Elastic Potential Energy - Energy

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If the force you apply to a spring is directly proportional to stretch, how many times would you have to increase your force to get 4 times the stretch?

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Fill in the Blank Notes

Elastic P.E.

work stored in a deformed spring

a) General Equation

PE_s = F x

F = Average Force needed to deform the spring

x = distance deformed (stretch or compress)

Noble and Greenough School HS

Ideal Spring

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b) Hooke's Law

Force needed to deform an ideal spring a given amount is directly proportional to its displacement (deformation).

Hooke's Law illustration
Graphic from Learning-Connections Clipart

Hooke's Law Through History

Explore five ways springs have helped people measure time, weight, motion, and position. Select an image to enlarge it and read the explanation.

Mechanical Watches illustrated explanation — A hairspring keeps a balance wheel oscillating to measure equal intervals of time.

Navigation at Sea illustrated explanation — Accurate spring-controlled clocks helped sailors find their longitude.

Spring Scales illustrated explanation — A spring stretches until its upward force balances the object's weight.

Earthquake Recorders illustrated explanation — A suspended mass and spring help record vibrations of the ground.

Spacecraft Accelerometers illustrated explanation — The displacement of a small test mass helps measure acceleration.

Robert Hooke

'Newton's bitter rival'

Robert Hooke Website

Ex) Ideal Spring

F (Force to Deform) (N)	x (distance deformed) (m)
2.0	0.30
4.0	0.60
6.0	0.90
8.0	1.2

Hooke's Law
University of Colorado PhET

slope = F / x

x = stretch

c) Spring Constant (k) — Ideal Spring

quantifies the stiffness of a spring

k = F / x

(N/m)

Question

What is the spring constant of the spring that produced the data above?

F (N)	x (m)
2.0	0.30
4.0	0.60
6.0	0.90
8.0	1.2

↓

k = 2.0 N / 0.30 m

k = 6.7 N/m

Which is the stiffer spring?

Spring A is the stiffer spring

More force is required for the same stretch

Two springs hold the same 2.0 kilogram mass; spring A stretches a small amount and spring B stretches much more

↓

Spring A is stiffer

Same force, smaller stretch → larger k = F/x

Ex) Plots and k

Pick any point on the line

k = F / x

↓

k = 20. N / 0.40 m

k = 50. N/m

Force versus stretch graph: a straight line from the origin to point A, the elastic limit, whose slope equals the spring constant k; past A the spring is permanently stretched

Slope of F vs. x = k

d) Elastic P.E.

= area of the triangle under the F vs x graph

PE_s = (½) base × height

PE_s = 6 Joules

Try This at Home!

What crushed this can?

S.M.U. Physics

The can implodes because the atmospheric pressure is greater than the pressure inside the can.

When the can was submerged in the water it caused the water vapor inside the can to rapidly cool. When a gas is cooled, its pressure goes down. This drop in pressure causes the pressure inside the can to be lower than the atmospheric pressure.

Note: Use less than 1 cm of water in this experiment (about ½ inch).

Check Your Understanding

5 quick questions — pick an answer to see if you’re right.

1. For an ideal spring, how many times must you increase the applied force to get 4 times the stretch?

2. A spring stretches 0.60 m when a 4.0 N force is applied. What is its spring constant?

3. Hooke's Law states that for an ideal spring, the force needed to deform it is

4. Spring A needs more force than spring B to stretch the same distance. Which is true?

5. A spring is stretched steadily from 0 to 0.40 m, with the force rising from 0 to 20. N. How much elastic PE is stored?

NYSSLS Regents Practice · Energy · Hooke’s Law

Testing a Suspension Spring

Engineers tested a spring for a mountain bike’s suspension by applying forces and measuring how much it compressed. The data are shown in Table 1.

Table 1: Spring Test Data

Compression (m)	0	0.020	0.040	0.060
Applied Force (N)	0	100.	200.	300.

What is the spring constant of this spring?

▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)

Challenge score: 0 / 6

Multiple choice

C1. What does the slope of a graph of applied force versus compression represent?

Support or Refute · Justify

C2. An engineer makes the following claim.

Claim: Compressing the suspension spring 0.040 m stores four times as much elastic potential energy as compressing it 0.020 m.

Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.

Support Refute

Substitute values

C3. Using the spring constant k = 5000 N/m, substitute numerical values into a mathematical representation that can be used to determine the elastic potential energy stored when the spring is compressed 0.060 m. Then calculate the energy.

PE_s = ½( N/m)( m)² = J

Determine the value

C4. When a rider sits on the bike, the suspension spring supports a force of 350 N. Determine how far the spring compresses (assume it obeys Hooke’s law).

x = m

Multiple choice

C5. When the spring was compressed beyond 0.15 m, the data no longer followed F = kx. What is the best explanation?

Constructed response

C6. A heavier rider asks for a stiffer spring with k = 10,000 N/m. Identify one benefit and one drawback of this design change.

PE and k →