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,