1. Aim: How do we explain the motion of a particle attached to a spring?

2. Oscillations and Simple Harmonic Motion
Simple Harmonic Motion- when a particle moves under the influence of a linear restoring force.

3. Hooke’s Law and Simple Harmonic Motion
F = -kx and F = ma So –kx = ma and a=-k/m The acceleration is proportional to the displacement of the particle from equilibirum and is in the opposite direction

4. Thought Question 1
A block on the end of a spring is pulled to position x – A and released. In a fully cycle of its motion, through what total distance does it travel? a) A/2 b) A c) 2A d) 4A

5. Mathematical Representation of Simple Harmonic Motion
Since a = -(k/m) x, we can write d2x/dt2 = -(k/m) x This is a differential equation for simple harmonic motion

6. Solving the differential equation
d2x/dt2 = -(k/m) x We let ω2 = k/m So we can write d2x/dt2 = –ω2x and so x(t) = Acos(ωt + φ)

7. Analyzing the cosine function for simple harmonic motion
x(t) = Acos(ωt + φ) A= amplitude of the motion which is the maximum value of the position of the particle in either the positive or negative direction ω= the angular frequency and has the units rad/s (ω=√(k/m)) φ=the phase constant which describes the phase of the oscillation at t= 0s

8. More oscillation vocabulary
T= the period of motion which is the time required for the particle to go through one full cycle of motion T = 2π/ω f= the frequency which represents the number of oscillations per second f= 1/T = ω/2π

9. For spring-mass systems
T = 2π√(m/k) f=1/2π √(k/m)

10. Velocity and Acceleration Functions for Simple Harmonic Motion
x(t) = Acos(ωt + φ) Can you use calculus to find v(t) and a(t) for a particle undergoing simple harmonic motion?
v = -ωAsin(ωt + φ) a=-ω2Acos(ωt + φ)

11. Expressions for maximum velocity and maximum acceleration
If the amplitude A is the maximum position, what are the maximum velocities and accelerations of a particle undergoing simple harmonic motion? Vmax = ωA amax = ω2A
Vmax = ωA amax = ω2A

12. Thought Question 2
Is a bouncing ball an example of simple harmonic motion? Is the daily movement of a student from home to school and back simple harmonic motion? NO AND NO because neither can be represented by a sine or cosine curve

13. Block-Spring System Problem 1
A block with a mass of 200 g is connected to a light horizontal spring of force constant 5 N/m and is free to oscillate on a horizontal, frictionless surface.
If the block is displaced 5 cm from equilibrium and released from rest, find the period of its motion.
Determine the maximum speed and maximum acceleration.
Express the position, velocity, and acceleration of this object as functions of time, assuming that the phase constant φ=0
a) 1.26 s b) .250 m/s m/s c) x= 0.05 cos5t v=-.25sin5t a=-1.25cos5t

14. Problem 1 Solutions T=2π√(m/k) m=.2 k=5 so T=1.26s Vmax = Aω
ω=√(k/m)=√(5/.2) = 5 rad/2
A=0.05
So vmax = .25 m/s
amax =Aω2=0.05(5)2=1.25m/s2
c) x= 0.05 cos5t v=-.25sin5t a=-1.25cos5t

15. Problem 2: An oscillating particle
A particle oscillates with simple harmonic motion along the x axis. Its position varies with time according to the equation,
X= (4.0m) cos(πt - π/4) where t is in seconds
Determine the amplitude, frequency, and period of the motion
Calculate the velocity and acceleration of the particle at any time t.
What are the position and velocity of the particle at time t = 0?
Determine the position, velocity, and acceleration of the particle at t = 1.0 s
Determine the maximum velocity and the maximum acceleration of the particle
a) 4m, .5Hz,2s b) -4πsin(πt + π/4) π^2cos(πt + π/4) c) 2.83 m and m/s d) m, v= 8.89 m/s,a=27.9m/s^2 e) 12.6 m/s 39.5m/s^2

16. Problem 2 Solutions A=4m ω = π rad/s T=2π/ω=2π/π = 2s f=1/T = 0.5 Hz
b) v(t)=-4πsin(πt + π/4) a(t)=-4π2cos(πt + π/4)
c) x(0)=2.83 m and v(0)= m/s a(0)=-27.9m/s2
d)x(1)=-2.83 m v(1)=8.89 m/s a(1)=27.9m/s2
e) Vmax =Aω=4π=4(3.14)=12.6m/s
amax = Aω2=4(3.14)2=39.5 m/s2

17. Thought Question 3
Which of the following relationships between the acceleration a and the displacement x of a particle involve SHM:
a=0.5x
a=400x2
a=-20x
a=-3x2

18. Thought Question 4
Given x = (2.0m)cos(5t) for SHM and needing to find the velocity at t=2s, should you substitute for t and then differentiate with respect to t or vice versa

19. Thought Question 5
Which of the following points corresponds to the position –xm ? 2
At point 4 is the velocity negative, positive, or zero? Positive
At point 5, is the particle at:
a) -xm , b) at +xm , c) at 0, d) between –xm and 0, e) between xm and 0

20. Thought Question 6
Which of the following phase constants is represented by the x-t graph below?
–π<Ф<-π/2
π<Ф<3π/2
-3π/2<Ф<-π

21. Thought Question 7
Is the particle stationary at point A or point B? If not, what direction is the particle moving at point A and/or point B? A moves in the – direction. B moves in the + direction
Where in its oscillation is the particle located at point A?
In between –xm and 0
c)Where in its oscillation is the particle located at point B?
d)Is the speed of the particle increasing or decreasing at point A?
The speed is decreasing
e)Is the speed of the particle increasing or decreasing at point B?
The speed is increasing