Group Work Predict the motion of a mass acted on only by a Hooke’s law spring. Express your prediction as a position-time graph. Explain why you believe that the mass will move in the manner you predicted.

Boing! Things that vibrate 13.1–13.5

Objectives Analyze force, acceleration, velocity, and position at any point in a vibration cycle. Trace the evolution of kinetic and potential energy, velocity, and displacement in an oscillation. Identify the factors determining the period of a spring and a pendulum.

What’s the Point? The Hooke’s law model describes the essential features of many types of oscillation.

Springs Hooke’s law: F = –kx F = force exerted by the spring k = spring constant (characteristic of the particular spring) x = distance spring is displaced from equilibrium

How does a spring mass move? Newton’s second law: F = ma Force F depends on position by Hooke’s law: F = –kx

Poll Question The spring’s force on an oscillating object is zero at the extreme positions maximum at the extreme positions minimum but not zero at the extreme positions

Poll Question The acceleration of an oscillating object is zero at the extreme positions maximum at the extreme positions minimum but not zero at the extreme positions

Poll Question The velocity of an oscillating object is maximum at the equilibrium position maximum at the extreme positions maximum midway between equilibrium and an extreme positions

Uniform Circular Motion Centripetal force F = mv2/r inwards Constant magnitude F0; direction depends on position F0 Force in y-direction is proportional to –y

Uniform Circular Motion Angle changes at a steady rate. Projection on y-axis has Hooke’s law force. So, projection on y-axis must have Hooke’s law motion too! What is the projection of an angle on the y-axis?

Hooke’s Law Motion

Class Work What is velocity (min, max, > 0, < 0, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom? What is acceleration (min, max, > 0, < 0, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom?

Class Work What is the object’s position when it is not accelerating? What is the object’s velocity when it is not accelerating? What is the object’s position when it is not moving? What is the object’s acceleration when it is not moving?

Group Work What is the object’s kinetic energy when it is not accelerating? What is the object’s potential energy when it is not accelerating?

Energy Potential energy of a stretched spring : kx2 PE = 1 2 Conservation of energy: PE + KE = constant (This of course ignores the nasty reality of energy dispersal by friction and drag.)

Group Work Show that the total energy E of an oscillating Hooke’s law spring is always positive. kx2 1 2 + mv2 E = PE + KE = > 0

Energy KE = 0 PE = max KE = max PE = 0 y time PE = max

Period and Frequency Period T Frequency f f = 1/T time of one cycle (units: s) Frequency f cycles per unit time (units: 1/s = Hz) f = 1/T

Poll Question Increasing the spring constant k makes the period shorter. longer. k has no effect on the period.

Poll Question Increasing the mass m makes the period shorter. longer. m has no effect on the period.

Period m Period T = 2p k So increased mass m  longer period increased k  shorter period period does not depend on amplitude

Effect of Gravity Less than you might expect: Changes equilibrium position x = 0 Does not change k

Spring + Gravity spring alone gravity force position

Spring + Gravity spring alone gravity force spring + gravity position

Spring + Gravity different equilibrium position same k net force position