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10/17/2011 Announcements Clicker Question • CAPA Set #9 due Friday at 10 pm • This week in Section: Lab #3 – Momentum Room Frequency BA Two paths lead to the top of a big hill. Path #1 is steep and direct and Path #2 is twice as long but less steep. Both are rough paths and you push a box up each. How much more potential energy is gained if you take the longer path? • Reading – Chapter 7 on Momentum A) none B) twice as much C) four times as much D) half as much d h #1 2d h #2 θ d 2 h2 φ 4d 2 h2 PE mgh in both cases. Clicker Question Room Frequency BA Springs Consider the same two paths as in the previous problem. Is the amount of work done the same along each path? A) Yes B) No, more work is done on the steeper path. C) No, more work is done on the longer path. D) Probably not, but the details are needed to know which is larger. d h #1 d h h #2 mg φ 4d 2 h2 2 | W f | Ff d Nd (mg cos )d d 2 h2 d mg d 2 h 2 d Force Compressed spring pushes back Force Stretched spring pulls back 2d θ 2 Spring in “relaxed” position exerts no force | W f | Ff d N(2d) (mg cos )(2d) mg Hooke’s “Law” Force = -kx (2d)2 h 2 2d mg 4d 2 h 2 2d Clicker Question Stored Energy in Springs Compressed spring has elastic potential energy Stretched spring also has elastic potential energy If no change in any other energy, then PE = Wext =Fspring x d However Fspring = -kx changes as we stretch the spring Use average Fspring = -½ kx Elastic potential energy: k = “spring constant” units = N/m x = displacement of spring PE = ½ kx2 Room Frequency BA A force of magnitude F stretches a spring through a displacement x. The force is then increased so that displacement is doubled. F = -kx, PE = ½kx2 What is the magnitude of force needed to double the displacement and what will be the effect on the potential energy of doubling displacement? A) B) C) D) E) Force doubles, PE doubles. Force doubles, PE quadruples. Force quadruples, PE doubles. Force doubles, PE halves. Force quadruples, PE quarters. 1 10/17/2011 Clicker Question Room Frequency BA Which of the following types of potential energy can be negative? A) Gravitational Potential Energy PEgravity = mgy B) Elastic Potential Energy PEelastic 1 2 kx 0 2 C) Both of the above D) Neither of the above 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 (i.e. x)? 0 0 KEi + PEi + Wfrict + Wexternal = KEf + PEf 0 + mgH + 0 + x 2mgH k = 0 + (0 + ½ kx2) 0 gravitational elastic Elastic Potential Energy – Does Not Have to Look Like a Spring What is Power? Power is a rate of energy per time P average power = work time Units are Joules / second = Watts The average power can be written in terms of the force and the average velocity: 2 10/17/2011 Raise a box of mass 2 kg at constant velocity by a height of 6 meters over a time interval of 2 seconds. How much energy is required to do the work? Wext = PE = mgh 6m Also Wext = Fext x d Fext = 117J/6m = 19.6 N How much power is required to do the work in that amount of time? Power = Wext / time = 117 J / 2 s = 59 Watts Also Power = Fext v = 19.6 N x (6m/2s) = 59 Watts Collisions m1 Approximately how many 100 Watt lightbulbs would need to be left on all the time to amount to that energy consumption? xx A) 1 lightbulb B) 10 lightbulbs C) 100 lightbulbs D) 1000 lightbulbs Energy = Power x time = (10 x 100 W) x (30x24x60x60 s) E) 10,000 lightbulbs ≈ 2.6 x 109 Joules Elastic Collision Before the collision: v2i v1i Room Frequency BA A kiloWatt-hour is a unit of energy (Power x time). 1 kW-hour = 3.6 x 106 Joules The average household in the USA uses 2.7 billion Joules of electrical energy each month. since KE=0 Wext = 2 kg x 9.8 m/s x 6 m = 117 Joules 2 kg Clicker Question m2 Total kinetic energy remains constant during the collision KE is conserved or ΔKE = 0 After the collision: v1f m1 v2f No energy converted to thermal or potential energy m2 v1f = final velocity of mass 1 v2f = final velocity of mass 2 These collisions should obey Newton’s Laws Demonstration KE1i KE2i KE1 f KE2 f 1 1 1 1 m1v1i 2 m2 v2i 2 m1v1 f 2 m2 v2 f 2 2 2 2 2 New Conserved Quantity Momentum momentum p mv (a vector) Kind of like kinetic energy (associated with motion). Macroscopic objects never have perfectly elastic collisions but it’s often a good approximation. Inelastic Collision: KE is not constant; some is converted to another kind of energy; typically heat or deformation. Totally Inelastic Collision: KE is not conserved, and the objects stick together after collision. However, not in units of Joules (instead kg m/s) and so not an energy. In addition to total energy being conserved in every collision, the total momentum vector is also conserved! p1i p2i p1 f p2 f mv1i mv2i mv1 f mv2 f Conservation of Momentum (in one-dimension) 3