PHY 107 1st EditionExam # 1 Study Guide Lectures: 1 - 10Lecture 1 (January 26)Mechanics – base quantities: length m, time s, mass kgKinematics – quantities: velocity m/s, acceleration m/s2, force kg·m/s2 or N (Newton), energy kg·m2/s2 or J (Joule)Lecture 2 (January 28) Significant figures:- All nonzero numbers are significant- In multiplying or dividing, answer may have no more sig figs than the number with the least amount of sig figs - Leading zeros are not significant (ex. 0.00001 only has one sig fig)- Any zero between two nonzero numbers is significant (ex. 504 has 3 sig figs)- If the number has a decimal point, all zeros to the right of the last nonzero number are significant (ex. 0.01000 has 4 sig figs, 1000. has 4 sig figs) Vectors have magnitude and direction while scalars only have magnitude. A vector is denoted asa or ´a and its magnitude is denoted as a or |´a|. Every vector has x and y components (ax and ay respectively) which may be found such that: - ax = acosθ, ay = asinθ. Unit vectors are special vectors of length 1 centered at the origin and pointing in the directions of the x, y, and z axes: denoted as ^i,^j ,and ^k or ´i, ´j, and ´k respectively. Adding and subtracting vectors: draw the vectors tip to tail and draw a resultant vector OR add/subtract the x, y, and z (if applicable) components of each for a resultant vector. IMPORTANT: ´a−´b=´c → ´a+(−´b)=´cTo get the magnitude of the resultant vector, |´a∨¿, use the equation:- |´a∨¿ = √ax2+ay2To get the angle the resultant vector makes with the x-axis, use the equation:- tanθ = ayaxLecture 3 (January 30)Multiplying vectors: Multiplying a vector by a scalar may be done by finding the components of a vector and multiplying each component by the scalar to generate the components of a new vector. Dot product (scalar product): - If the angle between vectors is given →´a · ´b = abcosθ - Calculating with components → ´a · ´b = axbx + ayby + azbz Cross product (vector product) – creates a new vector perpendicular to the plane the other 2 vectors lay in: - Calculating with components → ´a = ax´i + ay´j + az´k, ´b = bx´i + by´j +bz´k, ´c = cx´i + cy´j + cz´k- cx = aybz – azby cy = azbx – axbz cz = axby - aybx- Order IS important in cross product: ´b x ´a = – (´a x ´b)Lecture 4 (February 2)Unit Vector Multiplication – any unit vector crossed with itself gives a 0 vector´i x ´i = ´j x ´j = ´k x ´k = 0 Motion along a Straight Line: - Displacement – an objects change in position if it moves from position x1 to position x2o ∆x = x2 – x1 (vector quantity, depends only on final and initial position) - Average velocity – in a graph, plot position (x(t)) vs timeo vavg = x2−x1t2−t1 = ∆ x∆ t - Average speed – total distance traveled in a time intervalo savg = total distance∆ tLecture 5 (February 4)- Instantaneous velocity – first derivative of x(t) with respect to to v = lim∆t → 0∆ x∆ t = dxdt- Speed – magnitude of an object’s velocity- Average acceleration (m/s2) o aavg = v2−v1t2−t1 = ∆ v∆ t- Instantaneous acceleration – slope of the v vs t plot o a = d2xd t2- Motion with constant acceleration: o Eq. 1: v = v0 + ato Eq. 2: x(t) = x0 + v0t + a t22o Eq. 3: v2 = v02 = 2a(x – xa) Lecture 6 (February 6) Free fall motion has a constant acceleration of 9.8 m/s2 (gravitational acceleration: g). o Eq. 1) v = v0 – gto Eq. 2) y = y0 + v0t - g t22o Eq. 3) v2 – v02 = – 2g(y – y0) Graphical Integration in Motion Analysis (non-constant acceleration) - must use integration to determine velocity (v(t)) and the position (x(t)) at any given time- Analytically: o a = dvdt → dv = adt → ∫t0t1dv = ∫t0t1adt → v1 – v0 = ∫t0t1adt- Graphically:o Change in velocity = ∫t0t1adt = [area under a vs t curve between t0 and t1]Lecture 7 (February 9)Two-dimensional motion is considered motion in a plane. Three-dimensional motion is considered motion through space. Relative motion is the transformation of velocities between 2reference systems. - Position vector (´r) of a particle is a vector whose tail is at a reference point (usually the origin, O) and its tip is at the particle at point: ´r = x^i + y^j + z^k- Displacement vector – for a particle that changes its position vector from ´r1 to ´r2, the displacement vector ∆´r is defined as: ∆´r = ´r2 - ´r1- Speed = |´v| = √vx2+vy2+vz2- Average and instantaneous velocity:o Average velocity = displacementtime interval´vavg = ∆ ´r∆t = ∆ x^i+∆ y^j+∆ z^k∆ t = ∆ x^i∆t + ∆ y^j∆ t +∆ z^k∆ to Instantaneous velocity as the limit: (∆t → 0)´v = lim∆ ´r∆t = d ´rdt - Average and instantaneous accelerationo Average acceleration = change∈velocitytimeinterval ´aavg = ´v2− ´v1∆ t = ∆ ´v∆ to Instantaneous acceleration as the limit: (∆t → 0)´a = lim∆ ´v∆ t = d ´vdt = ddt(vx^i + vy^j + vz^k) =d vxdt^i + d vydt^j + d vzdt^k = ax^i + ay^j + az^kProjectile motion – motion of an object in a vertical plane under the influence of gravitational force. It is launched with initial velocity v0. Motion along the x-axis has zero acceleration, motionalong the y-axis has uniform acceleration (ay = -g). Velocity along the x-axis does not change (air-resistance is ignored) and the object is in free fall along the y-axis. - Components: v0x = v0cosθ0v0y = v0sinθ0- Horizontal motion o ax = 0o vx = v0cosθ0 (eq 1)o x = xo + (v0cosθ0)t (eq 2)- Vertical motiono ay = -go vy = v0sinθ0 - gt (eq 3)o y = y0 + (v0sinθ0)t - g t22 (eq 4)- Equation of the path: x = (v0cosθ0)t (eq 2), y = (v0sinθ0)t - g t22 (eq 4)- Horizontal range – has a maximum value when θ = 45°, Rmax = v02g- Maximum height H → H = v02sin2θ02 gUniform Circular Motion – a particle moves on a circular path of radius r with constant speed v. Though speed is constant, velocity is not because the direction of the velocity vector changes from point to point along the path. Acceleration is not zero and it is directed towards the center of the circle. -´v = vx^i + vy^j + vz^k, speed =|´v|, s = |´v| = √vx2+vy2+vz2- a = v2r- Period – time, T, taken to complete one full revolution: T = 2 π rvLecture 8 (February 11)Galilean Transformation – equations that connect the velocities seen by two different observers- The velocity of a particle, P, determined by 2 different observers, A and B, varies from observer to observer.Relative motion