Physics 121 Exam Sheet

Physics 121 Exam Formula Sheet Acceleration: a  dv/dt = (dvx/dt, dvy/dt, [dvz/dt]) = (ax, ay, [az]) You may take this sheet into the Testing Center. You are allowed to put additional handwritten information on this sheet if you so desire. Kinematic relationships valid iff a = constant: 2 Conversion Factors v = v0 + at, r = r0 + v0t + ½ at , r = r0 + ½ (v + v0)t, Mass vv = v0v0 + 2aΔr 1 kg = 1000 g, 1 u = 1.66  10-27 kg For free fall with a = (0, -g), v0 = (vx0, vy0) = (v0 cos θ, v0 sin θ) a = 0, a = -g, v = v , v = v - gt, x = x + v t, y = y + v t - ½ gt2 Length x y x x0 y y0 0 x0 0 y0 Projectile motion over level terrain with negligible air resistance 1 m = 100 cm = 3.28 ft = 39.37 in, 1 mi = 1.61 km = 5280 ft = 1609 m 2 2 2 Trajectory: y = (tan θ0)x – [g / (2v0 cos θ0)]x (a parabola) 2 2 2 Time Range: R = (v0 /g) sin 2θ0, Maximum height: h = v0 sin θ0 / 2g 2 2 2 2 2 1 day = 86 400 s, 1 yr = 3.156  107s Speed v as a function of time: v = vx + vy = vx0 + (vy0-gt) 2 2 Speed v as a function of y: v = v0 – 2g (y – y0) Speed Acceleration in Uniform Circular Motion: 2 60 mi/h = 88.0 ft/s = 26.83 m/s, 100 km/h = 27.78 m/s = 62.14 mi/h ar = v / r, directed toward the center of the circle Radial (centripetal) and Tangential Acceleration: a = ar + at, with 2 Angle ar = v / r, directed toward the center of the circle, 1 rev = 360° = 2π rad at = dv/dt, directed toward the direction of motion Relative Motion: Relative velocity: Force v = v + v , and, by induction, vBA = v + v + v + . . . + v 1 N = 0.225 lb, 1 lb = 4.45 N BA BC CA BC CD DE XA Relative acceleration: aBA = aBC, if aCA = 0 Energy Vector Differentiation: Rectangular form: 1 eV = 1.60  10-19 J, kcal = 1 Cal = 103 cal = 4.19 kJ, 6 dV dv dv y dv 1kWh = 3.6  10 J = 3.6 MJ  x ˆi  ˆj  z kˆ  (a , a , a ) dt dt dt dt x y z Physical Constants Polar form with v = (v, θ): speed of light c 2.998  108 m/s dv dv d ˆ ˆ -11 2 2  u||  v u gravitational constant G 6.670  10 Nm /kg dt dt dt For Earth: 2 2 free-fall acceleration g 9.80 m/s = 32.15 ft/s Chapters 5 and 6 – The Laws of Motion 24 Newton’s First Law – The First Law of Motion: mass ME 5.98  10 kg In the absence of a force (a free object) moves with a = 0, i.e., if at rest, it mean radius RE 6370 km remains at rest. If moving, it continues to move in a straight line at a constant speed. This is a law describing an inertial reference frame. Metric Prefixes If it appears to be violated, the observer’s reference frame is not in an 12 9 6 3 tera- T 10 giga- G 10 mega- M 10 kilo- k 10 inertial reference frame. milli- m 10-3 micro- μ 10-6 nano- n 10-9 pico- p 10-12 Definitions: dv Acceleration: a  . Chapter 2 - Motion in One Dimension dt Position: x(t)  x-coordinate at time t, y(t)  y-coordinate at time t, etc. Force: F, a push or pull, the cause of any true acceleration, a vector Average speed: average speed  total distance moved/Δt (Δt = elapsed time) quantity. In SI units forces are measured in Newtons (N); in British Displacement: Δx  xfinal – x initial units forces are measure in pounds (lbs). Average velocity:  Δx/Δt Inertial mass: m, a measure of how an object responds to a force, a scalar t quantity. In SI units masses are measured in kilograms (kg); in British Velocity: v  dx/dt = slope on x(t) plot. Hence Δx =  v(t)dt units masses are measured in slugs. t0 Newton’s Second Law – The Second Law of Motion: Acceleration: a  dv/dt = d2v/dt2 = slope on v(t) plot. In an inertial reference frame: t 1. Accelerations are caused by forces. Hence Δv = a(t)dt 2. a = F/m, or F = ma. The force in this relationship is the net force  acting upon the accelerating body, i.e., it is the sum of all forces t0 Kinematic relationships valid iff a = constant: acting upon that body, F  Fnet  F i. 2 2 2 i v = v0 + at, x = x0 + v0t + ½ at , x = x0 + ½ (v + v0)t, v = v0 + 2a (x - x0) Free fall: 3. a is in the same direction as F, always. a = -g (assuming that displacement is taken as positive upward and Newton’s Third Law – The Third Law of Motion: ignoring air resistance and other smaller effects) If body A exerts a force on body B, then body B exerts a force, equal in magnitude, but opposite in direction, on body A, i.e.., FAB =  FBA, where Chapter 3 - Vectors FAB is the force exerted on body B by body A and FBA is the force exerted Vector notation: A is a vector, A is its magnitude (which includes units) and on body A by body B. This law is sometimes called the Law of Action and Reaction. This is a somewhat misleading title because it implicitly uˆ is a unit vector in the direction of A. Hence A  Auˆ. The implies a cause-effect relation between the two forces which are associated direction of uˆ is given by the right-hand rule. with any interaction. In reality, neither force of a force pair is more Vector components in two dimensions: A = Axî+ Ayĵ = (Ax, Ay) fundamental than the other and neither should be viewed as the cause of r is a position vector in two dimensions: r = xî+ yĵ = (x, y), r = (x2 + y2)½ the other. All forces occur in pairs. There are no isolated forces. Dot (scalar) product: AB  AB cos θ (θ is the angle between A and B) Fundamental Forces: There are only four kinds of forces: 1. Gravity: all objects with mass attract all other objects with mass. Also A  B  AxBx  AyBy  AzBz. 2. Electromagnetic Force: all forces experienced at the macroscopic Cross (vector) product: AB  AB sin θ uˆ ( uˆ is a unit vector normal to level, except gravity, are electromagnetic, including tension, the plane of A and B in the direction given by the right-hand rule. ) compression, friction, buoyancy and viscous drag. 3. Nuclear Strong Force: this is the force that binds atomic nuclei ˆi ˆj kˆ together by overcoming the repulsive electromagnetic forces exerted Also A  B  Ax Ay Az . by the protons on each other. This is a very short-range force. Bx By Bz 4. Nuclear Weak Force: this is also a short-range force manifest in certainly nuclear reactions, including the emission of beta radiation. Chapter 4 - Motion in Two [Three] Dimensions Definitions: Chapter 7 – Energy of a System Position: r  (x, y, [z]) Displacement: Δr  (Δx, Δy, [Δz]) Velocity v  dr/dt = (dx/dt, dy/dt, [dz/dt]) = (vx, vy, [vz])  Wi. Work-Kinetic Energy Theorem: for any object ΔK = Wnet  Wi is i the work done on the object by the ith force acting upon it.

