Forces and elasticity

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Changing Shape

To change the shape of a stationary object, more than one force must be applied.
Objects can be stretched, bent, or compressed, or a combination of these.
Stretching: forces act in opposite directions, away from the object (e.g., weight of a mass and tension in a spring).
Compression: forces act in opposite directions, towards the object (e.g., weight of a mass on a spring and reaction force from the surface).
Bending: forces act in opposing directions at different points on the object (e.g., weight of a swimmer and reaction force from the block on a diving board).

A thrust force can cause a car to speed up, a gravitational force can cause a comet to change direction, compression forces can cause a spring to change shape.

Elastic and Inelastic Deformation

Elastic deformation: the object returns to its original shape and length when the deforming force is removed.
Examples of elastic materials: rubber bands, fabrics, steel springs.
Inelastic deformation: the object does not return completely to its original shape when the force is removed (permanent distortion).
Examples of inelastic materials: plastic, clay, glass.
In an exam, always state what happens when the force is removed to distinguish between elastic and inelastic.

Hooke's Law

Hooke's Law: the extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded.
Directly proportional means if force doubles, extension doubles; if force halves, extension halves.
The limit of proportionality is the point beyond which force and extension are no longer directly proportional.
Equation: F = k × e (or F = k × x), where F = force (N), k = spring constant (N/m), e = extension (m).
The spring constant k is a measure of stiffness; a higher k means a stiffer spring.
Extension = final length – original length. Compression is treated similarly, with e representing compression.

Force–Extension Graphs

A straight line through the origin shows a linear relationship (Hooke's law obeyed).
A curve shows a non-linear relationship (limit of proportionality exceeded).
The spring constant is the gradient of the straight-line region if force is on the y-axis and extension on the x-axis.
If extension is on the y-axis and force on the x-axis, the spring constant is 1 ÷ gradient.
A steeper straight line means a larger spring constant when force is on the y-axis; but if axes are reversed, a steeper line means a smaller spring constant.
Always check which variable is on which axis before interpreting the gradient.

Force-extension graph for a spring

Work Done and Elastic Potential Energy

When a spring is stretched or compressed, work is done and energy is transferred to its elastic potential energy store.
Provided the limit of proportionality is not exceeded, work done on the spring equals the elastic potential energy stored.
Equation: Ee = ½ × k × e² (or E = ½ k x²), where Ee = elastic potential energy (J), k = spring constant (N/m), e = extension (m).
The extension is squared, so doubling the extension quadruples the work done (since 2² = 4).
This equation is only valid for springs that have not been stretched beyond their limit of proportionality.

Required Practical: Investigating Force and Extension

Aim: investigate the relationship between force and extension for a spring.
Independent variable: force (weight of masses); dependent variable: extension; control variable: spring constant (same spring used).
Equipment: spring, ruler, mass hanger, 100 g masses, clamp stand, G clamp, fiducial marker.
Method: measure original length; add masses one at a time; record new length; repeat for each mass; remove masses and repeat to get averages.
Calculate force using W = mg (take g = 10 N/kg); calculate extension as final length – original length.
Plot a graph of force against extension; a straight line through the origin shows Hooke's law is obeyed.
Improve accuracy: use a fiducial marker, take readings at eye level to avoid parallax, wait for spring to stop moving before reading.
Safety: wear goggles, secure clamp stand with G clamp, place a mat below masses, stand up and keep feet clear.

Equipment for investigating the extension of a spring

Common Misconceptions and Exam Tips

Do not confuse extension with total length: extension = final length – original length.
The limit of proportionality is not the same as the breaking point.
A stiffer spring has a larger spring constant, not smaller.
Not all deformation is elastic; some materials deform inelastically.
When using F = k e or E = ½ k e², ensure extension is in metres (convert from cm by ÷ 100).
The unit of spring constant is N/m; do not forget it in answers.

