Material Properties, Forces, and Collisions

Material Properties and Forces

Shape Restoration and Rupture:
- When a force is applied to a body, its shape is distorted.
- If the original shape is not restored after the force is removed, or if a sufficiently large force is applied, the body will rupture.
- Examples:
  - A long sausage balloon stretches and returns to its shape when pulled, but too much force will cause it to pop or rupture.
  - A soda can can be pressed slightly, distorting its shape, but it returns to its original form. Squeezing harder will permanently bend it (overcoming its elastic limit), and further crushing or twisting can tear the aluminum.
Internal Resistance:
- An applied stretching force on a bar is transmitted throughout, pulling the material apart.
- This force is resisted by the cohesive forces holding the material together.
- The material breaks when the applied force exceeds the cohesive force.

Stress, Strain, and Young's Modulus

Stress ( $S$ or $\sigma$ ):
- Definition: The internal force per unit area acting on a material.
- Formula: $S = \frac{F}{A}$
  - $F$ is the applied force.
  - $A$ is the area over which the force is applied.
- Units: Force per unit area (e.g., dynes per centimeter squared, Pascals).
Longitudinal Strain ( $\epsilon$ ):
- Definition: The fractional change in the length of a material due to an applied force.
- Formula: $\epsilon = \frac{\Delta L}{L}$
  - $\Delta L$ (delta L) is the change in length (elongation or compression).
  - $L$ is the original length of the bar.
- Strain is a dimensionless quantity.
- Compression: A decrease in the volume or length of an object or substance from an applied force. Similar to stretching, initial compression is elastic. A sufficiently large compressive force will cause permanent distortion and then breakage (e.g., crushing a soda can).
Hooke's Law (Elastic Limit and Young's Modulus):
- Discovery: Robert Hooke (1676) discovered that while a body remains elastic (returns to its original shape after force removal), the ratio of stress to strain is constant.
- Young's Modulus ( $Y$ or $E$ ):
  - Definition: The constant of proportionality relating stress to strain within the elastic limit of a material.
  - Formula: $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$
  - It is a property of the material indicating how stiff it is or how much it can stretch and deform while remaining elastic.
  - Significance: Young's Modulus (elastic limit) represents the maximum stress a material can withstand and still return to its original shape. Exceeding this limit causes permanent deformation. The rupture strength is the force per unit area at which the material breaks.
- Examples of Young's Modulus and Rupture Strength (conceptual values mentioned):
  - Steel: High Young's Modulus and rupture strength (e.g., $200 \times 10^{10}$ dynes/cm $^2$ elastic limit, $450 \times 10^7$ dynes/cm $^2$ rupture strength).
  - Aluminum: More pliable, lower Young's Modulus and rupture strength than steel.
  - Bone: Less than metals but still high. Rupture strength for compression (e.g., $100 \times 10^7$ dynes/cm $^2$ ), stretching (e.g., $83 \times 10^7$ dynes/cm $^2$ ), and twisting (even lower).
  - Tendon: Intermediate between bone and muscle.
  - Muscle: Significantly lower Young's Modulus and rupture strength than bone or tendon.
  - Organic Substances: Calcaneous bone (heel bone, strong), cartilage (low), meniscus (low), anterior cruciate ligament (ACL).

Spring Constant and Energy Storage

Principle: The force required to stretch or compress a spring is directly proportional to the amount of stretch or compression ( $\Delta L$ or $x$ ).
Hooke's Law for Springs: $F = -kx$
- $F$ is the applied force.
- $k$ is the spring constant (stiffness of the spring).
- $x$ (or $\Delta L$ ) is the displacement (stretch or compression) from equilibrium.
- The negative sign indicates the restorative force is opposite to the displacement.
- Analogy: A stretched Slinky demonstrates that a small force causes a small stretch, and a larger force causes a larger stretch proportionally.
Potential Energy in a Spring:
- A stretched or compressed spring contains potential energy, capable of doing work.
- Formula: $E = \frac{1}{2} k (\Delta L)^2$
- Connection to Young's Modulus: An elastic body under stress is analogous to a spring. By rearranging the Young's Modulus equation for force ( $F = \frac{YA}{L} \Delta L$ ), the spring constant $k$ can be identified as $k = \frac{YA}{L}$ .
- Substituting this into the energy equation: $E = \frac{1}{2} \left( \frac{YA}{L} \right) (\Delta L)^2$

