Topics - - C6: Conservation of Angular Momentum - Radians - Angular Position 𝑠 = 𝑟𝜃 𝛥𝜃 - Angular Velocity |𝜔 ⃗ | = 𝑙𝑖𝑚𝛥𝑡 −> 0 | 𝛥𝑡 | - |𝑣| = 𝑟|𝜔 ⃗| - ⃗ = 𝐼𝜔 Angular Momentum: 𝐿 ⃗ - Twirl: 𝛥⃗𝐿 - Torque: 𝜏 = - ⃗𝑖 = 𝐿 ⃗𝑓 𝐿 ⃗ 𝑑𝐿 𝑑𝑡 C7: More About Angular Momentum - Cross Product (Right Hand Rule) - Moment of Inertia - ⃗𝑖 = 𝐿 ⃗𝑓 Conservation of Angular Momentum 𝐿 ⃗ =𝑟 ×𝑝 - For Point Objects: 𝐿 - ⃗ = 𝐼𝜔 For Rigid Bodies 𝐿 ⃗ ⃗ =𝐿 ⃗ 𝑟𝑜𝑡 + 𝐿 ⃗ 𝑐𝑚 Rotating and Moving Bodies: 𝐿 - Torque exerted at a specific point:: 𝜏 = ⃗𝑟 × ⃗⃗𝐹 - - C8: Conservation of Energy - Basic Energy Model 1 - Kinetic Energy: 𝐾 = 2 𝑚|𝑣|2 - Potential Energy Gravitational Potential Energy (General): 𝑉𝐺 = − - Gravitational Potential Energy (Earth): 𝑉𝑔 = 𝑚𝑔|𝑧| - Electrostatic Potential Energy: 𝑉𝑒 = 𝑘 - 𝐺𝑚1 𝑚2 - 𝑞1 𝑞2 𝑟 1 Spring Potential Energy: 𝑉𝑠𝑝 = 2 𝑘𝑠 (𝑥 − 𝑥0 )2 𝐸𝑖 = 𝐸𝑓 C9: Potential Energy Graphs - Finding 𝐸𝑡𝑜𝑡 , 𝐾, and 𝑉 on a Potential Energy Graph 𝑑𝑉(𝑥) - 𝐹𝑥 (𝑥) = − - Finding Stable and Unstable Equilibrium points 𝑑𝑥 (for motion in one line) 𝑟 - - C10: Work - Dot Product 𝑥 ⃗⃗⃗⃗ (Variable Force) 𝐹 (𝑥) ⋅ 𝑑𝑟 - 𝑊 = ∫𝑥 2 - 𝑊 = 𝐹 ⋅ 𝛥𝑟 (constant force) 𝑊𝑛𝑒𝑡 = 𝑊1 + 𝑊2 +. . . +𝑊𝑛𝑒𝑡 - Momentum Requirement: 𝑑𝐾𝑐𝑚 = 𝐹𝑒𝑥𝑡 ⋅ 𝑟𝑐𝑚 1 C14: Collisions - Elastic Collisions - Inelastic Collision - 1-D and 2-D collisions - Conservation of Momentum, Angular Momentum, and Energy - Quadratic Formula: 𝑥 = −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 Physics Homework 1) Two girls of mass 𝑚 are standing on a small sleigh that is of mass 𝑀. Assume that the sleigh is initially stationary, and that the frozen pond is frictionless. One of the girls jumps from one side of the sleigh while the other girl remains standing on the sleigh. a) Is momentum conserved in this interaction? Why or why not? Please make your reasoning clear. b) If the girl jumps so that her horizontal speed is |𝑣⃗| with respect to the ground, then what is the final speed of the sleigh with respect to the ground? a) If the second girl then jumps from the same side of the sleigh as the first girl, this time with a relative horizontal speed of 2|𝑣⃗| with respect to the sleigh, then what is the final speed of the sleigh? b) Now the girls attach a (massless) scoop to the front of the sleigh and push the sleigh so that it is moving with an initial velocity of 𝑣⃗ with nobody onboard. How fast is the sleigh moving after it has scooped up a total mass 𝑀 of snow from the surface of the frozen pond? 2) You are on a safari, conducting research on local Bengal tigers. You receive a call that a tiger has been spotted 25 km 32° north of west relative to your homebase. You then travel in a truck 10 km 40∘ east of north along the flatlands, and then climb 0.5 km vertically to a cliff to observe the tiger. a) How far is the tiger relative to you? b) Looking from the top down, what direction is the tiger relative to you? (I am looking for an angle.) 3) Scientists have been studying the expansion of the universe for decades. It was discovered that the density of matter in the universe plays a critical role in this expansion. In order for the gravitational attraction between the universe’s matter to be strong enough to prevent it from expanding forever, there must exist a critical density of matter 𝜌𝑐 . This density depends on the Hubble constant H, which determines the universe’s current expansion rate (as a fractional expansion per unit time, so its units are 1/s). It will also depend on the speed of light c, which along with the Hubble constant H helps reveal the radius of the universe we currently can receive light from, and thus how much of the universe might contribute to the attraction. a) What potential third quantity can you think of that should appear in this formula and why? (Hint: What attractive interaction is at play here, and what is a fundamental constant that is present in that interaction? Check the inside front cover of your book for a list of constants.) b) Up to a unitless constant, use dimensional analysis to come up with the formula for the critical density 𝜌𝑐 . (Assume that the third quantity referred to in a) has units of N·m2/kg2) c) Ignoring the unitless constant, what is the critical density for our universe? (𝐻 = 2.28 ∗ 10−18 𝑠 −1 )
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