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1.Which of the following statements is correct about inelastic distortion?
Easy
- AThe object returns to its original length when the force is removed
- BThe object does not return to its original shape when the force is removed
- CThe object requires only one force to be distorted
- DThe object returns to its original shape but not its original length
2.What is the correct unit for the force applied to a spring which makes it elastically distort?
Easy
- Am
- BN m
- CN / m
- DN
3.Which of the following options is the correct equation linking force (F), spring constant (k) and extension (x)?
Easy
- AF = kx²
- BF = ½ kx
- CF = kx
- DF = 2kx
4.A spring has a spring constant k. The force exerted on the spring is increased so that the extension doubles. How does this affect the work done on the spring?
Medium
- AWork done doubles
- BWork done halves
- CWork done increases by a factor of 4
- DWork done decreases by a factor of 4
5.A single force is enough to change the shape of a stationary object.
Easy
True or false?
6.A stiffer spring has a smaller spring constant.
Easy
True or false?
7.The limit of proportionality is the same as the breaking point of a spring.
Medium
True or false?
8.Which of the following statements about elastic deformation are correct? (Select all that apply.)
Medium
- AThe object returns to its original shape when the deforming force is removed
- BThe object remains permanently stretched when the force is removed
- CA steel spring can undergo elastic deformation
- DA plastic bag can undergo elastic deformation
- EA rubber band can undergo elastic deformation

