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Springs

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Restoring Forces and Hooke’s Law

The figure shows how a hanging mass stretches a spring of equilibrium length L0 to a new length L.

The mass hangs in static equilibrium, so the upward spring force balances the downward gravity force.

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Restoring Forces and Hooke’s Law

The figure shows measured data for the restoring force of a real spring.

s is the displacement from equilibrium.

The data fall along the straight line:

The proportionality constant k is called the spring constant.

The units of k are N/m.

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Hooke’s Law

One end of a spring is attached to a fixed wall.

(Fsp)s is the force produced by the free end of the spring.

s = s – se is the displacement from equilibrium.

The negative sign is the mathematical indication of a restoring force.

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Explore Learning – Gizmo - http://www.explorelearning.com/Gizmo - Determining the spring constant for a hanging spring Record in your lab notebookTitle: Spring ConstantPurpose: To determine the value of the spring constant for a hanging springProcedureChoose a spring – if your birthday is in months: Jan – Mar –> spring 1

Apr – Jun ->spring 2, Jul – Sept -> spring 3, Oct – Dec -> spring 4Collect data for at least 5 different masses – record in notebookExport the data – open excel and paste data – label the dataCreate two graphs

Plot Force vs stretch Plot Force vs length

Draw a sketch of both graphs – be sure to use good graphing techniquesDetermine slope of both graphs

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Conclusion Questions:1.How does the slope of the Force vs stretch graph compare the slope of the Force vs length graph.2.What can be determined from the slope.3.Draw a FBD for the hanging mass. How does the restoring force of the spring relate to the mass of the hanging mass4.Choose another spring. Collect data and find the slope. What does a steeper slope tell you about the spring?

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The restoring force of three springs is measured as they are stretched. Which spring has the largest spring constant?

QuickCheck 10.9

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The restoring force of three springs is measured as they are stretched. Which spring has the largest spring constant?

QuickCheck 10.9

Steepest slope. Takes lots of force for a small displacement.

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Example 10.5 Pull Until It Slips

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Elastic Potential Energy

Springs and rubber bands store potential energy that can be transformed into kinetic energy.

The spring force is not constant as an object is pushed or pulled.

The motion of the mass is not constant-acceleration motion, and therefore we cannot use our old kinematics equations.

One way to analyze motion when spring force is involved is to look at energy before and after some motion.

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Elastic Potential Energy

The figure shows a before-and-after situation in which a spring launches a ball.

Integrating the net force from the spring, as given by Hooke’s Law, shows that:

Here K = ½ mv2 is the kinetic energy.

We define a new quantity:

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Elastic Potential Energy

An object moving without friction on an ideal spring obeys:

Because s is squared, Us is positive for a spring that is either stretched or compressed.

In the figure, Us has a positive value both before and after the motion.

where

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A spring-loaded gun shoots a plastic ball with a launch speed of 2.0 m/s. If the spring is compressed twice as far, the ball’s launch speed will be

A. 1.0 m/s.

B. 2.0 m/s.

C. 2.8 m/s

D. 4.0 m/s.

E. 16.0 m/s.

QuickCheck 10.10

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A spring-loaded gun shoots a plastic ball with a launch speed of 2.0 m/s. If the spring is compressed twice as far, the ball’s launch speed will be

A. 1.0 m/s.

B. 2.0 m/s.

C. 2.8 m/s

D. 4.0 m/s.

E. 16.0 m/s.

QuickCheck 10.10

Conservation of energy:Double x double v

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Example 10.6 A Spring-Launched Plastic Ball

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Period of a spring

Play around with the gizmoMark the time to complete one oscillation after making changes to k, g, and m

Answer the 5 assessment problems at the bottom. I record the results as a classwork grade

If the mass is increase, how is the period affected?

If the spring constant is increased, how is the period affected?