Collisions and Impulsive Force

Definition of Collision: A sudden event where a large force is exerted for a short period of time on colliding objects.
Impulsive Force Characteristics:
- Starts at zero, increases to a maximum value, and then decreases back to zero.
- The time interval ( $t<em>2 - t</em>1$ or $\Delta t$ ) during which the force acts is the duration of the collision.
- Such a short-duration force is called an impulsive force.
Average Impulsive Force ( $F_{avg}$ ):
- Difficult to determine the exact instantaneous force, but the average value can be calculated.
- Momentum-Force Relationship: Based on the relationship $F = ma$ and $a = \frac{\Delta v}{\Delta t}$
- Formula: $F<em>{avg} = \frac{m v</em>f - m v_i}{\Delta t}$
  - $m$ is the mass of the object.
  - $v_f$ is the final velocity.
  - $v_i$ is the initial velocity.
  - $\Delta t$ is the duration of the collision.
- Final Velocity (after impact): Often $v_f = 0$ (e.g., a car hitting a pole and stopping).
Inverse Proportionality of Force and Collision Time:
- For a given change in momentum, the magnitude of the impulsive force is inversely proportional to the collision time ( $F_{avg} \propto \frac{1}{\Delta t}$ ).
- A shorter collision time ( $\Delta t$ is small) leads to a larger impulsive force and greater damage.
- A longer collision time ( $\Delta t$ is large) leads to a smaller impulsive force and less damage.
- The type of collision (hard vs. soft objects) determines collision time:
  - Hard/Rigid Collisions: (e.g., car hitting a tree, billiard balls, person hitting concrete) have very short collision times (e.g., $\approx 10^{-2}$ seconds = $0.01$ seconds).
  - Soft/Yielding Collisions: (e.g., fallen person landing in soft snow, diver hitting water, person falling with bent knees) have longer collision times, which reduces the impulsive force.
- Examples:
  - Falling into soft sand is less damaging than falling on hard concrete.
  - Diving from a high board: hitting water involves a longer collision time (slowing down as you go deeper) than hitting the concrete deck directly (very short stop).
  - The saying: "It's not the fall that kills you, it's the sudden stop at the end."
Deceleration: Refers to acceleration in the direction opposite to the direction of velocity.
Calculating Momentum on Impact (from a fall):
- Velocity on impact: $v = \sqrt{2gh}$ , where $g$ is acceleration due to gravity and $h$ is height.
- Momentum on impact: $p = mv = m\sqrt{2gh}$ (or $p = \frac{\text{weight}}{g} \sqrt{2gh} = \text{weight} \sqrt{\frac{2h}{g}}$ ).
- Average impact force: $F_{avg} = \frac{m\sqrt{2gh}}{\Delta t}$

Real-World Applications and Implications

Airbag Safety Systems:
- Designed to deploy and deflate quickly (short time necessary to cushion).
- Function: To lengthen the collision time between the occupant and the car's interior (steering wheel/dashboard).
- Result: A longer collision time reduces the impulsive force on the occupant, minimizing injury.
- Rapid deflation prevents impeding driver control and avoids injury from unexpected deployment.
Whiplash Injury:
- Occurs in sudden impacts (e.g., rear-end collisions).
- The body is accelerated forward by the car seat, but the unsupported head lags due to inertia, then is violently yanked backward.
- Neck muscles don't respond fast enough, and the energy is absorbed by delicate neck bones, causing fracture.
- Headrests: Crucial safety features designed to prevent hyperextension of the neck, thus reducing whiplash severity.
Survival from Falls:
- Reports of people surviving parachute failure by landing on soft snow.
- The body creates a deep depression (approx. $1$ meter) in the snow, significantly lengthening the collision time.
- This longer duration reduces the impact force, keeping it below the threshold for serious injury, even at terminal velocities (e.g., $62.5$ cm/s for this example).
Osteoarthritis:
- A joint disease characterized by the degenerative wearing out of joint components (synovial membrane, cartilage).
- Leads to loss of flexibility, strength, pain, stiffness, and loss of synovial fluid lubrication.
- Underlying bone erosion can occur.
- Major cause of disability in older age (e.g., $60\%$ of women and $75\%$ of men over $65$ are affected).
- Impact on Joints: Repetitive impacts contribute to wear and tear.
  - Running Shoes: Proper running shoes are crucial to absorb impact and reduce stress on joints.
  - Running Surfaces: Running on softer surfaces (e.g., track) provides more cushion and less impact than concrete.