1.1. To recognise that a loaded To recognise that a loaded spring is an application of spring is an application of SHMSHM

2.2. To review Hooke’s lawTo review Hooke’s law

3.3. To establish the period of To establish the period of an oscillating loaded springan oscillating loaded spring

4.4. To review springs in parallel To review springs in parallel & series& series

Book Reference : Pages 41-42Book Reference : Pages 41-42

What can we remember from Y12?What can we remember from Y12?

What does Hooke’s law state?What does Hooke’s law state?

How do we express this as a proportionality?How do we express this as a proportionality?

How do we turn this into an equation we can How do we turn this into an equation we can use?use?

http://www.rdg.ac.uk/acadepts/sp/picetl/publish/ISEs/Forces2.htm

Hooke’s Law states that...the Hooke’s Law states that...the change in length produced by a change in length produced by a force on a wire or spring is force on a wire or spring is directly proportional to the force directly proportional to the force applied.applied.

L (m)

Hooke’s law only applies within limitsHooke’s law only applies within limits

Extension Extension Force Applied Force AppliedL L F F

To turn a proportionality into an equation To turn a proportionality into an equation introduce a constant of proportionality...introduce a constant of proportionality...

L L F F F = F = kkLL

We call k the We call k the spring constant spring constant and it defines and it defines how stiff the spring ishow stiff the spring is

How can we find K experimentally?How can we find K experimentally?

As a mass As a mass mm on a spring oscillates, the on a spring oscillates, the increase in spring tension (increase in spring tension (TTss) provides ) provides the the “restoring force” “restoring force” required for SHMrequired for SHM

At displacement At displacement xx from the equilibrium from the equilibrium position we can calculate the restoring position we can calculate the restoring force from Hooke’s lawforce from Hooke’s law

TTss = -kx = -kx (minus sign shows force acting towards the equilibrium position)(minus sign shows force acting towards the equilibrium position)

So the restoring force is So the restoring force is –kx –kx and we can and we can relate this to acceleration via (f=ma) (a=f/m)relate this to acceleration via (f=ma) (a=f/m)

Acceleration = restoring force / massAcceleration = restoring force / mass

a = -kx/ma = -kx/m

For SHM we also know that the acceleration For SHM we also know that the acceleration can be written as can be written as

a = -(2a = -(2f)f)22 x x

Equating the two views of aEquating the two views of a

-(2-(2f)f)22 x = -kx/m x = -kx/m (then cancel (then cancel xx & times by -1 to kill - & times by -1 to kill - sign)sign)

(2(2f)f)22 = k/m = k/m (then square root)(then square root)

22f = f = (k/m) (k/m) (then ÷ 2(then ÷ 2))

f = f = (k/m) / 2(k/m) / 2 (find period T by knowing it is 1/f)(find period T by knowing it is 1/f)

(÷ both sides by f, times both sides by 2(÷ both sides by f, times both sides by 2, ÷ throughout by , ÷ throughout by (k/m) (k/m) (actually we times by inverted devisor (actually we times by inverted devisor (m/k)(m/k)

1/f = 21/f = 2 (m/k) = T (m/k) = T (find this in the crib sheet & highlight)(find this in the crib sheet & highlight)

When will the tension in the spring be at a maximum?When will the tension in the spring be at a maximum?

What will the maximum tension be equal to?What will the maximum tension be equal to?

When will the tension in the spring be at a minimum?When will the tension in the spring be at a minimum?

What will the minimum tension be equal to?What will the minimum tension be equal to?

Equilibrium position

Equilibrium position

Equilibrium position

x = -A x = -A x = -A

x = +A x = +A x = +A

Tension varies between Tension varies between mg + kA mg + kA and and mg - kAmg - kA

At a maximum when spring is fully stretched at At a maximum when spring is fully stretched at -A-A

At a minimum when spring is stretched as little as At a minimum when spring is stretched as little as possible at possible at +A+A

Equilibrium position

Equilibrium position

Equilibrium position

x = -A x = -A x = -A

x = +A x = +A x = +A

What happens to the frequency when the mass What happens to the frequency when the mass increases?increases?

What happens to the frequency when the spring What happens to the frequency when the spring constant increases, (stiffer spring)?constant increases, (stiffer spring)?

How would moving the experiment to the moon change How would moving the experiment to the moon change the period or frequency?the period or frequency?

f = f = (k/m) / 2(k/m) / 2

If the mass increases the frequency decreases, more If the mass increases the frequency decreases, more inertiainertia

The frequency increases if the spring constant increasesThe frequency increases if the spring constant increases

The equation for frequency/period of a loaded spring The equation for frequency/period of a loaded spring does not depend upon g. Therefore moving it to the does not depend upon g. Therefore moving it to the moon will change nothing (unlike the simple pendulum)moon will change nothing (unlike the simple pendulum)

Thinking about this experimentally, and compare with Thinking about this experimentally, and compare with the simple pendulum experimentthe simple pendulum experiment

What could we change?What could we change?

What could we measure?What could we measure?

What could we plot?What could we plot?

What could we find from the graph?What could we find from the graph?

The mass m can be our independent variableThe mass m can be our independent variable

We can measure the period by finding the time for n full We can measure the period by finding the time for n full oscillations and dividing appropriatelyoscillations and dividing appropriately

We want a straight line graph... TWe want a straight line graph... T22 against m, (usual against m, (usual dependent vs independent variable)dependent vs independent variable)

Gradient will be 4Gradient will be 422/k/k

What should we do to ensure the best quality value for What should we do to ensure the best quality value for this or any gradient?this or any gradient?

Springs in parallel share the load Springs in parallel share the load & have the same extension & have the same extension acting like a single spring with a acting like a single spring with a combinedcombined spring constant spring constant

Derive an equation for the Derive an equation for the combined spring constant in combined spring constant in terms of the two spring terms of the two spring constants kconstants kpp & k & kqq

p q

L

Hints : identical extensions, different tensions, Hints : identical extensions, different tensions, resolve verticallyresolve vertically

The force needed to stretch The force needed to stretch springs p & q respectively is:-springs p & q respectively is:-FFpp = = kkppL & FL & Fqq = = kkqqLL

W = W = kkppL + L + kkqqL which can be considered equal to L which can be considered equal to kkL where k is the L where k is the effectiveeffective spring constant spring constant

kkL = L = kkppL + L + kkqqL remove L remove L L kk = = kkpp + + kkqq

p q

L The tension is given by W = FThe tension is given by W = Fp p + + FFq q and so... and so...

Springs in series share the same Springs in series share the same tension which is equal to Wtension which is equal to W

Derive an equation for the Derive an equation for the combined spring constant in terms combined spring constant in terms of the two spring constants kof the two spring constants kpp & k & kqq

p

qL

Hints : different extensions, identical tensions, Hints : different extensions, identical tensions, equate the total extension to the sum of the two equate the total extension to the sum of the two extensionsextensions

Springs in series share the same Springs in series share the same tension which is equal to Wtension which is equal to W

The extensions in the two springs The extensions in the two springs is given by:-is given by:-LLpp = W/ = W/kkpp & & LLqq = W/ = W/kkqq

The total extension is The total extension is LLpp + + LLqq

= W/= W/kkpp + W/ + W/kkq q = = WW/k/k = = 1/1/kkpp + 1/ + 1/kkq q = = 11/k/k

where k is the where k is the effectiveeffective spring constant spring constant

p

qL