5.1.1 Hooke's Law

When you stretch a spring or a rubber band, you might notice that the more you pull, the more it stretches. There is a relationship between the force applied and the amount of stretch. This relationship was discovered by Robert Hooke in the 17th century and is known as Hooke's Law. It is a fundamental principle in physics that describes the elastic behavior of materials.

Definition

Hooke's Law states that within the elastic limit, the extension (or compression) of an elastic object is directly proportional to the applied force (or the restoring force).

Mathematically: F_res ∝ x or F_res = -kx where: - F_res is the restoring force (the force exerted by the spring to return to its original shape). - x is the extension (or compression) from the original length. - k is the spring constant (or force constant), a measure of the stiffness of the spring.

The negative sign indicates that the restoring force is always directed opposite to the displacement.

Main Idea

The main idea is that the deformation of an elastic object is proportional to the force applied to it, up to a certain limit. This means that if you double the force, you double the extension. The constant of proportionality, k, is a property of the object and indicates how stiff it is. A stiff spring has a large k, while a soft spring has a small k. Hooke's Law is valid only as long as the material remains within its elastic limit.

Core Concept

The Formula The text states: "When a spring is stretched or compressed (with in elastic limit), the extension or compression is directly proportional to the applied force (Figure 5.4). This relationship is known as Hooke's law which states that within elastic limits the extension (or compression) x is directly proportional to the restoring force F_res i.e. F_res ∝ x or F_res = -kx therefore k = -F_resx where 'k' is the ratio of restoring force to the extension and is known as the force constant or spring constant having units Nm^-1."

• A large k means a stiff spring (requires a lot of force to stretch).
• A small k means a soft spring (requires little force to stretch).
• The SI unit of k is newton per metre (N/m or Nm^-1).

The Negative Sign The negative sign in the equation F_res = -kx indicates that the restoring force is always in the opposite direction to the displacement. If you stretch a spring to the right (positive displacement), the spring pulls back to the left (negative force). If you compress it to the left (negative displacement), the spring pushes back to the right (positive force).

The Force-Extension Graph The text states: "This relationship is also true for a wire under tension. Provided that the limit of proportionality is not exceeded, a graph of stretching force against extension is a straight line through the origin, as shown in figure 5.4."

The graph of force versus extension is a straight line through the origin. The gradient (slope) of this line is the spring constant k. Gradient = Fx = k

The Limit of Proportionality The text states: "The gradient of the line Fx is the spring constant 'k'. Hooke's Law is obeyed up to the limit of proportionality. Beyond this point, stretching force and extension are no longer directly proportional and the graph begins to curve."

The limit of proportionality is the point up to which Hooke's Law is obeyed. Beyond this point, the graph is no longer a straight line, and the relationship between force and extension is no longer linear.

Detailed Explanation

Hooke's Law is a fundamental principle that describes the elastic behavior of materials. It is valid for many materials, including springs, wires, and bones, as long as the deformation is within the elastic limit.

The Restoring Force The restoring force is the force exerted by the deformed object to return to its original shape. It is equal in magnitude and opposite in direction to the applied force. For example, if you pull a spring with a force of 10 N to the right, the spring exerts a restoring force of 10 N to the left.

The Spring Constant The spring constant k is a measure of the stiffness of an object. It depends on the material and the geometry of the object. For a spring, k depends on the wire diameter, coil diameter, and number of coils. For a wire, k depends on the material, length, and cross-sectional area.

The Force-Extension Graph The force-extension graph is a powerful tool for understanding Hooke's Law. The straight-line portion of the graph (up to the limit of proportionality) indicates that the material obeys Hooke's Law. The gradient of this line is the spring constant. Beyond the limit of proportionality, the graph curves, indicating that the material is no longer obeying Hooke's Law and is undergoing plastic deformation.

• Spring scales: Used to measure weight (force) by measuring the extension of a spring.
• Car suspensions: Springs are used to absorb shocks and keep the car stable.
• Mattresses: Springs provide support and comfort.
• Mechanical watches: Hairsprings are used to regulate the timekeeping mechanism.
• Seismographs: Used to detect and record earthquakes.