Definitions: 2 2 ΔK  Kfinal Kinitial  ½ mvf  ½ mvi

rf Wi   Fi  dr , Wi is the work done by any force Fi in a displacement ri

from ri to r f. If Fi is (is not) conservative, the integral and therefore Wi does not (does) depend upon the path taken in evaluating the integral.

rf rf Wnet  Fnet  dr  Wi  Fi  dr, where Fnet  Fi .  i i  i ri ri Special Cases: 1. F = constant: w  F  d  F  r , ˆ e.g., Wg  mg (y f  yi ) for Fg  mg j

x f 2. One-dimensional motion: W   Fxdx. xi 1 2 2 ˆ e.g., WH   k(xv  xi ) for FH  kx i. (Hooke's law) . 2 3. W = 0, if F  r . Power: lim W P   F  v. t  o t Chapter 8 - Conservation of Energy If a physical quantity X is conserved for an isolated system, then dX X  0 or X final  Xinitial or X  cons tan t or  0. dt Conservation of Energy: dE For an isolated system: E  0 or E final  Einitial or  0. dt Definitions:

E   K  U  Uint  kineticenergy  potential energy  int ernal energy 1 1 function of rest mass, temperature, physical K  mv 2  mv  v, U  mc 2  2 2 int state, chemical state and nuclear structure Conservative force: is a force which does not change the total mechanical energy of a system when it does work on the system. i.e., K  U  0 as

a consequence of work done on a system by such a force. For a conservative force the associated potential energy can be defined by r U ˆ for a one - dimensional U  Wc  Fc  dr and therefore Fc  U (r)   i r x force. Potential Energy: ˆ U g  mgy for Fg  -mg j. (near - earth gravity) , 1 2 ˆ UH  kx for FH  kx i (Hooke's law) , 2 k k U   for F   rˆ (long - range gravity and electrostatic forces) r r2

For gravity k = GMm for electrostatic force k = kCQ. U Force from a radial potential energy: Fr   rˆ . r from stable equilibrium exhibit SHM.