Physics 1 PSet 7 Summary Sheet PDF

# Inclines & Machines (L13A) - W = mg - Wy = WCosθ - Wx = WSinθ For there to be equilibrium: - Fnet = 0 => - Fnet, x = 0 => T = WSinθ => T = mgSinθ - Fnet, y = 0 => FN = WCosθ => FN = mgCosθ ## Cord cut - T = 0 - Fnet, x = WSinθ = mgSinθ => Fnet = mgSinθ - (FN & Wy cancel) - a = F...

# Inclines & Machines (L13A) - W = mg - Wy = WCosθ - Wx = WSinθ For there to be equilibrium: - Fnet = 0 => - Fnet, x = 0 => T = WSinθ => T = mgSinθ - Fnet, y = 0 => FN = WCosθ => FN = mgCosθ ## Cord cut - T = 0 - Fnet, x = WSinθ = mgSinθ => Fnet = mgSinθ - (FN & Wy cancel) - a = Fnet/m = gSinθ # The Atwood Machine - Frictionless pulley ## For m₁: - T - m₁g = m₁a ## For m₂: - m₂g - T = m₂a + - m₂g - m₁g = (m₁ + m₂)a - a = (m₂ - m₁)g / (m₁ + m₂) ## m₂ ≥ m₁ a ≥ 0 - a = (m₂ - m₁)g / (m₁ + m₂) ## m₂ = m₁ a = 0 * Can solve for Tension by plugging in a to T = m₁a + m₁g* # Double Atwood ## Constraints - y₁ + yp = L₁ => yp = L₁ - y₁ - y₂ = yp + y₂' => y₂ = y₂' - yp - y₃ = yp + y₃' => y₃ = y₃' - yp - (y₂ - yp) + (y₃ - yp) = L₂ - y₂ + y₃ - 2yp = L₂ - y₂ + y₃ - 2(L₁ - y₁) = L₂ - y₂ + y₃ - 2L₁ + 2y₁= L₂ - y₂ + y₃ + 2y₁ = L₂ - 2L₁ - d/dt (y₂ + y₃ + 2y₁) = 0 - (v₂ + v₃ + 2v₁) = 0 (equation 1) - a₁ = m₁g - T₁ = g - (T₁/m₁) - a₂ = m₂g - T₂ = g - (T₂/m₂) - a₃ = m₃g - T₂ = g - (T₂/m₃) - 2T₂ - T₁ = 0 ## Equation 1: - (a₁ + a₂ + 2a₁) = 0 - (g - T₁/m₁) + (g - T₂/m₂) + 2(g - T₁/m₁) = 0 - 4g - (1/m₁)T₁ - (1/m₂)T₂ - (2/m₁)T₁ = 0 - 4g - (3/m₁)T₁ - (1/m₂)T₂ = 0 - T₁ = 4g - (1/m₂)T₂ / (3/m₁) - T₁ = 4m₂m₃g / (3m₁m₂ + m₃) ## T₁ = 2T₂ - T₁ = 8m₂m₃g / (4m₂m₃ + m₁(m₂ + m₃)) ## T₂ = (4m₁m₂m₃g) / (4m₂m₃ + m₁(m₂ + m₃)) # L13B: Systems with Friction ## Static Friction: - (tendency for motion) - some push or tendency for an object to slide with respect to a tuble, but it doesn't move - fs,max = μsFN ## Kinetic Friction: - now a force is applied and the object slides with respect to the table, it moves - fn = μkFN - Normal force = friction - Friction opposes the "tendency of motion" - Coefficient of kinetic friction is typically smaller than the coefficient of static friction # L13A: Centripetal Force & Banked Roads ## Uniform Circular Motion: - a = (rCosθ)i + (rSinθ)j - ==>*acp = v²/r* - Fcp = -mv²/r ## Static Friction: - don't want the car to slide - allows circular motion between the tires and a cor and the road (round about) - Fcp = fs - mu = Fcp = fs / FN - mu = v²/r # * If the static friction is above fs,max = μsFN - then the car will slide and the curve will not be successful. - If the static friction is anything less than fs,max = μsFN, then the car taking the curve will be successful. # Fcp needed < fs,max ==> makes curve w/o skidding - Fcp = μsFN = μsmg # General Formulas: - Fnet, x = Fcp - Fnet, y = 0 - Fcp = mv²/r - Fcp = μsFN = μsW - Fcp = FN Sinθ + fsCosθ - FN Cosθ = mg + fsCosθ # L14 B: Velocity-dependent Forces - Drag Force - Terminal Speed - *Free full predictions cannot be the whole story* In freefall, air resistance fr opposes velocity (mg=w). # Air resistance opposes velocity in freefall. fr opposes W=mg ## Linear air resistance (simplest model) - fr =-bv - F=ma a = F/m - a= m(mg-bu) - As v increases "bv" will increase and a will eventually come to match mg. Then a = 0 and mg -bu = 0, - so V= mg/b = VT (terminal velocity). - v₁ = v (1 - e ^ -bt/m) - a = g - bv/m - y(t) = v₁t + (m/b²)( e^ -bt/m -1) ## Freefall: - y(t)= 1/2gt² ## Quadratic air resistance (more sophisticated) - fr = -bu² - pragforce: IFRI = F = 1/2ρAv² - a = g - bv²/m - a = (mg - bv²) /m