Intermediate Level

In physics, Hooke's Law is expressed as F = -kx. The negative sign indicates that the restoring force is opposite to the displacement. The spring constant k is a measure of the stiffness of the spring. The force-extension graph is a straight line through the origin, with a gradient equal to k. This linear relationship holds only up to the limit of proportionality. Beyond this limit, the material undergoes plastic deformation, and the graph curves. The elastic limit is the maximum deformation from which the material can recover. Hooke's Law is fundamental to understanding the elastic behavior of materials and is used in the design of springs, suspension systems, and many other mechanical devices.

Important Points

• - Hooke's Law: F = -kx (within elastic limit).
• - Restoring Force: The force exerted by the deformed object to return to its original shape.
• - Spring Constant (k): A measure of stiffness. Unit: N/m.
• - Negative Sign: Indicates that the restoring force is opposite to the displacement.
• - Force-Extension Graph: A straight line through the origin up to the limit of proportionality. The gradient is k.
• - Limit of Proportionality: The point up to which Hooke's Law is obeyed.
• - Elastic Limit: The maximum deformation from which the material can recover.

Remember

• - Hooke's Law: F = -kx.
• - Spring Constant (k): Stiffness. Unit: N/m.
• - Negative Sign: Restoring force is opposite to displacement.
• - Graph: Straight line (gradient = k) up to limit of proportionality.
• - Limit of Proportionality: Hooke's Law obeyed up to here.
• - Elastic Limit: Maximum recovery point.

Common Mistakes

• - Forgetting the negative sign: The negative sign is important—it indicates that the restoring force is opposite to the displacement.
• - Thinking Hooke's Law applies to all deformations: It only applies within the elastic limit.
• - Confusing the limit of proportionality with the elastic limit: The limit of proportionality is the point up to which the force-extension graph is a straight line. The elastic limit is the maximum deformation from which the material can recover. The elastic limit is usually slightly beyond the limit of proportionality.

Quick Revision

Hooke's Law states that within the elastic limit, the extension (or compression) of an elastic object is directly proportional to the applied force. Mathematically, F = -kx, where F is the restoring force, k is the spring constant, and x is the extension. The negative sign indicates that the restoring force is opposite to the displacement. The force-extension graph is a straight line through the origin up to the limit of proportionality, with a gradient equal to the spring constant.

60-Second Revision

• k: Spring constant (stiffness). Unit: N/m.
• Negative sign: Restoring force opposite to displacement.
• Graph: Straight line (gradient = k) up to limit of proportionality.
• Limit of Proportionality: Hooke's Law obeyed up to here.
• Elastic Limit: Maximum recovery point.

Frequently Asked Questions (FAQs)

Q1. What is Hooke's Law? Answer: Hooke's Law states that within the elastic limit, the extension (or compression) of an elastic object is directly proportional to the applied force (or restoring force). Mathematically, F = -kx.

Q2. What is the spring constant? Answer: The spring constant k is a measure of the stiffness of a spring. It is defined as the force required to produce a unit extension. Its SI unit is newton per metre (N/m).

Q3. Why is there a negative sign in Hooke's Law? Answer: The negative sign indicates that the restoring force is always directed opposite to the displacement. If you stretch a spring to the right, the restoring force is to the left.

Q4. What is the limit of proportionality? Answer: The limit of proportionality is the point up to which Hooke's Law is obeyed. Beyond this point, the force-extension graph is no longer a straight line, and the relationship between force and extension is no longer linear.

Q5. What is the difference between the limit of proportionality and the elastic limit? Answer: The limit of proportionality is the point up to which the force-extension graph is a straight line (Hooke's Law is obeyed). The elastic limit is the maximum deformation from which the material can recover. The elastic limit is usually slightly beyond the limit of proportionality. Beyond the elastic limit, the material undergoes permanent (plastic) deformation.

• Q6. How can you find the spring constant from a force-extension graph?
• Answer: The spring constant is the gradient (slope) of the straight-line portion of the force-extension graph: k = Fx.

• Q7. What are the applications of Hooke's Law?
• Answer: Applications include spring scales, car suspensions, mattresses, mechanical watches, and seismographs.

