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A mass slides down a frictionless ramp of height h. Its initial speed is zero. Its final speed at the bottom of the ramp is v. +y h 0 As the mass descended, its KE, PE, and total Energy: A: B: C: D: KE increased decreased increased decreased L22 W 10/15/14 a+er lecture PE increased decreased decreased increased total Energy increased decreased stayed the same stayed the same 1 Trials and tribulations of friction – a skater encounters a carpet. https://www.youtube.com/watch? annotation_id=annotation_3308747645&feature=iv&src_vid=SRHoPL0WtdM&v=e pSrFV2Wtjs Or a shorter version at: https://www.youtube.com/watch?v=SRHoPL0WtdM L22 W 10/15/14 a+er lecture 2 Assignments For this week: • You should have read up through Ch. 6 and Ch. 7 of Wolfson and Prof. Dubson’s notes. • Read Ch. 8 of Wolfson by Friday. • Complete HW 7 this week. CAPA 8 is now live. • Midterm 2 will be Thursday of next week: two old exams on D2L. • LA applications will close Oct 24: lacentral.colorado.edu Today: • Continue: Conservation of Energy: mechanical and total. • Introduce: Power, today or Friday. L22 W 10/15/14 a+er lecture 3 1. Work: WF = ∫ F ⋅dr (in some circumstances: F ⋅ Δr ) f i Component of “specific” force in direction of displacement 2. Kinetic Energy: 3. Work – KE Principle: 1 2 KE = mv 2 Wnet = WFnet = ΔKE = KEf - KEi (Point-like object.) L22 W 10/15/14 a+er lecture 4 4. Potential energy: PE is the amount of work done on a system by an external conservative force when KE does not change and no heat flows (no friction, dissipation): PEgrav = mgΔy ΔPE = Wext when ΔKE = 0 1 2 PEspring = kx 2 The work done by some other forces depends on the path taken: e.g., friction. Under the action of these forces, mechanical energy is lost. L22 W 10/15/14 a+er lecture 5 Claim: F"internal" = − 4. Potential energy: dU dx Definition: Fexternal = force “external” to the system (e.g., my hand lifting an eraser against gravity or stretching a spring). Definition: Finternal = force “internal” to the system (e.g., gravity or the force of the spring). Notation: U ≡ PE dU grav d Fgrav = − = − ( mgx ) = −mg ✔ dx dx For Example: dU elastic d ⎛ 1 2⎞ Felastic = − = − ⎜ kx ⎟ = −kx ⎠ dx dx ⎝ 2 L22 W 10/15/14 a+er lecture ✔ 6 5. Conservation of “Mechanical Energy”: Emechanical = KE + PE = constant (no friction, no heat dissipation) or KEi + PEi = KEf + PEf L22 W 10/15/14 a+er lecture 7 A hockey puck slides without friction along a frozen lake toward an ice ramp and plateau as shown. The speed of the puck is 4m/s and the height of the plateau is 1m. Will the puck make it all the way up the ramp? +y 0 A) Yes B) No C) Impossible to tell without knowing the mass of the puck. To make it up the ramp: KEi ≥ PE f 1 2 KEi = mv = (8m)J 2 PE f = mgh = mg = (9.8m)J ∴ KEi < PE f so the puck will not make it up the ramp. L22 W 10/15/14 a+er lecture 8 Example. A block of mass m is released from rest at height H on a frictionless ramp. It strikes a spring with spring constant k at the end of the ramp. How far will the spring compress, x? Ethermal = 0 KEi + PEigrav = KE f + PE elastic f 1 2 0 + mgH = 0 + kx 2 2mgH x= k L22 W 10/15/14 a+er lecture 9 A mass m is at the end of light (massless) rod of length R, the other end of which has a frictionless pivot so the rod can swing in a vertical plane. The rod is initially horizontal and the mass is pushed down with an initial speed vo . What minimum speed is required for the mass to pivot 270o to the vertical position? A) v0 = gR B) v0 = 2gR Ei = E f C) v0 = 2 gR 1 2 mv0 = mgR 2 D) v0 = 6gR L22 W 10/15/14 a+er lecture PE = 0 → v0 = 2gR 10 5. Conservation of “Mechanical Energy”: Emechanical = KE + PE = constant (no friction, heat, dissipation) or KEi + PEi = KEf + PEf 6. General Statement of the Conservation of Energy: Etotal = Emechanical + Ethermal = constant Although we’ll use this statement, it is somewhat confusing because it’s important to remember that although thermal energy (TE) is produced as a physical system evolves, it cannot be converted to mechanical energy (ME). Therefore, ME can covert to TE but TE cannot convert to ME directly. L22 W 10/15/14 a+er lecture 11 5. Conservation of “Mechanical Energy”: Emechanical = KE + PE = constant (no friction, heat, dissipation) or KEi + PEi = KEf + PEf 6. General Statement of the Conservation of Energy: Etotal = Emechanical + Ethermal = constant or KEi + PEi = KEf + PEf + Ethermal (Typically: Wfriction = -Ethermal) L22 W 10/15/14 a+er lecture Initial thermal energy is typically considered = 0, so Ethermal is placed on the RHS of the eqn and considered to be the thermal energy produced during the evolution of the system. 12 A mass slides down a rough ramp of height h. Its initial speed is zero. Its final speed at the bottom of the ramp is v. +y h 0 As the mass descended, Mechanical Energy and Total Energy: A: B: C: D: Mechanical Energy increased decreased increased decreased L22 W 10/15/14 a+er lecture Total Energy increased decreased stayed the same stayed the same 13 KE + PE + Etherm = constant A block of mass m slides down a rough ramp of height h. Its initial speed is zero. Its final speed at the bottom of the ramp is v. m h Which of the following expressions would give the amount of thermal energy, Ethermal, released from the block’s motion down the ramp? KE + PE = KE + PE + E i A) mgh + 1 2 mv 2 1 2 mv − mgh 2 E) mgh C) B) mgh − D) 1 2 mv 2 L22 W 10/15/14 a+er lecture 1 2 mv 2 i f f thermal 1 2 dU 0 + mgh = mv =+−0 + Ethermal F"internal" 2 dx 1 2 Ethermal = mgh − mv > 0 2 14 KE + PE + Etherm = constant A block of mass m slides down a rough ramp of height h. Its initial speed is zero. Its final speed at the bottom of the ramp is v. m Ethermal 1 2 h = mgh − mv > 0 2 What is the Work done by friction, Wfriction? 1 2 A) mgh − mv 2 C) mgh L22 W 10/15/14 a+er lecture 1 2 B) mv − mgh 2 1 2 D) mv 2 W friction = −Ethermal 1 2 = mv − mgh < 0 2 15 A small track, starting at rest, slides without friction along a frictionless rail in the shape of a loop. The maximum height of the track is the same as the initial height of the ball. ASSUME NO FRICTION. Will the ball make it to the top of the loop? A) Yes, the ball will make it to the top of the loop. B) No, the ball will not make it to the top. C) Not enough info to say. L22 W 10/15/14 a+er lecture 16