1

08 m

ar 2

019

Today:• SHO• Electric force and field

SHM (the SHO)

Simple harmonic motion approximates many, many physical systems!

Terms associated with SHM: Hooke’s Law = F = -kx, where k is the ... frequency = f

period = T = 1/f amplitude = xm

angular frequency = T = 2B/T = 2Bf phase constant = n displacement = x = xm cos(T t + n)

2

SHO

The solution to the SHO differential equation,

is

2

2

d xm k x

dt

( ) cos( )x t A t

SHO - Oscillation Relate to potential energy “wells” and

turning points.

With no friction (“damping”), mechanical energy is conserved: K + Us= constant

Linear oscillations (motion in regions where Hooke’s Law is valid) depend on k and m:

21

2sU k x

k

m 2

mT

k

3

Visualizing E conservation via potential energy curves

E

U(-2)

K(-2)

Turning pointsU(J)

Animation of Circular/SHM

SHO – relation to circular motion

4

The Equation of Motion

Using Hooke’s Law for the net force on a horizontal oscillator (mass on a spring):

Fnet = m a

- k x = m d2x/dt2

d2x/dt2 = -(k/m) x

This is the “differential equation” of SHM.It’s “solutions” are sines (or cosines).

They’re really the same; just differ by phase!

Phase Shifts

If Φ is positive, the shift is to the left.

In the plot, the increments are in 0.1 radians

x = xm cos(T t + n)

ωt

5

Phase Shifts

If Φ is negative, the shift is to the right.

In the plot, the increments are in 0.1 radians

x = xm cos(T t + n)

ωt

For

sol

ns

of f

orm

x =

xm

cos(T

t + n

)

6

Which of the following describe φfor the SHM of the figure?

A. 0 < φ < π/2

B. π/2 < φ < π

C. π < φ < 3π/2

D. - π/2 < φ < 0

E. - π < φ < - π/2

F. - 3π/2 < φ < - π

Assume x = xm cos(T t + n)

ωt

7

Assume x = xm cos(T t + n)

Which of the following describe φ for the SHM of the figure?

ωt

A. 0 < φ < π/2

B. π/2 < φ < π

C. π < φ < 3π/2

D. - π/2 < φ < 0

E. - π < φ < - π/2

F. - 3π/2 < φ < - π

Now, a little tangent …

8

Newton’s Law of Gravity

A uniform spherical shell of mass exerts no net gravitational force on a particle inside.

A uniform spherical shell of mass attracts a particle outside the shell as if all the shell’s mass were concentrated at its center.

rr

mmGF ˆ

221

The Shell Theorems

whereG = 6.67 x 10-11 Nm2/kg2

The superposition principle holds for the gravitational force!

j

i

m1

m3m2

x

y

12F 13F

1 22

ˆgrav

m mF G r

r

θ

j

i

q1

q3q2

x

y

12F 13F

1 22

1ˆ

4Coulo

q qF r

r

θ

[for q1 opposite in sign to q2 & q3]

Gravitation! Electrostatics!

In both cases, the free-body diagrams and the general net forceExpressions are identical:

12 13 13 13 12ˆ ˆsin ( cos )netF F F i F j F F

we just have to calculate the magnitudes of the forces from the respective force laws.

9

Happy weekend!