11.2 - springs

what does hooke’s law state?

• force needed to stretch a spring ∝ extension of the spring from its natural length

• F = k Δ L

force needed to stretch a spring ∝ ?

force needed to stretch a spring ∝ extension of the spring from its natural length

extension of the spring from its natural length ∝ ?

extension of the spring from its natural length ∝ force needed to stretch a spring

what is the equation for hooke’s law?

F = k Δ L

• F = force needed to stretch a spring

• k = spring (/ stiffness) constant

• Δ L = extension from natural length L

F = k Δ L

hooke’s law

what is tension?

• a pull on the object attached to the string

• equal and opposite to the force needed to stretch the spring

how can we prove hooke’s law?

• using a stretched spring supporting a weight at rest

• by increasing the masses, and therefore tension, we can plot the force against tension

• hooke’s law states force needed to stretch a spring ∝ extension of the spring from its natural length

• therefore a force-time graph for hooke’s law will be a straight line through the origin

here

what will the force-time graph for hooke’s law look like?

straight line through the origin

how does spring stiffness vary with k?

the greater the value of k, the more stiff the spring is

what is the unit of k?

N m^-1 (newtons per metre)

what is the value of k?

constant for the spring

is the value of k constant?

yes, for the spring

what happens when a spring is stretched beyond its elastic limit?

it doesn’t regain its initial length when the applied force is removed, i.e., the point the spring is plastically / elastically deformed

what is the elastic limit?

• the point the spring doesn’t regain its initial length when the applied force is removed

• the point the spring is plastically / elastically deformed

when is a spring elastically deformed?

when the spring passes its elastic limit

springs in parallel picture here

springs in parallel

where is the spring extension for springs in parallel?

here

where is the weight positioned for springs in parallel?

adjusted along the rod to make extension of both springs the same

here

what is the spring extension (Δ L) for a spring in parallel?

the same for each spring, because the weight is adjusted until the extension is the same

for springs in parallel, what is the force needed to stretch each spring?

here

here

F_P = k_P Δ L

F_Q = k_Q Δ L

how do you find the weight held up by springs in parallel?

W = k Δ L

• W = weight

• k = k_P + k_Q

for springs in parallel, what is the tension in each spring?

half the weight

for springs in parallel, why was the tension in each spring half the weight?

the weight is supported by both springs, and as the tension is equal and opposite to the weight, the tension in each spring is half the weight

W = k Δ L

weight held up by springs in parallel

derive W = k Δ L

• weight is supported by both springs

• W = F_P + F_Q

• F_P = k_P Δ L

  F_Q = k_Q Δ L

• therefore W = k_P Δ L + k_Q Δ L

• k = k_P + k_Q

• therefore W = k Δ L

k + k_P + k_Q

spring constant for springs in parallel

what is the spring constant for springs in parallel?

k + k_P + k_Q

springs in series here

springs in series

where is the spring extension for springs in series?

here

for springs in series, what is the tension in each spring?

tension in each spring is equal to each other, and equal to the weight W

for springs in series, is the tension in the each spring each or different to each other?

equal

for springs in series, is the tension equal or different to the weight?

equal

for springs in series, why is the tension in each spring equal to the weight?

the tension is equal and opposite to the force needed to stretch the spring, therefore tension in each spring is equal to weight

for springs in series, what is the extension of each spring?

different to each other

• Δ L_P = W / k_P

• Δ L_Q = W / k_Q

what is the total extension for springs in series?

• Δ L = W / k

• Δ L = Δ L_P + Δ L_Q

Δ L = Δ L_P + Δ L_Q

total extension for springs in series

Δ L = W / k

total extension for springs in series

derive Δ L = W / k

• Δ L_P = W / k_P

  Δ L_Q = W / k_Q

• Δ L = Δ L_P + Δ L_Q

• so Δ L = W / k_P + W / k_Q

• therefore Δ L = W / k

what is the spring constant for springs in series?

1 / k = 1 / k_P + 1 / k_Q

1 / k = 1 / k_P + 1 / k_Q

spring constant for springs in series

derive 1 / k = 1 / k_P + 1 / k_Q

• Δ L = W / k

• = W / k_P + W / k_Q

• therefore 1 / k = 1 / k_P + 1 / k_Q

do the general information thing for each spring type

what energy type is stored in a stretched spring?

elastic potential energy

what is the energy transfer in a stretched spring released from taut?

elastic potential energy = spring’s kinetic energy

what is the work done to stretch the spring by extension Δ L?

W = ½ F Δ L

• W = work done

• F = force needed to stretch the spring to extension Δ L

what is the work done stores as in a spring?

elastic potential energy

W = ½ F Δ L

work done to stretch the spring by extension Δ L

what is the elastic potential in a stretched spring?

• the area under a force-time graph, E_p = ½ F Δ L

• E_p = ½ k Δ L²

what is the area under a force-time graph for a stretched spring following hooke’s law?

elastic potential energy stores in a stretched spring

where is the elastic potential energy on a force-time graph for a stretched spring following hooke’s law?

area under the graph

E_p = ½ F Δ L

elastic potential energy stores in a stretched spring

derive E_p = ½ F Δ L

• the work done to stretch a spring to Δ L is stored as elastic potential energy, therefore E_p = ½ F Δ L

• as the elastic potential energy is the area stored under the graph, which is in a triangle shape, area = 1/2 x base x height, therefore E_p = ½ F Δ L

E_p = ½ k Δ L²

elastic potential energy stores in a stretched spring

derive E_p = ½ k Δ L²

• the work done to stretch a spring to Δ L is stored as elastic potential energy, therefore E_p = ½ F Δ L

• hooke’s law states F = k Δ L

• E_p = ½ (K Δ L) Δ L

• therefore E_p = ½ k Δ L²