Multiple Choice Questions (MCQs)

• Q1. Hooke's Law states that the extension of an elastic object is directly proportional to:
• A. Its mass
• B. The applied force
• C. Its length
• D. Its volume

• Correct Answer: B
• Explanation: The text states that within elastic limits, the extension is directly proportional to the restoring force (or applied force).

• Q2. The formula for Hooke's Law is:
• A. F = kx
• B. F = -kx
• C. F = kx
• D. F = xk

• Correct Answer: B
• Explanation: The text gives the formula F_res = -kx.

• Q3. The spring constant k has the unit:
• A. N
• B. m
• C. N/m
• D. J

• Correct Answer: C
• Explanation: The text states that the spring constant has units Nm^-1 (or N/m).

• Q4. The negative sign in Hooke's Law indicates that:
• A. The force is negative
• B. The restoring force is opposite to the displacement
• C. The extension is negative
• D. The spring constant is negative

• Correct Answer: B
• Explanation: The negative sign shows that the restoring force is directed opposite to the displacement.

• Q5. A force-extension graph is a straight line through the origin up to the:
• A. Breaking point
• B. Elastic limit
• C. Limit of proportionality
• D. Yield point

• Correct Answer: C
• Explanation: The text states that Hooke's Law is obeyed up to the limit of proportionality, beyond which the graph curves.

• Q6. The gradient of the force-extension graph gives:
• A. The mass
• B. The spring constant
• C. The extension
• D. The force

• Correct Answer: B
• Explanation: The text states that the gradient of the line Fx is the spring constant 'k'.

• Q7. A spring with a large spring constant is:
• A. Easy to stretch
• B. Stiff (hard to stretch)
• C. Very long
• D. Very short

• Correct Answer: B
• Explanation: A large spring constant means a lot of force is required to stretch the spring, so it is stiff.

• Q8. Hooke's Law is valid:
• A. For all deformations
• B. Only beyond the elastic limit
• C. Only within the elastic limit
• D. Only for plastic materials

• Correct Answer: C
• Explanation: The text states that Hooke's Law is valid within elastic limits.

• Q9. If a spring is stretched by 2 cm with a force of 10 N, what is its spring constant?
• A. 5 N/m
• B. 500 N/m
• C. 20 N/m
• D. 0.2 N/m

• Correct Answer: B
• Explanation: k = Fx = 100.02 = 500 N/m.

• Q10. Beyond the limit of proportionality, the force-extension graph:
• A. Remains a straight line
• B. Curves
• C. Becomes horizontal
• D. Becomes vertical

• Correct Answer: B
• Explanation: The text states that beyond the limit of proportionality, the graph begins to curve.

Frequently Asked Questions (FAQs)

Q1. What is the difference between the spring constant and the elastic limit? Answer: The spring constant k is a measure of the stiffness of a spring (how much force is needed to stretch it by a unit length). The elastic limit is the maximum deformation from which the material can recover. They are different properties: k describes the stiffness, while the elastic limit describes the maximum safe deformation.

Q2. Can Hooke's Law be applied to materials other than springs? Answer: Yes. Hooke's Law applies to any elastic material within its elastic limit, including wires, bones, and rubber bands. The relationship between force and deformation is linear for these materials within the elastic limit.

Q3. What happens if you exceed the limit of proportionality but stay below the elastic limit? Answer: If you exceed the limit of proportionality but stay below the elastic limit, the force-extension graph is no longer a straight line (Hooke's Law is not obeyed), but the material will still return to its original shape when the force is removed. The deformation is still elastic, but not proportional.

Q4. How is the spring constant related to the stiffness of a spring? Answer: A larger spring constant means the spring is stiffer (requires more force to stretch). A smaller spring constant means the spring is softer (requires less force to stretch).

Q5. What is the difference between the restoring force and the applied force? Answer: The applied force is the force you exert to deform the object. The restoring force is the force exerted by the object to return to its original shape. They are equal in magnitude and opposite in direction (by Newton's third law).

Q6. How can you determine the spring constant experimentally? Answer: You can determine the spring constant by hanging known weights from a spring and measuring the extension. Plot a graph of force vs. extension. The gradient of the straight-line portion of the graph gives the spring constant.

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