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Pulley System with Rоtаtiоnаl Inertiа PHY 2048C Cumulative Final Examinatiоn Points: 10 Suggested Time: 20–25 minutes Instructions: Show all work. Begin each derivation with an appropriate physics principle. Clearly define any additional symbols you introduce. Block 1 of mass m1 rests on a frictionless incline of angle θ. It is connected by a light cord over a pulley to a hanging block of mass m2. The pulley has radius R and moment of inertia I. The cord does not slip on the pulley. Assume block 2 moves downward and block 1 moves up the incline. Tasks Part A. Write Newton's second-law equation for each block and the rotational equation for the pulley.Part B. Derive the magnitude of the acceleration a of the system.Part C. Determine the two cord tensions T1 and T2.Part D. Starting from rest, determine the speed v of the blocks after block 2 descends a distance s. You may use either Newton's laws or conservation of energy.
Sаtellite Orbit Chаnge PHY 2048C Cumulаtive Final Examinatiоn Pоints: 10 Suggested Time: 20–25 minutes Instructiоns: Show all work. Begin each derivation with an appropriate physics principle. Clearly define any additional symbols you introduce. A satellite of mass m is initially in a circular orbit of radius r0 around a planet of mass M. At one point in the orbit, the satellite performs a brief prograde tangential burn that increases its speed by Δv. Define μ = GM and neglect atmospheric drag. Reference Information vc = √(μ/r0), ε = v2/2 − μ/r, ε = −μ/(2a) Tasks Part A. Determine the satellite's speed immediately after the burn. Part B. Determine the satellite's new specific orbital energy ε1. Part C. Determine the semimajor axis a of the new orbit. Part D. Determine the period T of the new orbit. Part E. Determine the minimum value of Δv required for the satellite to escape from radius r0.
Lаb Dаtа Questiоn — Rоlling Object Investigatiоn PHY 2048C Cumulative Final Examination Points: 10 Suggested Time: 20–25 minutes Instructions: Show all work. Clearly label graphs, axes, units, and calculated quantities. Support conclusions using the experimental data. A student investigates the motion of a rigid object that rolls without slipping down a ramp. The object has mass m, radius R, and moment of inertia I = βmR2, where β is a dimensionless constant that depends on how the object's mass is distributed. The object is released from rest at several vertical heights h above the bottom of the ramp. A photogate measures its center-of-mass speed v at the bottom. The measured uncertainty is ±0.002 m for each height and ±0.03 m/s for each speed. Experimental Data Trial Release height, h (m) Measured speed, v (m/s) v2 (m2/s2) 1 0.080 1.05 2 0.120 1.29 3 0.160 1.48 4 0.200 1.66 5 0.240 1.82 6 0.280 1.96 Tasks Part A. Starting with conservation of mechanical energy, derive a linear relationship between v2 and h. Express the slope in terms of g and β. Part B. Complete the final column of the data table. On graph paper, plot v2 on the vertical axis and h on the horizontal axis. Draw a best-fit line. Part C. Determine the slope of the best-fit line. Do not calculate the slope using only one pair of adjacent data points. Part D. Use the experimental slope to determine β. Take g = 9.80 m/s2. Part E. Based on the value of β, identify which object is most consistent with the data: solid sphere: β = 2/5 solid cylinder: β = 1/2 thin hoop: β = 1 Justify your selection quantitatively. Part F. The best-fit line has a small positive vertical intercept instead of passing exactly through the origin. Identify one plausible experimental cause and explain how it could produce a positive intercept. Reminder: A strong experimental conclusion must reference both the calculated value of β and its agreement with a theoretical model.
Nоnlineаr Spring Oscillаtоr PHY 2048C Cumulаtive Final Examinatiоn Points: 10 Suggested Time: 25–30 minutes Instructions: Show all work. Begin each derivation with an appropriate fundamental physics principle. Clearly define any additional symbols you introduce. Unsupported answers may not receive full credit. A cart of mass m moves without friction along a horizontal track. It is attached to a nonlinear spring whose restoring force isFs(x) = −k0x(1 + αx)where x is measured from equilibrium, k0 > 0, and α is a constant with units of inverse length. The cart is released from rest at x = A, where 1 + αx remains positive over the entire motion. Tasks Part A. Derive the spring potential-energy function U(x), choosing U(0) = 0.Part B. Derive the speed of the cart as a function of position, v(x), while it moves from x = A toward equilibrium.Part C. Determine the cart's speed as it passes through x = 0.Part D. Write, but do not evaluate, a definite integral that gives the time required for the cart to move from x = A to x = 0.Part E. Linearize the equation of motion for sufficiently small displacement and determine the corresponding small-amplitude angular frequency ω0.Part F. Determine the magnitude of the cart's acceleration at the instant it is released. State how this compares with the acceleration predicted by the small-amplitude model.