1 + 12 ρ v12 = P2 + 12 ρ v 2 2