Plus de questions

9.Match each key term with its correct definition.
Medium
- Elastic deformation
- Inelastic deformation
- Limit of proportionality
- Spring constant
- The point beyond which force and extension are no longer directly proportional
- The object returns to its original shape when the force is removed
- The object does not return completely to its original shape when the force is removed
- A measure of how stiff a spring is, measured in N/m
10.A spring has a spring constant of 40 N/m. What force is needed to extend it by 0.20 m?
Medium
- A8 N
- B0.005 N
- C200 N
- D80 N
11.Which of the following is the correct unit for the spring constant?
Medium
- AN
- BN/m
- CJ
- Dm
12.In an experiment to investigate the relationship between force and extension for a spring, which of the following are control variables? (Select all that apply.)
Medium
- AThe spring used
- BThe force applied
- CThe extension of the spring
- DThe temperature of the room
- EThe mass added
13.A spring obeys Hooke's law. If the force applied is tripled, what happens to the extension?
Medium
- AIt triples
- BIt stays the same
- CIt increases by a factor of 9
- DIt decreases to one third
14.A spring is stretched beyond its limit of proportionality. What happens to the relationship between force and extension?
Medium
- AIt becomes non-linear
- BIt remains directly proportional
- CThe extension becomes zero
- DThe force becomes zero
15.Place the steps of the required practical to investigate force and extension in the correct order.
Medium
- Set up the apparatus with no masses hanging from the spring
- Record the initial length of the spring
- Add a 100 g mass hanger to the spring
- Record the new mass and position after the spring extends
- Repeat until all masses have been added
16.Match each type of shape change to the description of the forces that cause it.
Medium
- Stretching
- Compressing
- Bending
- Two forces acting in opposite directions, away from the object
- Two forces acting in opposite directions, towards the object
- Forces acting at different points on the object, often towards each other
17.Which of the following materials typically undergo inelastic deformation when stretched? (Select all that apply.)
Medium
- APlastic
- BClay
- CRubber band
- DGlass
- ESteel spring
18.A force-extension graph is plotted with extension on the y-axis and force on the x-axis. The straight-line region has a gradient of 0.005 m/N. What is the spring constant of the spring?
Medium
- A0.005 N/m
- B200 N/m
- C0.005 N/m²
- D5000 N/m
19.Place the following steps in the correct order to find the extension of a spring when a mass is added.
Medium
- Measure the original length of the spring with no mass attached
- Add the mass to the spring and wait for it to stop moving
- Measure the new length of the spring
- Subtract the original length from the new length
20.Which of the following are control variables in the required practical to investigate the relationship between force and extension for a spring? (Select all that apply.)
Medium
- AThe spring constant of the spring
- BThe mass added to the spring
- CThe extension of the spring
- DThe same spring used throughout
- EThe gravitational field strength
21.The spring constant is equal to the gradient of a force-extension graph when force is on the y-axis and extension is on the x-axis.
Easy
True or false?
22.A spring obeys Hooke's law. A force of 6 N produces an extension of 2 cm. What force is needed to produce an extension of 5 cm?
Medium
- A15 N
- B2.4 N
- C30 N
- D12 N
23.A spring is stretched beyond its limit of proportionality. Which statements are correct? (Select all that apply.)
Hard
- AThe force-extension graph becomes a curve
- BThe spring may not return to its original length
- CThe spring constant increases
- DThe relationship between force and extension is no longer directly proportional
- EThe spring obeys Hooke's law
24.A spring obeys Hooke's law. A graph of extension (y-axis) against force (x-axis) is plotted. Which statement about the spring constant k is correct?
Medium
- Ak is equal to the gradient of the straight-line region
- Bk is equal to 1 divided by the gradient of the straight-line region
- Ck is equal to the gradient multiplied by the extension
- Dk is equal to the area under the graph
25.A spring of original length 12 cm is stretched to a final length of 18 cm. What is the extension of the spring?
Medium
- A6 cm
- B18 cm
- C12 cm
- D30 cm
26.A spring has a spring constant of 300 N/m. It is stretched by 0.15 m. What is the elastic potential energy stored in the spring?
Medium
- A3.375 J
- B6.75 J
- C45 J
- D22.5 J
27.A force-extension graph for a spring is a straight line through the origin up to a force of 8 N. At 8 N the extension is 4 cm. What is the spring constant of the spring?
Medium
- A2 N/m
- B0.5 N/m
- C200 N/m
- D32 N/m
28.A spring is stretched by a force of 10 N, giving an extension of 0.20 m. The force is then increased to 30 N. Assuming the limit of proportionality is not exceeded, what is the new extension?
Medium
- A0.60 m
- B0.20 m
- C0.067 m
- D1.80 m
29.A student stretches a spring and then releases it. The spring does not return to its original length. Which term best describes this change?
Medium
- AElastic deformation
- BInelastic deformation
- CElastic potential energy
- DLimit of proportionality
30.A student investigates the extension of a spring when masses are added. Which of the following are control variables that should be kept constant? (Select all that apply.)
Medium
- AThe spring constant of the spring
- BThe mass added to the spring
- CThe temperature of the spring
- DThe extension of the spring
- EThe gravitational field strength
