Forces and Deformations: Hooke's Law

A spring has a spring constant $k = 10\ \text{N/cm}$, as shown in the figure.

Calculate the force that must be applied to stretch it by $3\ \text{cm}$. Remember Hooke's law: $F = k \times x$.

Complete the sentence: "Hooke's law states that the extension of a spring is $\dots$ proportional to the $\dots$ applied to it." Then write the formula $F = k \times x$ and state the meaning and unit of each letter.

A spring extends by $4\ \text{cm}$ when a force of $2\ \text{N}$ is applied to it. Without doing detailed calculations, and considering Hooke's law: how much will it extend with $4\ \text{N}$? And with $1\ \text{N}$? Justify your answers.

A force of $20\ \text{N}$ is applied to a spring with constant $k = 5\ \text{N/cm}$. How many centimeters does it extend by?

A spring has a spring constant of $k = 200\ \text{N/m}$ and extends by $5\ \text{cm}$.

1. a) Express the extension in meters.
2. b) Calculate the applied force.

The spring constant of a spring is $k = 2\ \text{N/cm}$.

1. a) Express it in $\text{N/m}$.
2. b) Calculate the force required to stretch the spring by $0.1\ \text{m}$.

A spring measures $10\ \text{cm}$ at rest and its constant is $k = 5\ \text{N/cm}$. When a weight is hung from it, the spring length becomes $14\ \text{cm}$, as shown in the figure.

1. a) How much has it extended?
2. b) What is the value of the weight?

A spring with constant $k = 50\ \text{N/m}$ measures $20\ \text{cm}$ at rest. We hang a weight from it and the spring length becomes $30\ \text{cm}$.

1. a) Express the extension in meters.
2. b) Calculate the weight of the weight hanging from it.
3. c) If we hung twice as much weight, what length would the spring measure?

Spring A has a spring constant of $k = 2\ \text{N/cm}$ and spring B has $k = 8\ \text{N/cm}$. Without performing calculations:

1. a) Which one is "stiffer", that is, which one is harder to stretch?
2. b) Which one will extend more if we apply the same force to both?

For spring A, we know that with $6\ \text{N}$ it extends by $2\ \text{cm}$; for spring B, with the same $6\ \text{N}$ it extends by $6\ \text{cm}$.

1. a) Calculate the spring constant of each spring.
2. b) Which of the two is stiffer?

We hang the same weight from spring A and it extends by $2\ \text{cm}$; from spring B and it extends by $6\ \text{cm}$.

1. a) Which of the two springs has a larger spring constant?
2. b) How many times larger?
3. c) If we hang a weight from spring B that weighs half as much, how much will it extend?

What is a newton meter? What quantity does it measure and in which unit? What is the main part inside it?

We hang a bag of oranges from a newton meter and it reads $15\ \text{N}$, as shown in the figure.

1. a) What force is the newton meter measuring?
2. b) The bag is stationary: what force does the newton meter exert on the bag?

When a weight of $8\ \text{N}$ is hung from a spring, it extends by $2\ \text{cm}$, as shown in the figure.

Calculate the spring constant of the spring, in $\text{N/cm}$.

When a force of $10\ \text{N}$ is applied to a spring, it extends by $5\ \text{cm}$.

1. a) Calculate its spring constant.
2. b) What force will be required to extend it by $8\ \text{cm}$?

When a weight of $6\ \text{N}$ is hung from a spring, it extends by $3\ \text{cm}$.

1. a) What is its spring constant?
2. b) How much will it extend if we hang a weight of $10\ \text{N}$?

You want to build a homemade newton meter using a spring with constant $k = 0.5\ \text{N/cm}$ capable of measuring forces up to $10\ \text{N}$.

1. a) How much will the spring extend under the maximum force?
2. b) What minimum length must the ruler where you draw the scale have?
3. c) What distance will there be between the $1\ \text{N}$ mark and the $2\ \text{N}$ mark?

You have two springs: spring A, with $k = 10\ \text{N/cm}$, and spring B, with $k = 0.5\ \text{N/cm}$. You want to measure small forces, of around $2\ \text{N}$, using a school ruler.

1. a) Calculate how much each spring extends with $2\ \text{N}$.
2. b) Which one should you use to clearly read the measurement? Give reasons for your answer.

In the laboratory we measure the extension of a spring with different forces: with $1\ \text{N}$ it extends by $2\ \text{cm}$; with $2\ \text{N}$, $4\ \text{cm}$; and with $3\ \text{N}$, $6\ \text{cm}$.

1. a) Calculate the spring constant for each pair of data.
2. b) Does this spring satisfy Hooke's law? Justify your answer.

We stretch a spring with increasingly larger forces and, beyond a certain point, when released it no longer returns to its original length: it remains deformed.

1. a) What is the name of the point beyond which this happens?
2. b) Is Hooke's law satisfied before or after this point?
3. c) Why must we not exceed it when using a newton meter?

When stretching a spring with different forces we obtain these extensions: with $2\ \text{N}$, $1\ \text{cm}$; with $4\ \text{N}$, $2\ \text{cm}$; with $6\ \text{N}$, $3\ \text{cm}$; and with $8\ \text{N}$, $4.5\ \text{cm}$. The data are represented in the graph:

1. a) Calculate the spring constant of the spring in the region where Hooke's law is satisfied.
2. b) Above what force is Hooke's law no longer satisfied?
3. c) What is happening to the spring beyond that point?

The force–extension graph of a spring is a straight line passing through the origin and through the point corresponding to an extension of $4\ \text{cm}$ and a force of $12\ \text{N}$, as shown in the figure.

1. a) What force corresponds to an extension of $1\ \text{cm}$?
2. b) What is the spring constant of the spring?
3. c) What extension corresponds to a force of $15\ \text{N}$?

A spring scale at a market stall uses a spring with $k = 20\ \text{N/cm}$ that must not extend by more than $5\ \text{cm}$ to avoid damage.

1. a) What is the maximum force the scale can measure?
2. b) Can a bag of oranges weighing $120\ \text{N}$ be weighed on it? Justify your answer.
3. c) What will happen to the spring if someone insists on weighing the bag?

A spring extends by $2\ \text{cm}$ when a force of $5\ \text{N}$ is applied to it.

1. a) Calculate its spring constant.
2. b) Predict the extension for $12\ \text{N}$ assuming Hooke's law is satisfied.
3. c) When tested in the laboratory with $12\ \text{N}$, the spring extends by $6\ \text{cm}$ instead of the predicted amount. Give two possible explanations for the difference.