assignment

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Mass on a Spring

Walker, Chapter 13 Simulations: https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html http://physics.bu.edu/~duffy/HTML5/mass_on_spring_graphs.html https://phet.colorado.edu/sims/html/hookes-law/latest/hookes-law_en.html

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Vibrations, Oscillations,� and Periodic motion

•  Some everyday examples: pendulum, object on a spring, swing, bridge, vibrations of guitar string

•  Goal: describe the motion, forces, and energy of the system

•  Last week: pendulum à what did we learn?

•  This week: mass on a spring à examples?

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Object (Mass) on a Spring

Horizontal •  x is the displacement from equilibrium

Vertical •  gravity also comes into play

Can change: •  mass (m) •  spring constant (k): stiffness of spring •  amplitude (A): max that you stretch it

Equilibrium position (spring unstretched)

Table is frictionless

x = 0 at

equilibrium position

+x -x

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Elastic Potential Energy •  Everyday example: launching a ball across the

room with a compressed spring (ping pong ball gun). Energy in compressed spring converts to kinetic energy of ball.

•  Other examples: archery, kangaroo, bungee cord game at fairs, compression of femur

•  Conservation of energy: Efinal = Einitial o  Total energy remains constant, energy is transformed

•  Elastic potential energy: U = (1/2)kx2

o  Energy stored in spring when stretched or compressed o  Energy is zero when x = 0 o  More potential energy when displacement is bigger o  Energy is a scalar, and always positive here

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Energy of Object on a Spring Given what you know about energy, where do you think each of the following will be highest? Why? • Gravitational potential energy highest:

When object is at greatest height U = mgy

• Kinetic energy highest when: When object is at greatest speed K = ½ mv2

• Elastic potential energy highest when: When object is at greatest displacement U = ½ kx2

• Total energy: Stays the same Einital = Efinal

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Period of Mass on Spring We will determine how various factors affect the motion of the mass on a spring. Factors we might test: • Mass? • Spring constant? • Amplitude? • Anything else?

Equilibrium position (spring unstretched)

Table is frictionless

x = 0 at

equilibrium position

+x -x

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Characteristics of Motion

•  Amplitude (A): max displacement relative to equilibrium. Object reaches +/- A during each cycle. (Units are meters.)

•  Period (T): time to complete a cycle (seconds) +A à -A à +A

(in simulation it goes from middle to ends back to middle)

•  Frequency (f): number of cycles per second (units are 1/s = s-1 = Hz)

•  Relationship between frequency and period: f = 1/T

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Period of a Mass on a Spring A mass on a spring has a period:

T = 2π √(m/k)

o  Bigger mass à _____ T à oscillation takes _____ o  Stiff spring (____ k) à ______ T à _____ oscillation o  Does / does not depend on amplitude

What you observe in the simulations should agree with this formula

bigger longer big shorter rapid

does not

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Spring Force: Hooke’s Law •  Force is proportional to displacement •  Force is in the opposite direction of displacement:

F = - kx

F = _______ x = ____________ k = ______________ (stiff spring = large k) “-” means _______________

•  When is spring force biggest? Smallest? •  Spring force is a restoring force: object is pushed or pulled

towards equilibrium. What’s the direction at various points in its motion?

force

displacement spring constant

opposite direction