Review and then some…

Work & Energy Conservative, Non-conservative, and non-constant Forces

Energy Defined again Total Mechanical Energy Total Mechanical Energy E total = PE + KE PE = mgh PE = mgh KE = ½ mv 2 KE = ½ mv 2 It is always transferred and so the total energy in a closed system is conserved. The conservation of Mechanical Energy mgh 1 + ½ mv 1 2 = mgh 2 + ½ mv 2 2 mgh 1 + ½ mv 1 2 = mgh 2 + ½ mv 2 2

Work Energy Theorem Work to get an object to change velocity is the work-energy theorem Work =  KE, F  d = ½ mv f 2 - ½ mv i 2 Since Work, Potential, and Kinetic energies, are measured in Joules, they can transfer.

Conservative Forces Total mechanical energy can usually be solved with these two main equations. F  d = ½ mv f 2 - ½ mv i 2 F  d = ½ mv f 2 - ½ mv i 2 mgh i + ½ mv i 2 = mgh f + ½ mv f 2 mgh i + ½ mv i 2 = mgh f + ½ mv f 2

Constant Force – distance Graph Consider the equation for work… In your notes, predict what a Force vs. Displacement (Work) graph looks like if the force is constant. Can you tie the graph to the equation?

Springs What would happen to force as you pull on a spring?

Spring – Work Graph The force changes as the displacement changes. It takes more force to stretch the spring a big distance rather than a small distance… so we get a triangle.

Graph to Equation What do you think the equation for work done by a spring is? Think about the graph. What does the area under the curve represent? What shape does this make? Spring force (F s ) is equal to what? (use k for slope) W s = ½F s x F s = kx, so W s = ½kx 2

Work and Springs Springs do work but they don’t have a constant force. For a spring: F = -kx (Hooke’s law) k is the spring constant for that spring (N/m) x is how much the spring is stretched/compressed (aka Δx) F is the restoring force

Spring Fun A 1.50kg object hangs motionless from a spring with a force constant of k = 250N/m. How far is the spring stretched from its equilibrium length? m

Spring Fun x2 A backpack full of books weighing 52.0N rests on a horizontal table. A spring with a force constant of 350N/m is attached to the backpack and pulled horizontally. If the spring is pulled until it stretches 2.00cm and the pack remains at rest, what is the magnitude of the force of friction keeping the backpack in place? 7.00N The backpack begins to slide just as the spring is stretched beyond 2.00cm. What is the coefficient of static friction? 0.135

Spring Work A 1.20kg block is held against a spring with spring constant (k) = 1.00x10 4 N/m, compressing it a distance of 0.150m. How fast is the block moving after it is released and the spring pushes it away (the instant it is no longer touching the spring)? 13.7m/s

Spring Potential Energy If you compress or stretch a spring and hold it there, it is considered PE (stored energy). Soooooooooooooo: W s = ½ F s x = ½ kx 2 = PE s

Spring Potential Practice When a force of 120.0N is applied to a certain spring, it causes a stretch of 2.25cm. What is the potential energy of this spring when it is compressed by 3.50cm? Clue? Solve for k k = 5330N/m PE s = 3.26J

Conservation of Energy Your total initial energy MUST equal your total final energy. There is no way around this fact! E = E’ Without Friction: PE + KE = PE’ + KE’ With Friction: PE + KE = PE’ + KE’ + Loss to Friction (in J)

Graduation For graduation, you install a spring with k=100.0 N/m under your 0.120kg cap. The spring is compressed 15.5cm. When released your cap is launched straight upward with what initial velocity? How high above the release point will your cap travel, assuming no friction? If it reaches a height of only 0.85m, what energy was “lost” to air friction? v = 4.47m/s h = 1.02m E start =1.20J, E top = 1.00J, E friction = 0.20J

More? A 1.9-kg block slides down a frictionless ramp, as shown. The top of the ramp is 1.5 m above the ground; the bottom of the ramp is 0.25 m above the ground. Suppose the ramp is not frictionless. Find the distance d for the case in which friction on the ramp does -9.7J of work on the block before it becomes airborne.

Solution Determine the horizontal velocity at point B (end of ramp) using energy. v = 3.78 d = 0.85m Now use 2-d motion to determine d.