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Final: 2 hour long.
Final 221
Rough estimate of a 120-125 points allocation:
1. Problem from Flipit homework: 35-45 pts
2. Lecture slides: Proofs/problems: 60-70 pts
3. Cumulative section: questions from previous exams (mostly): 15-20 pts
Final 221
Ch14 Rotational Kinematics (Lecture slides):
• Define a Rigid body (4pts)
• Proof: Prove that for pure rotation around a fixed axis (8pts)
K= 12 I/o w/o2 where I/o =Smiri2
• Problem: Compute the moment of inertia of a rod around a perpendicular
axis through its end (12pts)
Ch15 Parallel Axis theorem and Torque (Lecture slides):
1
1
2
2
• Proof: Starting 𝐾𝑇𝑜𝑡 = 2 𝑀𝑣𝐶𝑀
+ 2 𝑛𝑖=1 𝑚𝑖 𝑣𝑖/𝐶𝑀
(a Ch13 result or see “Optional
1
1
2
proof” in CH15 slides) prove that 𝐾𝑇𝑜𝑡 = 2 𝑀𝑣𝐶𝑀
+ 2 𝐼𝐶𝑀 𝜔2 for a rigid body. Use this
result to prove the Parallel Axis Theorem. (14pts)
• Know how to compute the torque from τ=r Χ 𝐹
• Optional proof: proof mentioned above (bonus points)
Ch16 Rotational Dynamics: No required proofs, only problems (Lecture slides):
• Problem: Atwood machine problem (disk with strings and 2 masses hanging) (16pts)
• Problem: Disk rolling down incline: know the 3 different approaches to the solution:
point of contact solution, CM solution and energy solution. (14-18pts)
• Optional proof: proof that the CM torque equation is valid even when the CM is
accelerated. (bonus points)
Final 221
Ch17/18 Rotational Statics (Lecture slides):
• If a rigid body is static then τ=0 and 𝐹 = 0 around point of choice
• Problem: hanging sign (14 pts)
• Problem: Ladder against wall (16 pts)
• Problem: Box in truck problem (16 pts)
Final Review
Ch 19/20 (Lecture slides):
dL
• Proof: Prove that dt   starting from L  r  p and then prove that angular
momentum is conserved when no external torque is present. (8pts)
• Problem: Use 𝐿 = 𝐶𝑆𝑇 to solve the following problems (you must justify and
demonstrate that the net torque is zero to get any credit):
• person jumping on a merry-go-round (8pts)
• skaters (4pts)
• orbits (8 pts)
• the mass and string through table problem (14 pts)
• Problem: Derive the precession angular frequency for a torque constantly
perpendicular to L (14 pts)
Final Review
Ch21 Simple Harmonic motion (Lecture slides):
• Problem: Starting from F=ma derive the differential equation satisfied by a mass
m, attached at the end of a horizontal spring of constant k. Show that: 𝑥 𝑡 =
𝑥0 cos 𝜔𝑡 + 𝜑 is a solution provided that 𝜔 takes a special value. Derive that
value in terms of 𝑚 and 𝑘. (12 pts)
• Problem: Vertical spring of constant k with hanging mass m. Derive, from F=ma
the differential equation satisfied by ξ=x-xequil. and find solution, and period in
terms of m and k.
• Using a second order Taylor series expansion for the potential energy 𝑢 𝑥
associated with a force F in one dimension, prove that any system will behave
like a spring when disturbed from a stable equilibrium position. Derive the value
of the Spring constant k in terms of: 𝑢′′ 𝑎 , the second derivative of the
potential evaluated at the equilibrium position x=a. (12pts)
Final Review
Ch 22 Physical Pendulum (Lecture Slides):
• Problem: Derive, from the torque equation, the solution for 𝜃 t and the
angular frequency of a torsion pendulum subject to a torsion torque 𝜏 = 𝐾𝜃
where K is a constant. (16 pts)
• Problem: derive from the torque equation, the equation satisfied by 𝜃, the
angle from the vertical of a physical pendulum of inertia 𝐼0 . Find the solution
for θ for small oscillations, explaining why the approximation is needed. Then
derive the angular frequency of the pendulum in terms of I and other physical
parameters (16pts)
Ch 23 Harmonic Waves and Wave Equation (Lecture Slides):
• Be able to justify if a function f(x,t), of time and space, depicts a wave travelling
down the x axis. Know how to determine the velocity of the wave given the
function. (8pts)
• Proof: Prove that if a wave is periodic in time with period T, it must also be
periodic in space with wavelength 𝜆. In doing so derive the relationship between
𝜆 and 𝑇 (10 pts)
• Problem: Derive, starting from F=ma, the wave equation describing a small kink
travelling down a string of mass density 𝜇 under tension T Show that the
equation admits solutions of the form 𝑦 𝑥, 𝑡 = 𝐴 𝑐𝑜𝑠 𝑘 𝑥 − 𝑣𝑡 provided 𝑣 =
𝑇
𝜇 . (20 pts)
Final Review
Ch 24 Superposition of waves (Lecture Slides):
• Proof: Show that two harmonic waves of the same amplitude travelling in opposite
direction(say 𝑞 = 𝐴 cos 𝑘 𝑥 ± 𝑣𝑡 ), can generate standing waves. Derive the position
of the nodes in terms of k. (16 pts)
Ch 25 Static fluids (Lecture slides):
𝐹
• Proof: By considering a static fluid element prove that d𝑃 = 𝜌𝑔d𝑦 where P = 𝐴𝑁 and
𝑦 is the vertical position (12 pts)
• Problem: Derive the max height that a pump can lift a fluid of density 𝜌 at the surface of
the earth (12 pts)
Ch 26 Moving Fluids (Lecture Slides):
• Proof: Use the work energy theorem to derive Bernoulli’s equation (20 pts)
• Problems (each 6 pts; use Bernoulli to explain the phenomenon):
• Airplane wings
• Roofs in high wind
• Lift/drop on tennis balls