Relationship between Force and Potential Energy

Relationship between Force and Potential Energy Energy Diagrams for Mechanical Systems Etot Etot = U(y1) + KE(y1) U(y) = mgy KE(y1) U(y1) y1 Fy...
Author: Allen McCarthy
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Relationship between Force and Potential Energy

Energy Diagrams for Mechanical Systems Etot

Etot = U(y1) + KE(y1) U(y) = mgy

KE(y1)

U(y1)

y1

Fy   dU  mg dy Note: since U = U(y)), we can determine all of the components of F:  U F iˆ  U ˆj  U kˆ  mgˆj x y z

y

Energy Diagrams for Mechanical Systems Etot

U(y) Etot = U(y1) + KE(y1) At this point, Etot = U(y), so KE = 0  v=0; 0 maximum excursion in y. Motion is bounded in y.

½ mv2

mgy gy1

y1

y

Springs E

U(x) = ½ kx2

Etot KE(x1) = ½ mv2

F   dU   kx d dx

U(x1) = ½ kx12 x1 d 2U  0  Stable Equilibrium; Force always opposite displacement dx 2

x

Stable Equilibrium? E dU/dx > 0

dU/dx < 0

F0

x d2U/dx2 < 0 everywhere  Unstable Equilibrium

Arbitrary Energy Diagrams E

U(x)

x

Quantization E E4 E3 E2 E1 E0 x

Quantization and Atomic Effects

The Work-Energy Theorem

A 2-kg block slides down a frictionless curved ramp, starting from rest at a height of 3 m. The block then slides 9 m on a rough horizontal surface before coming to rest. ( ) What (a) Wh t is i the th speedd off the th block bl k att the th bottom b tt off the th ramp?? (b) What is the energy dissipated by friction? ( ) Wh (c) What iis the h coefficient ffi i off kinetic ki i friction f i i between b the h block bl k and the horizontal surface, assuming that we can ignore any energy that goes into heating the block?

A mass m attached to a spring of spring constant k executes uniform circular motion on a frictionless horizontal table. The spring has an unstretched length of L0, and the radius of the circle of motion is R. R

((a)) Fi Find d th the potential t ti l energy stored t d in i the th spring i (b) Find the kinetic energy of the mass ( )N (c) Now suppose ffriction i i is i turnedd on between b the h mass andd the h table, and that the coefficients of kinetic and static friction are equal and small small, 

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