1. Harmonic Motion (III)
Physics 1D03 - Lecture 33

2. Recall: Velocity and Acceleration
Physics 1D03 - Lecture 33

3. where t is in seconds and the angles in radians.
Example
An object oscillates with SHM along the x-axis. Its displacement from the origin varies with time according to the equation: x(t)=(4.0m)cos(πt+π/4)
where t is in seconds and the angles in radians.
determine the amplitude
determine the frequency
determine the period
its position at t=0 sec
calculate the velocity at any time, and the vmax
calculate the acceleration at any time, and amax
Physics 1D03 - Lecture 33

4. When do we have Simple Harmonic Motion ?
A system exhibits SHM is we find that acceleration is
directly proportional to displacement:
a(t) = - w 2 x(t)
SHM is also called ‘oscillatory’ motion. Its is called ‘harmonic’ because the sine and cosine function are called harmonic functions, and they are solutions to the above differential equation – lets prove it !!! SHM is ‘periodic’.
Physics 1D03 - Lecture 33

5. Mass and Spring M Newton’s 2nd Law: F = -kx so x
This is a 2nd order differential equation for the function x(t). Recall that for SHM, a = -w 2 x : the above is identical except for the proportionality constant. Hence, we must have:
or:
Physics 1D03 - Lecture 33

6. Find the spring constant (aka force constant of the spring).
Example
A 7.0 kg mass is hung from the bottom end of a vertical spring fastened to the ceiling. The mass is set into vertical oscillations with a period of 2.6 s.
Find the spring constant (aka force constant of the spring).
Physics 1D03 - Lecture 33

7. The block is displaced 5.0cm from equilibrium and released from rest.
Example
A block with a mass of 200g is connected to a light spring with a spring constant k=5.0 N/m and is free to oscillate on a horizontal frictionless surface.
The block is displaced 5.0cm from equilibrium and released from rest.
find the period of its motion
determine the maximum speed of the block
determine the maximum acceleration of the block
Physics 1D03 - Lecture 33

8. The mass moves, and after 0.50 s, the speed of the mass is zero.
Example
A 1.00 kg mass on a frictionless surface is attached to a horizontal spring. The spring is initially stretched by 0.10 m and the mass is released from rest.
The mass moves, and after 0.50 s, the speed of the mass is zero.
What is the maximum speed of the mass ???
Physics 1D03 - Lecture 33