# Springs  A spring can exert a force, store energy, and do work.  Think about a dart gun – when a dart gun is loaded it compresses a spring inside the.

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Springs  A spring can exert a force, store energy, and do work.  Think about a dart gun – when a dart gun is loaded it compresses a spring inside the barrel. What happens when the trigger is pulled?  The spring exerts a force on the dart and fires it out of the gun!

Springs  The exerted force is an example of Hooke’s Law:  Hooke’s Law: F s = - kx F s = force exerted by a compressed or stretched spring (in N). k = spring constant, a characteristic of each spring. k is small for a flexible spring, big for a stiff spring (in N/m). x = distance the spring is compressed (negative) or stretched (positive) (in m).

Spring constant from a graph  Weights are hung from a spring and the elongation of the spring is measured.  Using a graph of force (y-axis) vs distance (x-axis), the spring constant can be calculated as the slope of the line.  For this graph, k = 60 N/m

Spring constant from data  A 3.0 kg mass is hung from a spring. As a result, the spring stretches 2.0 cm. What is the spring constant for this spring?  The spring stretches due to the weight of the 3.0 kg mass:  F s =- kx but F s is equal to mg  mg = -kx  (3.0 kg) (9.8 m/s 2 ) = - k (.020 m)  k = 1500 N/m (it’s not a vector so the sign doesn’t matter)  A bigger k corresponds to a stiffer spring

Springs  Hooke’s Law: F s = - kx The negative sign in the equation is due to the fact that a spring force is a restorative force – it tends to restore things back to equilibrium. If I stretch a spring and release it, it will go back to its original unstretched state. If I compress a spring and release it, it will also go back to its original unstretched state.

Spring force points in opposite direction  In this picture, the spring is stretched  in the direction of f  But the force felt by the spring is in the opposite direction   Hence the negative sign in the equation: F = -kx

Hooke’s Law example  What is the force exerted by a spring in a dart gun (k =15 N/m) when it is compressed 2.5 cm?  F s = -kx  F s = -(15 N/m) (-.025 m)  F s = +0.38 N  The positive sign tells you that the spring force points in the direction of stretch, not the direction of compression.

Energy of a spring  The reason the dart gun fired at all is because the spring stored energy.  PE s = ½ kx 2 PE s = potential energy stored in the spring (in J) k = spring constant (in N/m) x = distance spring is compressed or stretched (in m).

Energy stored in spring example  What is the energy stored in a spring with a spring constant of 15 N/m when it is compressed 2.5 cm?  PE s = 1/2 kx 2  PE s = ½ (15 N/m) (-.025 m) 2  PE s = 0.0047 J  We added this energy to the spring when we did work on it by compressing the spring (applying a force) through a distance.

Work done by a spring  Similarly, when the compression or stretching is released, the spring does work equal to the potential energy stored in the spring.  PE s = Work

Conservation of energy  If a spring is used, it must be included in total energy, and conservation of energy.

Conservation of energy - example  A 2.0 kg ball starts from rest at the top of a 3.0 meter hill. At the bottom of the hill, it hits a horizontal spring with a spring constant of 900 N/m. How far does the ball compress the spring before it comes to a rest?  Remember that energy must be conserved – what kind(s) of energy are present at the top of the hill?  What kind(s) of energy are present when the spring is done compressing?

Conservation of energy - example  At the top of the hill, all the energy is potential:  PE = mgh = (2.0 kg)(9.8 m/s 2 )(3.0 m) = 60 J  By the law of conservation of energy, this must also be the total energy stored in the spring when it is done compressing (since the ball isn’t moving).  PE s = 1/2 kx 2  60 J = ½ (900 N/m) x 2  x = -.36 m  This is the distance the spring compresses.

Centripetal Force  Acts on an object in circular motion.  Centripetal means “center-seeking”.  For an object in circular motion, centripetal force is the force that points towards the center of the circle.  Like all forces, it is measured in Newtons.

