Physics 11 Lesson #5

Elastic Potential Energy

• Elastic potential energy is associated with objects that can stretch or compress, such as elastic bands, springs, bungee cords, shocks, and non-rigid materials, including metals.
• If an object returns to its original condition after being stretched or compressed, it is considered elastic.

Hooke’s Law

• Robert Hooke (1660) discovered the Law of Elasticity.
• Hooke’s Law (1676) defines the relationship between the extension of a spring and the force exerted on it: F_{restore} = -kx
  • The negative sign indicates that the force acts in the opposite direction of the displacement.
  • The force acts to restore the spring to its original position.

Compression & Extension

*A visual representation of compression and extension:

• x < 0: Compression
• x = 0: Unstretched spring
• x > 0: Extension

Hooke’s Law Explained

• At x = 0, the spring is unstretched and exerts no force.
• When the spring compresses (x < 0) or stretches (x > 0), a restoring force exists.
• This restoring force causes the spring to want to return to its original position.
• The restoring force is opposite to the direction of stretch.
  F_{restore} = -kx

Direction of Restoring Force

• x < 0: Compression
• x = 0: Unstretched
• x > 0: Extension

Spring Constant

• F_{restore} = -kx
  • k = spring constant, which refers to the stiffness of an elastic object, measured in N/m.
  • High k values indicate a very stiff spring, requiring a large force to stretch it.
• Hooke’s Law can also be written as F = kx, with the understanding that the force is a restoring force opposite to x.

Forms of Hooke’s Law

• Hooke’s Law can be used in two different forms:
  • ‘Vector’ Emphasized: F_{restore} = -kx, where F and x are vectors.
  • ‘Scalar’ Emphasized: F_{restore} = kx, where F and x are scalars; signs are added later if needed.

Example Problems

• Example #1:
  • A 0.25 kg mass hangs on the end of a spring. What is its extension if it has a spring constant of 48 N/m?
• Example #2:
  • A 0.52 kg mass hangs on the spring and stretches 31.2 cm. What is the spring constant of the spring?

Work Done on a Spring

• The work done by a force acting on a spring involves a varied force.
• The standard work equation cannot be directly used to find the formula for EPE.
• Instead, determine it graphically.

Examination of “k” & Elastic Potential Energy

• The force vs. extension graph of a spring reveals specific information:
  1. Slope represents the spring constant, k.
  2. The area under the curve represents “work” (energy).
  3. The force increases at a linear rate.

Example #5

• Based on a given graph:
  • Determine the spring constant.
  • Determine the amount of elastic potential energy at 0.20 m (no SF needed).

Example #3

• A compound archery bow requires a force of 133 N to hold an arrow at “full draw” (pulled back 0.71 m).
  • Assuming the bow obeys Hooke’s Law:
    • a) What is the spring constant?
    • b) What is its maximum EPE?

Example #6

• Mr. Ngo conducts an experiment on a spring hanging on a ring stand, using masses on the spring to measure distances:
  • Force (N) | Extension (cm)
    ----------|----------------
    0.0 | 0.0
    2.0 | 1.5
    4.0 | 3.0
    6.0 | 4.5
  • What is the spring constant of the spring?
  • What is the elastic potential energy in the spring at 4.5 cm?

Example #7

• How much energy does a bow have when pulled back 8.0 cm if it has a force constant of 160 N/m?