Physics 101: Lecture 20, Pg 1

Lecture 20: Lecture 20: Ideal Spring and Simple Harmonic Ideal Spring and Simple Harmonic

MotionMotion New Material: Textbook Chapters 10.1 and 10.2

Physics 101: Lecture 20, Pg 2

Ideal SpringsIdeal Springs

Hooke’s Law:Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position.

FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality.

(often called “spring constant”)

relaxed position

FX = 0

xx=0

Physics 101: Lecture 20, Pg 3

Ideal SpringsIdeal Springs

Hooke’s Law:Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position.

FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality.

(often called “spring constant”)

relaxed position

FX = -kx > 0

xx 0

x=0

Physics 101: Lecture 20, Pg 4

Ideal SpringsIdeal Springs

Hooke’s Law:Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position.

FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality.

(often called “spring constant”)

FX = - kx < 0

xx > 0

relaxed position

x=0

Physics 101: Lecture 20, Pg 5

Concept QuestionConcept Question

In Case 1 two people pull on the same end of a spring whose other end is attached to a wall. In Case 2, the same two people pull with the same forces, but this time on opposite ends of the spring. In which case does the spring stretch the most?

1. Case 12. Case 23. Same

Case 1

Case 2

CORRECT

Physics 101: Lecture 20, Pg 6

What does moving along a circular path have to do with moving back & forth in a straight line (oscillation about equilibrium) ??

y

x

-R

R

0

1 1

2 2

3 3

4 4

5 5

6 62

R

8

7

8

7

23

x

x = R cos = R cos (t)since = t

Physics 101: Lecture 20, Pg 7

Concept QuestionConcept Question

A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The speed of the mass is constant

+A

t-A

x

CORRECT

Physics 101: Lecture 20, Pg 8

Concept QuestionConcept Question

A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The acceleration of the mass is constant

+A

t-A

x

CORRECT

Physics 101: Lecture 20, Pg 9

X=0

X=AX=-A

X=A; v=0; a=-amax

X=0; v=-vmax; a=0

X=-A; v=0; a=amax

X=0; v=vmax; a=0

X=A; v=0; a=-amax

Springs and Simple Harmonic MotionSprings and Simple Harmonic Motion

Physics 101: Lecture 20, Pg 10

Simple Harmonic Motion:Simple Harmonic Motion:

x(t) = [A]cos(t)

v(t) = -[A]sin(t)

a(t) = -[A2]cos(t)

x(t) = [A]sin(t)

v(t) = [A]cos(t)

a(t) = -[A2]sin(t)

xmax = A

vmax = A

amax = A2

Period = T (seconds per cycle)

Frequency = f = 1/T (cycles per second)

Angular frequency = = 2f = 2/T

For spring: 2 = k/m

OR