Centripetal Force Picture  Without centripetal force, these amusement park cars would move in the direction of the yellow arrows (with a velocity tangent to the curve).  Centripetal force (always pointing towards the center) helps to keep the cars on the track.

Centripetal Force equations  F c = mv 2 / r  Where F c = centripetal force (N) m = mass (kg) v = linear velocity (m/s) r = radius (m)

Centripetal Force and N 2 Law  Remember Newton’s 2 nd Law: F = ma.  If the force is centripetal, then the acceleration must be centripetal too.  F c = ma c  Where F c = centripetal force (N) m = mass (kg) a c = centripetal acceleration (m/s 2 ) Examples

Centripetal acceleration equations  a c = v 2 / r Where a c = centripetal acceleration (m/s 2 ) v = linear velocity (m/s) r = radius (m) examples

Centripetal acceleration aka…  Sometimes centripetal acceleration (a c ) is called radial acceleration, while linear acceleration (a) is called tangential acceleration.

Period  Period ( Τ ) – the time it takes for one revolution. Period is measured in seconds.  Remember that velocity = distance/time  For an object moving in a circle, the distance traveled is the circumference (2πr) and the time required to complete 1 revolution is Τ.

Another equation for velocity  Therefore, for an object moving in a circle:  v = 2πr / Τ  Where: v = linear velocity (m/s) r = radius (m) Τ = period (s)

Centripetal Force and N 2 Law - FBDs  When using Newton’s 2 nd Law to sum up forces on steadily rotating object: ΣF = F c  If that weren’t true, the object would not be steadily rotating.  Note that since F c is the summation of the forces, it is never drawn on a FBD of a steadily rotating object.

Example 1  What are the forces acting on the rider in the roller coaster on the left?  Weight and Normal force.  What would the FBD look like?

FBD for example 1  FBD:  Sum up the forces:  Σ F = N – W  Remember that the sum of the forces equals F c.  Therefore: N – W = Fc Normal Weight

Example 2  What are the forces acting on the rider in the top roller coaster on the right?  Weight and Normal force.  What would the FBD look like?

FBD for example 2  FBD:  Sum up the forces:  Σ F = - N – W  Remember that the sum of the forces equals F c.  Therefore: - N – W = Fc Normal Weight

Centripetal Force?  You and your huge older brother are going to ride the sleigh ride at Santa’s Village (the one that goes in a circle).  Where do you want your huge brother to be – on the inside or the outside?  Does that seem like a center-seeking force to you?

Centrifugal “Force”  That force is centrifugal (center-fleeing) and it is a false force.  It is due to Newton’s 3 rd Law – see drawing on the next slide.  What you felt on Santa’s sleigh ride is the car turning from under you and causing you to push up against the side of the car.  Most people have felt centrifugal “force”, but not centripetal force.

Centrifugal “Force” A body lies in the back of a car. Here is an overhead view of the car driving straight. Now the car turns to the right and the unrestrained body continues in a straight line. The result is the body gets shoved against the side of the car. That’s what you think of as centrifugal “force”!

References for images  www.worsleyschool.net/sciencefiles/ amusement/centripetal.html www.worsleyschool.net/sciencefiles/ amusement/centripetal.html  http://starphysics.dit.ie/questions/Circular% 20motion%20and%20SHM%20Q_files/ima ge003.gif http://starphysics.dit.ie/questions/Circular% 20motion%20and%20SHM%20Q_files/ima ge003.gif  http://www.ux1.eiu.edu/~cfadd/1350/06Cir Mtn/Images/RCoaster1.jpg http://www.ux1.eiu.edu/~cfadd/1350/06Cir Mtn/Images/RCoaster1.jpg

References for images  http://www.cellmigration.org/resourc e/modeling/res_resource_images/fig6.gif http://www.cellmigration.org/resourc e/modeling/res_resource_images/fig6.gif  http://www.onlinephys.com/hooke6.g if http://www.onlinephys.com/hooke6.g if

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