Physics 321
Hour 8
Potential Energy in Three Dimensions
Gradient, Divergence, and Curl
Bottom Line
We can use conservation of energy in three ways to
describe the motion of an object:
1)
1
𝑚𝑣 2
2
+𝑈 𝑥 =𝐸
2)
1
𝑚𝑥 2
2
+ 𝑈 𝑥 = 𝐸 – Differential equation (hard)
3) 𝑇 + 𝑈 = 0 – Differential equation (easier in 1D)
A Problem
We’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 Laws
A sphere rolls without slipping down an incline. Given
m, R, and θ, find the acceleration.
Conservation of Energy I
A 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 II
A sphere rolls without slipping down an incline. Given
m, R, and θ, find x(t).
Since 𝑣 = 𝑥, 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 𝑇 + 𝑈 = 0.
A Pendulum Problem
(a) Write T and U as functions of
theta.


R
m
A Pendulum Problem
(a) Write T and U as functions of
theta.


R
m
1 2 2 1 2
T  m   I
2
2
 1 2 1 2  2
    R m
5 
2
U   mg cos  constant
A Pendulum Problem
(b) Initial conditions: θ(0)=θ0, θ(0)=0
Find θ(t) = ω(t).
T0  U 0  E  T  U

 mg cos 0

R
m
1 2 2 1 2
 m   I  mg cos
2
2
2mgcos  cos 0 
2

 
m 2  I
A Pendulum Problem
(c) Using this equation in Mathematica,
solve for θ(t).


R
m
A Pendulum Problem
(d) Find an equation of motion using




1
T  m 2  I  2
2
U  mg cos
T+U = 0.


1

T  m 2  I 2
2
U  mg sin  
R
m
m
2

 I   mg sin   0
A Pendulum Problem
(e) Use Mathematica to solve this problem.
m


R
m
2

 I   mg sin   0