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: Fs = - kx

Fs = 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:

Fs =- kx but Fs is equal to mg mg = -kx (3.0 kg) (9.8 m/s2) = - 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: Fs = - 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?

Fs = -kx

Fs = -(15 N/m) (-.025 m)

Fs = +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.

PEs = ½ kx2

PEs = 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?

PEs = 1/2 kx2

PEs = ½ (15 N/m) (-.025 m)2

PEs = 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.

PEs = 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/s2)(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).

PEs = 1/2 kx2

60 J = ½ (900 N/m) x2

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

Fc = mv2 / r

Where Fc = centripetal force (N)

m = mass (kg) v = linear velocity (m/s) r = radius (m)

Centripetal Force and N 2 Law

Remember Newton’s 2nd Law: F = ma. If the force is centripetal, then the acceleration

must be centripetal too. Fc = mac

Where Fc = centripetal force (N)

m = mass (kg) ac = centripetal acceleration (m/s2)

Examples

Centripetal acceleration equations

ac = v2 / r Where ac = centripetal acceleration (m/s2)

v = linear velocity (m/s) r = radius (m) examples

Centripetal acceleration aka…

Sometimes centripetal acceleration (ac) 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 2nd Law to sum up forces on steadily rotating object: ΣF = Fc

If that weren’t true, the object would not be steadily rotating.

Note that since Fc 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 Fc.

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 Fc.

Therefore: - N – W = FcNormal 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 3rd 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

http://starphysics.dit.ie/questions/Circular%20motion%20and%20SHM%20Q_files/image003.gif

http://www.ux1.eiu.edu/~cfadd/1350/06CirMtn/Images/RCoaster1.jpg

References for images

http://www.cellmigration.org/resource/modeling/res_resource_images/fig6.gif

http://www.onlinephys.com/hooke6.gif