1. USSC3002 Oscillations and Waves Lecture 1 One Dimensional Systems Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

2. NEWTON’S SECOND LAW The net force on a body is equal to the product of the body’s mass and the acceleration of the body. Question: what constant horizontal force must be applied to make the object below (sliding on a frictionless surface) stop in 2 seconds? 2

3. STATICS Why is this object static (not moving) ? What are the forces acting on this object? What is the net force acting on this object? 3

4. VECTOR ALGEBRA FOR STATICS The tension forces are The gravity force is 4

5. TUTORIAL 1 1. Analyse the forces on an object that slides down a frictionless inclined plane. What is the net force? Compute the time that it takes for an object with initial speed zero to slide down the inclined plane. 5

6. WORK-KINETIC ENERGY THEOREM Consider a net force that is applied to an object having mass m that is moving along the x-axis The work done is Newton’s 2 nd Law Chain Rule Kinetic Energy 6

7. POTENTIAL ENERGY Definition and in that case we can also compute the work as is a potential energy function if so the total energyis constant since 7

8. VIBRATIONS IN A SPRING For an object attached to a spring that moves horizontally, the total energy is is conserved, therefore where is the angular frequency is the phase, and is the period. is the amplitude (where 8

9. NONLINEAR VIBRATIONS Since the energy for a pendulum L is the constant we can computefrom the nonlinear ODE 9

10. TUTORIAL 1 2. Show that the solution x(t) of the spring problem satisfies a second order linear ODE and derive this ODE directly from the equation E = constant. 4. Derive an approximation for E for pendulum if and use it to derive a linear approximation for the equation of motion for a pendulum. 3. Show directly that the set of solutions of the ODE in problem 2 forms a vector space (is closed under multiplication by real numbers and under addition). This is the case for all linear ODE’s. 10