Understanding Hooke's Law (David Corona)

Title: Understanding Hooke's Law

Principle(s) Investigated: Hooke's Law, Elasticity, Conservation of Energy, Parallel and Series Circuits

Standards : [SED] Using Mathematics and Computational Thinking, Engaging in Argument from Evidence, Analyzing and Interpreting Data; [CCC] Energy and Matter; [DCI] PS3.B Conservation of energy means that the total change of energy in any system is always equal to the total energy transferred into or out of the system; PS3.C When two objects interacting through a field change relative position, the energy stored in the field is changed

Materials:

-Springs (no longer than 1 m when vertically extended by 10 Newtons of force); 2 per lab group

-Ring stand assembly with clamp to extend a horizontal bar over the edge of the desk; 1 per lab group

-Metersticks

-Series of Weights (250g, 500g, and 1000g); 1 per lab group

Procedure:

Prior to demonstration, the teacher should prepare one setup for themselves at the front of the classroom. The set up should have a horizontal bar suspended over the edge of a lab desk by means of the ring stand and clamp assembly. The goal overall should be to provide enough vertical space from the horizontal bar to the floor to accommodate two fully stretched springs.

Teacher should have students pick up required supplies for their group from a location they have suggested and set up their own ring stands.

The teacher will demonstrate how a single spring suspended at one end on the horizontal bar can be stretched by adding one of the weights at the other end. Care should be taken that the weight does not fall off and the oscillations of the spring are not too drastic.

They will then do the same with a second spring, again taking care with the oscillations of the spring. The teacher will then demonstrate the direct relationship between force and spring elongation by adding more weight to the end of the second spring.

The teacher can then demonstrate Hooke's Law by taking measurements of the length of the spring with no added mass, with 250 grams added, 500 grams added, and 1000 grams added, and demonstrating a linear relationship between mass and spring elongation (change in length).

The teacher will then ask students to attempt the same series of progressive additions of mass to their own springs and record their data to note this relationship for themselves.

Student prior knowledge:

Students will be responsible for understanding the relationship between weight and mass prior to this activity. This entails understanding Newton's second law of motion relating the magnitude of an object's acceleration directly to the magnitude of the net force on an object and inversely to the mass of an object. As the resulting formula (F=m*a), applies to gravitational force ("weight"), an object's weight is a force directly proportional to its mass, where the constant of proportionality is the gravitational constant of acceleration (taken at earth's surface this would be 9.8 m/s² or 32.2 ft/s²), or put more simply W=m*g.

Students will further apply Newton's second law in understanding that the net force on the mass is zero since it is at rest at the end of the spring, which means the downward pull of gravity is countered equally by the upward pull of the spring; therefore, the force exerted by the spring is equal in magnitude to the weight added to it.

If the demonstration is done as an introduction to oscillations, students should understand the law of conservation of energy and the calculation of gravitational potential energy and work. Hooke's Law directly relates force (here weight) to elongation of a spring. Students should be able to recognize that the downward movement of the mass at the end of the spring is a change in gravitational potential energy or positive work done on the mass by gravity while that work is countered by the negative work done by the spring in equal magnitude.

Explanation:

A force is an interaction between two objects and is measured in Newtons (which in the KMS system, 1 Newton = 1 kg*m/s²). The first interaction occurs between the earth and the mass at the end of the spring, pulling them towards each other, but from the perspective of an observer, this merely appears as if the mass is being pulled downward (i.e. towards the center of the earth). The second interaction occurs as the spring is extended by the pull of the falling mass. The spring reacts in kind by pulling upward on the mass. The magnitude of the upward force exerted on the spring grows proportionally to the magnitude of its elongation, per Hooke's Law.

Hooke's Law: F_spring = k *Δx ; F_spring - force of spring _, k - spring constant , Δx - elongation

The magnitude of the pull of the mass on the spring is equal to the magnitude of the pull of the Earth's gravity on it and is relatively constant along its descent. Taking these two points into account, the downward force on the mass will stay constant and, at some degree of elongation, the magnitude of the force exerted by the spring will be equal to that exerted by gravity. Per Newton's second law of motion, as the net force acting on the mass will be zero at this degree of elongation (equilibrium point), the mass will not be accelerating. If the mass is at rest at this point, it will remain as such. This is why the mass has to be guided down manually and stopped at the equilibrium point to prevent the spring and mass from oscillating up and down continuously.