31.A spring is stretched beyond its limit of proportionality. Which of the following statements are correct? (Select all that apply.)
Hard
- AThe force-extension graph becomes non-linear.
- BThe spring may not return to its original length when the force is removed.
- CThe spring constant remains exactly the same as before the limit.
- DThe relationship between force and extension is no longer directly proportional.
- EThe spring obeys Hooke's law more strongly.
32.The extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded.
Easy
True or false?
33.A stiffer spring has a smaller spring constant than a less stiff spring.
Medium
True or false?
34.When a spring is stretched by a force, work is done on the spring and elastic potential energy is stored in it.
Medium
True or false?
35.A student hangs a 100 g mass hanger from a spring and records the new position of the bottom of the spring. Which equation should be used to calculate the force added to the spring?
Medium
- AF = k × x
- BW = m × g
- CE = ½ × k × x²
- DF = m / g
36.In the required practical to investigate force and extension, why is a fiducial marker (pointer) used?
Medium
- ATo measure the mass of the spring
- BTo increase the spring constant
- CTo make the extension measurement more accurate
- DTo prevent the spring from breaking
37.A student is investigating how the extension of a spring varies with the force applied. Which of the following are control variables that should be kept constant? (Select all that apply.)
Medium
- AThe spring constant of the spring
- BThe force applied to the spring
- CThe extension of the spring
- DThe gravitational field strength
- EThe original length of the spring
38.In the force and extension experiment, the extension is calculated by subtracting the original length from the final length.
Easy
True or false?
39.Match each step in the force and extension practical with the correct action.
Medium
- Step 1
- Step 2
- Step 3
- Step 4
- Record the initial length of the spring without masses
- Add a 100 g mass hanger to the spring
- Record the new mass and position after the spring extends
- Repeat the process until all masses have been added
40.Place the following steps in the correct order to analyse the results of the force and extension experiment.
Medium
- Calculate the force by multiplying mass by gravitational field strength
- Calculate the extension by subtracting original length from final length
- Plot a graph of force against extension
- Draw a line of best fit and check if it is a straight line through the origin
41.State the three types of force that can be applied to an object to change its shape.
Easy
The three types are ____, ____, and ____.
42.A spring extends by 0.15 m when a force of 3.0 N is applied. Calculate the spring constant in N/m.
Medium
43.A spring with spring constant 200 N/m is stretched by 0.10 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
44.What is the name of the point on a force-extension graph beyond which the relationship between force and extension is no longer directly proportional?
Easy
45.A spring has a spring constant of 250 N/m. Calculate the work done in stretching the spring by 0.30 m. Give your answer in joules.
Hard
46.A spring with spring constant 150 N/m is compressed by 0.08 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
47.Complete the sentence about Hooke's law.
Easy
The extension of an elastic object is directly proportional to the ____ applied, up to the limit of ____.
48.What is the name of the point on a force-extension graph beyond which force and extension are no longer directly proportional?
Medium
49.A spring has a spring constant of 120 N/m. Calculate the force needed to extend it by 0.25 m. Give your answer in newtons.
Medium
50.A spring with spring constant 80 N/m is compressed by 0.05 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
51.Complete the equation for elastic potential energy.
Medium
elastic potential energy = 0.5 × ____ × extension ____
52.A force of 0.50 N gives a spring an extension of 13 mm. Calculate the spring constant k in N/m.
Medium
53.Explain what is meant by the term elastic behaviour.
Hard
54.Describe how the reduction in the length of a spring when weights are added can be determined.
Hard
55.State two ways that the experimental procedure for investigating force and extension could be improved.
Hard
56.Explain how the design of a toy that launches a ball prevents the spring from becoming damaged.
Hard
57.Describe how the student can determine the extension of the spring when a mass is added.
Hard
58.A spring extends by 0.15 m when a force of 3.0 N is applied. Calculate the spring constant in N/m.
Medium
- A18 N/m
- B22 N/m
- C20 N/m
- D24 N/m
59.A spring with spring constant 200 N/m is stretched by 0.10 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
- A2 J
- B0 J
- C1 J
- D3 J
60.A spring has a spring constant of 250 N/m. Calculate the work done in stretching the spring by 0.30 m. Give your answer in joules.
Hard
- A10.25 J
- B13.25 J
- C12.25 J
- D11.25 J
61.A spring with spring constant 150 N/m is compressed by 0.08 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
- A0.48 J
- B-0.52 J
- C1.48 J
- D2.48 J
62.A spring has a spring constant of 120 N/m. Calculate the force needed to extend it by 0.25 m. Give your answer in newtons.
Medium
- A27 N
- B30 N
- C33 N
- D36 N
63.A spring with spring constant 80 N/m is compressed by 0.05 m. Calculate the elastic potential energy stored in the spring in joules.
Medium
- A1.1 J
- B0.1 J
- C-0.9 J
- D2.1 J
64.A force of 0.50 N gives a spring an extension of 13 mm. Calculate the spring constant k in N/m.
Medium
- A42.4 N/m
- B38.5 N/m
- C46.3 N/m
- D34.6 N/m

