Bell Ringer: What is a force? What is Newton’s 2nd Law? What is work? What is the equation for the Work-Kinetic Energy theorem and what does it mean?

Notes 6.1: Work done by a Spring Force An Adventure created by Billy J. Jenkins

Objectives: Describe the force in an elastic spring. Determine the energy stored in an elastic spring. Understand why a spring force is often referred to as a restoring force.

Vocabulary: Hooke’s Law Restoring Elastic Spring

Active Physics Book: Chapter 4 - Section 5: p. 396- 400 (Hooke’s Law) Chapter 4 – Section 3: p. 376 – 378 (Elastic Potential Energy)

Further Learning: Red Book in Class: Chapter 14 - Section 1: p. 375- 278      (Hooke’s Law) Physics Classroom: Hooke's Law Spring Potential Energy

Restoring Force (Spring Force): There are many objects in nature which take the form of a spring. Stretching a rubber band or a spring requires a force. Another example is an atom within the lattice of a molecule. Technically the forces that hold the atoms together in a molecule can be modelled as springs connecting each one another. Many springs have the property that the stretch of a spring is directly proportional to the force applied to it. - This means that if you double the force, the stretch of the spring doubles. If you triple the force, the stretch of the spring triples. - Once you let go of the object after stretching it or compressing it, the spring returns to its relaxed state which is why it is called a restoring force.

Hooke’s Law: Hooke’s Law explains the restoring force on a spring after it has been stretched or compressed. Basically, the more you stretch a spring, the larger the restoring force of the spring. A spring force is given to us by Hooke’s Law which states that the force needed to extend or compress a spring by some distance is proportional to that distance.   F s =−kx [Hooke’s Law] 𝐹 𝑠 is the force exerted by the spring. x is the stretch (or compression) of the spring. k is the spring constant.

Hooke’s Law: F s =−kx [Hooke’s Law]   F s =−kx [Hooke’s Law] 𝐹 𝑠 is the force exerted by the spring. x is the stretch (or compression) of the spring. k is the spring constant. Note: The negative sign in the equation indicates that the pull by the spring is opposite to the direction it is stretched or compressed. Note: The spring constant k is a measure of the stiffness of the spring.

Sample Problem 1:   A 3.0 N weight is suspended from a spring. The spring stretches 2.0 cm. Calculate the spring constant. Many bathroom scales work by compressing a spring. Inside the bathroom scale is a spring. When you step on the scale, the spring compresses just enough to provide an upward force equal to your weight. The more weight, the more compression of the spring is required. Write Newton’s 2nd Law for the forces acting on a person who is standing on this spring-loaded bathroom scale.

Work done by a Spring force: To find the work done by a spring force, we first make three assumptions about the spring: The spring is massless: The springs mass is negligible relative to the block’s mass. The spring is ideal; that is, it obeys Hooke’s law precisely. Assume that the contact between the block and the floor is frictionless.

Work done by a Spring force: IMPORTANT NOTE: You cannot solve spring equations by the formula W=Fdcosθ because that equation assumes a constant force; whereas, spring force is a variable force. For the purpose of high school, we will simply use the equation: 𝑊 𝑠 =− 1 2 𝑘( 𝑥 𝑓 2 − 𝑥 𝑖 2 )

Work done by a Spring force:

Work done by a Spring force: Work done by a spring force, 𝑊 𝑆 , is positive if the block ends up closer to the relaxed position (x=0) than it was initially.   Work done by a spring force is negative if the block ends up farther away from x=0 (the relaxed state). Work done by a spring force is zero if the block ends up at the same distance from x=0.

Sample Problem 2: A package of spicy Cajun pralines lies on a frictionless floor, attached to the free end of a spring in the arrangement pictured below. A rightward applied force of magnitude 𝐹 𝑎 =4.9 𝑁 would be needed to hold the package at 𝑥 1 =12𝑚𝑚. How much work does the spring force on the package if the package is pulled rightward from 𝑥 0 =0 𝑚𝑚 to 𝑥 2 =17 𝑚𝑚? Next the package is moved leftward from 𝑥 2 =17 𝑚𝑚 to 𝑥 3 =−12 𝑚𝑚. How much work does the spring do on the package during this displacement?

Sample Problem 2 - Answers: A package of spicy Cajun pralines lies on a frictionless floor, attached to the free end of a spring in the arrangement pictured below. A rightward applied force of magnitude 𝐹 𝑎 =4.9 𝑁 would be needed to hold the package at 𝑥 1 =12𝑚𝑚. k = 408 N/m. W = -0.059 J. 30 mJ

Sample Problem 3: Calculate the Spring Constant k from the graph of a stretched spring below:

Spring Potential Energy: Elastic potential energy is potential energy stored as a result of the deformations of an elastic object, such as the stretching of a spring. Work and Potential Energy have a relationship similar to the Work-Kinetic Energy theorem (with some limitations) where: ∆𝑈=−𝑊 Given this relationship and the fact that we previously found that Work done by a Spring force is: 𝑊 𝑠 = 1 2 𝑘 𝑥 𝑖 2 − 1 2 𝑘 𝑥 𝑓 2 Then we can find that Spring (Elastic) Potential Energy is: 𝑈 𝑠 = 1 2 𝑘( 𝑥 𝑓 2 − 𝑥 𝑖 2 )

PHET – Spring Lab: Show the Energy of the Spring. Use the stimulation to determine the Spring Constant. Use the stimulation to determine an unknown mass. Use the stimulation to determine the Elastic Potential Energy.

Exit Ticket: A 5 newton force causes a spring to stretch 0.2 meter. What is the potential energy stored in the stretched spring? A weight of 12 N causes a spring to stretch 3.0 cm. What is the spring constant (k) of the spring?  If the force to stretch a spring is given by F = (100 N/m)x, how much work does it take to stretch the spring 4 meters from rest?