As per the law of conservation of energy, energy is considered to be of finite quantity and is by definition a quantity that allows objects to do work and, in a system, the amount of energy is kept constant though it may change form. In this example with a vertical mass-spring-Earth system, the total energy is stored as either gravitational potential energy or elastic potential energy. The former is theoretical and created by virtue of the position of an object within a gravitational field, and the latter is based on the potential for the elastic object to do work. When the displacement along the field is relatively small compared to the average distance from the source of gravitational pull (the distance from the top of the spring to the floor is negligible compared to the distance from the midpoint between the bottom of the unextended spring and the floor to the center of the Earth), then the change in gravitational potential energy (U_g) can be expressed by the formula

ΔU_g = m*g *Δh ; ΔU_{g - change in gravitational potential energy , m*g- weight of the object moved ,} Δh - vertical displacement

As work is the product of change in distance and the force applied in the direction of the change in position, then the work done by a spring is equivalent to 0.5*k*Δx². This is calculated by integrating the spring force over the length of its elongation.

By the law of conservation of energy, the total energy of the mass-spring-Earth system remains the same before and after the spring is released. In the spring-mass-Earth system, the total gravitational potential energy has declined as the center of mass of the system has gotten closer to the center of the Earth and has been replaced in equal quantity by the elastic potential energy now stored in the spring. As Δh =Δx, m*g *Δh is really m*g *Δx. Nevertheless, the two work formulas can not be equated unless Δx = Δx/2. The reason the two sources of work are not equal is that, without the instructor guiding the mass down (therefore doing upward work on the mass) the mass would have gained more energy from gravity than is stored in the spring at the equilibrium point and kept moving as that excess energy would be expressed as kinetic energy (energy of a moving object). The guiding hand provides enough force, and therefore enough work, to keep the mass periodically at rest until the force from the spring can negate the force of gravity.

Elasticity is by definition the ability of an object to return to its original dimensions once a stress (σ = force/area) has been removed. Hooke's law describes the behavior of these perfectly elastic objects relating the stressing force to the degree of elongation. All materials are elastic, and therefore behave under Hooke's law under a certain amount of stressing force, the "yield stress" unique to each kind of material. After object's and materials experience the "yield stress," they cease to behave elastically and will never return to their original dimensions once the stress is removed. Springs are unique in that they have relatively low k values, thus they will have high degrees of elongation with relatively low applied forces, making them ideal for demonstrating Hooke's Law.

Questions & Answers:

1. Will a stiffer spring have a greater or lower k value? Will the stiffer spring perform more or less work on the mass than a less rigid spring?

A stiffer spring by definition, will exhibit less elongation with the same amount of force. By Hooke's law, under the same force, the k value of a spring will be inversely related to its change in length. Taking these two points into account, a stiffer spring will have a larger k value.

The amount of work performed by a spring is calculated through the formula (0.5*k*Δx² ). Assuming the same mass is used. As the spring has a greater k value, it's elongation will proportionately decrease. As the elongation will be squared, its factor of reduction will be squared as well. The net effect will then be that the amount of work done by a stiffer spring will decrease in the same ratio as the k value increased. For example, if the spring is 3 times stiffer, k will be 3 times as great and Δx will be a third as great for the same force; consequently, the amount of work will increase by a factor of 3 due to increased k but be reduced by a factor of 9 by the change in the spring's elongation, and the net effect will be a reduction in the magnitude of work done by a factor of 3.

2. If two springs are connected end to end, will the net stiffness of the two springs together increase or decrease?

As the two springs are lined up, they each experience the pull of gravity. By rearranging Hooke's Law for each individual spring and the springs as a whole, we can see the net k value will actually be less than either spring's individual k value. This means that stringing the springs continuously will make them less rigid and stiff overall.

3. If two springs of equal length are organized so they are side by side and share a load, will the net stiffness of the two springs together increase or decrease?

As the two springs share the load, the combined effect of their spring forces will match the pull of gravity. By rearranging Hooke's law for each individual spring and the springs as a whole, we can see the net k value is the sum of the two original springs.

Applications to Everyday Life:

Hooke's law is applicable also in compressing a spring. Due to the linear relationship between change in length and applied force on the spring, springs are useful in measuring forces, most commonly weight. The most commonplace example are the scales at use in grocery stores. The spring inside stretches proportionately to the amount of weight added to the scale. A gear within the scale turns a dial to show a reading of usually pounds. The gear is turned by a sliding bar attached to the spring that descends in the same amount as the spring is stretched.

Another use for springs under Hooke's law are the suspension springs within vehicles. As energy can easily be stored inside elastic materials such as springs, the introduction of springs to vehicle structure provides a structural component that absorbs excess force and energy, thereby reducing the rigidity and tendency toward yield or structural rupture of the entire vehicle structure.

Hooke's Law is also used in designing digital force meters. As the resistance of a wire changes in relation to the it's length and diameter, and the amount of current through that wire at a constant voltage will change inversely proportional to its resistance (per Ohm's law), the amount that a wire has been elongated or compressed can be directly calculated by the change in current in that wire. Electronic force meters use this method to relate the magnitude of a tensile or compressive force on a small component carrying current.

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