Cartes mémo

What are the three ways a stationary object's shape can be changed by forces?

Stretching, bending and compressing (or a combination of these).

Why must more than one force be applied to change the shape of a stationary object?

A single force would just move the object; changing its shape requires opposing forces acting on it.

Describe the forces involved in stretching an object.

Two forces act in opposite directions, away from the object.

Describe the forces involved in compressing an object.

Two forces act in opposite directions, towards the object.

Describe the forces involved in bending an object.

Forces act in opposing directions at different points on the object, or at an angle to each other.

Give an example of compression and state the two forces involved.

A mass resting on top of a spring on a table: the weight of the mass and the reaction force from the surface, acting towards each other.

Give an example of stretching and state the two forces involved.

A mass hanging from a vertical spring: the weight of the mass and the tension in the spring, acting away from each other.

Give an example of bending and state the two forces involved.

A diver standing on a diving board: the weight of the swimmer and the reaction force from the block, acting at different points on the board.

What is elastic deformation?

When an object returns to its original shape and length once the deforming force is removed.

What is inelastic deformation?

When an object remains permanently stretched and does not return to its original shape when the deforming force is removed.

Give examples of materials that deform elastically and materials that deform inelastically.

Elastic: rubber bands, fabrics, steel springs. Inelastic: plastic, clay, glass.

Does 'elastic' mean an object is stretchy? Explain.

No — elastic means the object returns to its original shape when the deforming force is removed, not that it stretches easily.

State Hooke's law.

The extension of an elastic object is directly proportional to the force applied, provided the limit of proportionality is not exceeded.

What does 'directly proportional' mean for force and extension?

If the force is doubled, the extension doubles; if the force is halved, the extension halves.

What is the limit of proportionality?

The point beyond which the force and extension are no longer directly proportional; it varies with the material.

Write the equation linking force, spring constant and extension, with units.

F = k × e, where F is force in newtons (N), k is the spring constant in newtons per metre (N/m) and e is extension in metres (m).

What does the spring constant tell you about a spring?

How stiff the spring is — the higher the spring constant, the stiffer the spring.

How is the extension of a spring calculated?

Extension = final length − original length.

What is the unit of the spring constant?

Newtons per metre (N/m).

On a force–extension graph with force on the y-axis and extension on the x-axis, what does the gradient of the straight region represent?

The spring constant, k.

On a force–extension graph with extension on the y-axis and force on the x-axis, how is the spring constant found?

It is 1 ÷ gradient of the straight-line region.

How does a force–extension graph show that a material obeys Hooke's law?

The graph is a straight line through the origin in that region.

What does a curve on a force–extension graph indicate?

A non-linear relationship — the material has passed its limit of proportionality and no longer obeys Hooke's law.

What is elastic potential energy?

The energy stored in an elastic object when work is done on it, e.g. when it is stretched or compressed.

Write the equation for elastic potential energy stored in a stretched spring.

E = ½ × k × e², where E is in joules (J), k in N/m and e in metres (m).

If the extension of a spring is doubled, what happens to the work done on it?

The work done increases by a factor of 4, because work done depends on extension squared.

For a spring not inelastically deformed, how does the work done compare with the elastic potential energy stored?

They are equal.

In the required practical investigating force and extension, what are the independent and dependent variables?

Independent variable = force, F; dependent variable = extension, e.

In the force and extension practical, what is the control variable?

The spring constant, k (the same spring is used throughout).

How is the force applied to the spring calculated in the force and extension practical?

Force is the weight of the mass: W = mg, using g = 10 N/kg.

Why is the experiment repeated and an average length calculated?

To reduce the effect of random errors and improve accuracy.

What is a fiducial marker and why is it used?

A pointer placed at the bottom of the spring to read its position more accurately, reducing random error.

Why should readings be taken at eye level in the force and extension practical?

To avoid parallax error, which is a systematic error.

Why should you wait a few seconds after adding a mass before taking a reading?

To let the spring fully extend so the reading is accurate.

State two safety precautions for the force and extension practical.

Wear goggles in case the spring snaps, and place a mat or soft material below the masses in case they fall.

Why is a G clamp used in the force and extension practical?

To secure the clamp stand to the desk so the apparatus and masses do not fall over.

Why must the spring not be stretched beyond its limit of proportionality in the practical?

Beyond that point it stops obeying Hooke's law, so the force–extension relationship is no longer linear.

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