Chapter 13 SHM? WOD are underlined

Remember Hooke’s Law F = - k Δx New Symbol: “k” Spring constant. “Stiffness” of the spring. Depends on each spring’s dimensions and material. In N/m

Question If I let go, what will happen to the mass? Then what? Then what?

Simple Harmonic Motion Repeating up and down motion, (like cos wave.) (Draw a picture.) Motion that occurs when the net force obeys Hooke’s Law The force is proportional to the displacement and always directed toward the equilibrium position Show Example with Spring The motion of a spring mass system is an example of Simple Harmonic Motion Are springs the only type of SHM?

Simple Harmonic Motion The motion of a spring mass system is an example of Simple Harmonic Motion Are springs the only type of SHM: No, Jump Rope, Sound Waves, Pendulum, Swing, up and down motion of an engine piston

Motion of the Spring-Mass System Initially, Δx is negative and the spring pulls it up. The object’s inertia causes it to overshoot the equilibrium position. Δx is positive now and the spring pushes it down. Again it will over shoot equilibrium.

Δx, v and a versus t graphs What type of curve is this? For Calculus, Derivative of sin is what? What happens if you bump the spring?

Δx, v and a

All three look like sinusoidal curves. V is shifted backwards from Δx a is shifted backwardwards from v.

Acceleration of an Object in Simple Harmonic Motion Remember F = - k x & F = ma Set them equal to each other: - k x = ma Solve for a: a = -kΔx / m The acceleration is a function of position Acceleration is not constant. So non-inertial frame of reference. So, the kinematic equations are not valid here.

Amplitude: New Symbol “A” Amplitude, A The amplitude is the maximum position of the object relative to the equilibrium position: (Max Height) In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A What friction is there?

Amplitude: New Symbol “A” Amplitude, A The amplitude is the maximum position of the object relative to the equilibrium position: (Max Height) In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A What friction is there? Air Resistance, Molecular Motion in Spring

Period: New Symbol “T” Period: T uppercase T stands for “period.” Amount of time for the oscillator to go through 1 complete cycle. (Time for 1 up and 1 down.) Often measured from Max to Max, But can be measured from start to start, etc. Measured in seconds.

Frequency: Another new symbol “ƒ” “ƒ” is for frequency. It is the number of cycles an oscillator goes through in one second. It is measured in 1/seconds 1/seconds => New unit “Hertz” or Hz. What is the frequency of revolutions of a new M16 bullet?

Frequency: Another new symbol “ƒ” “ƒ” is for frequency. It is the number of cycles an oscillator goes through in one second. It is measured in 1/seconds 1/seconds => New unit “Hertz” What is the frequency of revolutions of a new M16 bullet? Ans:5100 Hz or Rev per Second.

Period and Frequency The period, T, is the time per cycle. The frequency, ƒ, is cycles per time. Frequency is the reciprocal of the period ƒ = 1 / T

Quick Recap(Pic for WOD) A – maximum distance from rest postion. T – time for one complete cycle ƒ = 1 / T

In the table, label each +, -, or 0.

Question When you compress (or stretch) a spring, you have to do work on it. You apply a force over some distance. Can you get that energy back?

Elastic Potential Energy (Energy stored in a spring. Ability of a spring to do work.) Work done on a spring is stored as potential energy. The potential energy of the spring can be transformed into kinetic energy of the mass on the end.

Energy Transformations Suppose a block is moving on a frictionless surface. Before it hits the spring, the total mechanical energy of the system is the kinetic energy of the block. What happens next?

Energy Transformations, 2 The spring is partially compressed. The mass has slowed down. Σ ME = K.E. + P.E.

Energy Transformations, 3 The spring is now fully compressed The block momentarily stops The total mechanical energy is stored as elastic potential energy of the spring

At all times, total Mechanical Energy is constant = KE + PE (Put into notes) Equations for SHM Energy: KE = ½ mv 2 PE = ½ kx 2

Keep in mind. It takes the same energy to stretch a spring as compress it. PE = ½ kx 2 Is the same as = ½ k(-x) 2 So PE is same at Max or Min A.

Back to Period and Frequency Period Frequency What variable is not in these equations?

Back to Period and Frequency Period Frequency What variable is not in these equations? A. T and f do not depend on Amplitude.

Problem A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. How far will the spring compress?

Problem (revisited) A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. What will it’s frequency and period of oscillation be?

Problem A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. Q1. How far will the spring compress? Q2. What will it’s frequency and period of oscillation be?