Physics 321

Hour 8Potential Energy in Three Dimensions

Gradient, Divergence, and Curl

Bottom LineWe can use conservation of energy in three ways to describe the motion of an object:

1)

2) – Differential equation (hard)

3) – Differential equation (easier in 1D)

A ProblemWe’ll solve a simple problem using different methods. A sphere rolls without slipping down an incline. We are given m, R, and θ.

Newton’s LawsA sphere rolls without slipping down an incline. Given m, R, and θ, find the acceleration.

Conservation of Energy IA sphere rolls without slipping down an incline. Given m, R, and θ, find the velocity.

Identify all Ts, Us. ΣT+ΣU = E = E0. Gives v(y).

Conservation of Energy IIA sphere rolls without slipping down an incline. Given m, R, and θ, find x(t).

Since , solve a differential equation for x(t).

Conservation of Energy III (a)A sphere rolls without slipping down an incline. Given m, R, and θ, find x(t).

1) Write T and U.2) Write equations of constraint among variables.

Conservation of Energy III (b)A sphere rolls without slipping down an incline. Given m, R, and θ, find x(t).

Use constraints to write T and U in terms of independent variables, then solve

A Pendulum Problem

R

m

(a) Write T and U as functions of theta.

A Pendulum Problem

R

m

(a) Write T and U as functions of theta.

constantcos

5

1

2

12

1

2

1

222

222

mgU

mR

ImT

A Pendulum Problem

R

m

(b) Initial conditions: θ(0)=θ0, θ(0)=0Find θ(t) = ω(t).

Im

mg

mgIm

mg

UTEUT

202

222

0

00

coscos2

cos2

1

2

1

cos

A Pendulum Problem

R

m

(c) Using this equation in Mathematica, solve for θ(t).

A Pendulum Problem

R

m

(d) Find an equation of motion using

T+U = 0.

cos2

1 22

mgU

ImT

sin

22

1 2

mgU

ImT

0sin2 mgIm

A Pendulum Problem

R

m

(e) Use Mathematica to solve this problem.

0sin2 mgIm