Download Chapter 7 Conserva(on of Energy (cont`d) Recollect: Conserva(on of

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Mechanical energy is of two forms:
Chapter 7 Conserva.on of Energy (cont’d) kinetic (by motion)
potential (by relative positions)
kinetic  potential transformable.
Mechanical energy conserva.on Examples Both of the ability to do work (work-energy theorem).
Work by non‐conserva.ve forces Energies in Nature
come with several (known) forms:
Mechanical; Electromagnetic & Chemical (heat);
Atomic or nuclear …
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Recollect: Conserva.on of Energy •  If only conserva.ve forces are present, the total kine.c plus poten.al energy of a system is conserved.  Mechanical Energy = Poten.al Energy (U) + Kine.c Energy (K) Conserva.ve forces interchange U  K (work done), but E = K + U is a constant. ΔE = ΔK + ΔU = 0 Work‐kine.c energy theorem: ΔK = W thus ΔU = ‐W, for conserva.ve forces only. Phys 201, Spring 2011
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Example: The simple pendulum •  Suppose we release a mass m from rest a height h1 above its lowest possible eleva.on. Assuming no fric.on (air drag): –  What is the maximum speed of the mass and where does this happen? –  To what eleva.on h2 does it rise on the other side? m
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Example: The simple pendulum •  Total mechanical energy: E = ½mv2 + mgy –  Ini.ally, y = h1 and v = 0, so E = mgh1. –  Since E = mgh1 ini.ally, and energy is conserved, E = mgh1 at all .mes. y
Example: The simple pendulum E = ½mv2 + mgy Velocity is maximum where poten.al energy is lowest, at boYom of the swing. So, at y = 0, ½mv2 = mgh1  v2 = 2gh1  v = (2gh1)1/2 y
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Example: The simple pendulum Example: Airtrack and Glider •  To find maximum eleva.on on other side, note that maximum is reached when v=0. Since E = mgh1, and maximum poten.al energy on right is mgh2, h2=h1. The ball returns to its original height. •  A glider of mass M is ini.ally at rest on a horizontal fric.onless track. A mass m is aYached to it with a massless string hung over a massless pulley as shown. What is the speed v of M a^er m has fallen a distance d? v
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Example: Airtrack and Glider Problem: Toy car •  Kine.c + poten.al energy is conserved since all forces are conserva.ve. •  Choose ini.al configura.on to have U=0. ΔK = ‐ΔU •  A toy car slides on a fric.onless track shown below. It starts at rest, drops a height d, moves horizontally at speed v1, rises a height h, and ends up moving horizontally with speed v2. –  Find v1 and v2. v
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A projec.le of mass m is propelled from ground level with an ini.al kine.c energy of 450 J. At the exact top of its trajectory, its kine.c energy is 250 J. To what height, in meters, above the star.ng point does the projec.le rise? Assume air resistance is negligible. Problem: Toy car •  K+U is conserved, so ΔK = ‐ΔU •  When the eleva.on decreases a distance D, ΔU = ‐mgd, ΔK = ½mv12. •  Solving for the speed: A)  50/(mg)
B)  250/(mg)
C)  700/(mg)
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Ques.on •  A box sliding on a horizontal fric.onless surface runs into a fixed spring, compressing it to a distance x1 from its relaxed posi.on while momentarily coming to rest. –  If the ini.al speed of the box were doubled and its mass were halved, what would be the distance x2 that the spring would compress? A) x2=x1 B) x2=x1√2 C) x2=2x1 ½ kx2 = ½ mv2, Thus v = x(k/m)1/2  x = v(m/k)1/2 So, 2v and m/2 increases x by √2. x
Non‐conserva.ve forces: •  If the work done does not depend on the path taken, the force is said to be conserva.ve. •  If the work done does depend on the path taken, the force is said to be non‐conserva.ve. •  An example of a non‐conserva.ve force is fric.on. Work done is propor.onal to the length of the path! –  The mechanical energy is converted to heat. 3/3/11
Spring pulls on mass: with fric.on •  Suppose spring pulls on block, but now there is a nonzero coefficient of fric.on μ between the block and the floor. •  The total work done on the block is now the sum of the work done by the spring, Ws (same as before), and the work done by the fric.on Wf, (not related to either kine.c energy or poten.al energy) Work‐energy theorem now reads: Wnet = Wcons + Wf = ΔK m
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Many forms of energy: Ques.on Which statement is true? A. Mechanical energy (U+K) is always conserved B. Total energy is always conserved C. Poten.al energy is always conserved Energy of one form can be converted to
another form, but the total energy remains
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Newton’s laws  Conserva.on of energy F = m dv/dt = m dv/dx v
F dx = m v dv
When Fnc=0, then –ΔU = ΔEk
But Conserva.on of energy is more broadly applicable than Newton’s laws •  Newton’s laws do not apply to systems that are fast‐moving close to the speed of light (where Einstein’s theory of rela.vity applies) and to very small systems (where quantum mechanics applies), but conserva.on of energy is always valid. Conserva.on laws are the consequence of symmetries: Energy conserva.on  Time transla.on invariance. 3/3/11
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