Rolling Kinetic Energy and Material Deformation

Kinetic Energy of Rolling Motion

• Definition and Context: Rolling motion is a combination of translation (moving forward) and rotation. When an object exhibits a relationship between these two motions such that it does not slide, it is referred to as rolling without slipping.

• The Components of Rolling Energy: Kinetic energy in rolling is the sum of two distinct forms of energy we have previously encountered:

  • Rotational Kinetic Energy: Represented as $\frac{1}{2}I\times\text{omega}^2$, where $I$ is rotational inertia and $\omega$ is angular velocity.
  • Translational (Forward) Kinetic Energy: Represented as $\frac{1}{2}m\times v^2$, where $m$ is mass and $v$ is velocity.

• Velocity of the Axle: In the context of rolling, the velocity $v$ refers specifically to the velocity of the center of mass or the axle ($v_{axle}$). Even if the object does not have a physical axle, $v_{axle}$ represents the forward velocity of the object as a whole moving through space.

• Developing the Comprehensive Equation:

  • The variables $\omega$ (angular velocity) and $v_{axle}$ (forward velocity) are not independent. For rolling without slipping, they are related by the radius of the object ($r$) as follows:         $\omega = \frac{v_{axle}}{r}$
  • Substituting this into the rotational kinetic energy expression ($\frac{1}{2}I\times\omega^2$) yields:         $\frac{1}{2}I\times\left(\frac{v_{axle}}{r}\right)^2 = \frac{1}{2}I\times\frac{v_{axle}^2}{r^2}$

• Combining Transformations: The total rolling kinetic energy ($K_{rolling}$) is:     $K_{rolling} = \frac{1}{2}I\times\frac{v_{axle}^2}{r^2} + \frac{1}{2}m\times v_{axle}^2$

  • Factoring out common terms ($\frac{1}{2}$ and $v_{axle}^2$):         $K_{rolling} = \frac{1}{2}v_{axle}^2\times\left(\frac{I}{r^2} + m\right)$
  • Factoring out the mass ($m$) to achieve a mathematically interesting form:         $K_{rolling} = \frac{1}{2}m\times v_{axle}^2\times\left(\frac{I}{m\times r^2} + 1\right)$

• Physical Implications of the Formula:

  • The term $\frac{1}{2}m\times v_{axle}^2$ is the standard forward kinetic energy ($K$).
  • The term in parentheses, $\left(\frac{I}{m\times r^2} + 1\right)$, accounts for the additional energy required for rotation.
  • Inertia and Distribution: Objects with larger rotational inertia ($I$)—meaning they are heavy or have their mass distributed far from the center (not compact)—will have more rolling kinetic energy at the same speed. This is because they are harder to get rotating.
  • The Sliding Limit: If rotational inertia ($I$) were to become zero, the equation would simply become $K_{total} = K$. This represents pure sliding without rotation.

Introduction to Deformation (Chapter 10)

• Defining Deformation: Deformation is the change in the shape or size of an object when forces are applied. In the context of steady forces, at least two forces are required to deform an object. A single steady force would merely cause acceleration through space rather than a change in shape.

• Universality of Deformation: Though some objects are modeled as perfectly rigid, this is a simplification. Chemical bonds between atoms are not infinitely strong; therefore, every object deforms to some degree when force is applied, even if it is too small to notice.

• Key Specialized Terminology:

  • Strain: The normalized (generalized) deformation of an object. It represents the degree of deformation relative to a specific parameter.
  • Stress: The normalized (generalized) force causing the deformation.
  • The Cause-and-Effect Relationship: In physics, stress causes strain. This phrase helps distinguish the two terms: stress is the input (force applied), and strain is the output (physical change).

• The Modulus (Material Parameter):

  • Within the elastic limit (before an object breaks), the relationship between stress and strain is typically linear.
  • The general equation is: $\text{stress} = \text{modulus}\times\text{strain}$
  • A modulus is a coefficient or material parameter describing how well a material resists deformation. A higher modulus value indicates a more rigid material that deforms less under a given stress.

Tension and Compression

• Mechanism: These are two directions of the same type of linear deformation. Tension involves forces pulling away from each other (stretching), while compression involves forces pushing toward each other (squeezing along a line).

• Tensile Stress Calculation:

  • Tensile stress is defined as the force ($f$) divided by the cross-sectional area ($A$) where the force is applied.
  • $\text{Stress} = \frac{F}{A}$
  • The cross-sectional area depends on the shape of the object (e.g., $\pi\times r^2$ for a cylinder or $\text{width}\times\text{height}$ for a rectangle).

• Tensile Strain Calculation:

  • Tensile strain is defined as the change in length ($\Delta L$) divided by the original length ($L$).
  • $\text{Strain} = \frac{\Delta L}{L}$
  • Strain is unitless because it is a ratio of two lengths ($\text{meters}/\text{meters}$). It can be thought of as a percentage change.

• Young’s Modulus ($Y$):

  • The equation for linear deformation is: $\frac{F}{A} = Y\times\frac{\Delta L}{L}$
  • Named after Thomas Young, this modulus applies to both tension and compression for a given material.
  • Young's Modulus Values ($\times 10^9\,N/m^2$):
    • Rubber: $0.005$
    • Brick: $16$
    • Steel: $200$
    • Diamond: $1200$

• Units of Pressure and Modulus:

  • The standard unit for stress and modulus is the Pascal ($Pa$), which is defined as one Newton per square meter ($N/m^2$).
  • One common SI prefix used for moduli like steel or diamond is the Gigapascal ($GPa$), which equals $10^9\,Pa$.
  • In the United States, the everyday unit is PSI (pounds per square inch).

Shear Deformation and Bulk Compression

• Shear Deformation:

  • Occurs when forces are applied parallel to the surfaces but in opposite directions and misaligned (e.g., pushing the top of a stationary jell-o block sideways).
  • Shear Stress: Defined as force divided by the area of the top surface ($\frac{F}{A}$).
  • Shear Strain: Defined as the sideways displacement ($\Delta x$) divided by the height ($L$).
  • Shear Equation: $\frac{F}{A} = S\times\frac{\Delta x}{L}$, where $S$ is the Shear Modulus.
  • The Shear Modulus is a distinct value from Young's Modulus for the same material.

• Bulk Compression:

  • Occurs when an object is squeezed inward from all directions, typically by a fluid (air or water pressure).
  • Bulk Stress: Equivalent to the pressure ($P$) applied to the object ($\frac{F}{A} = P$).
  • Bulk Strain: Defined as the change in volume ($\Delta V$) divided by the original volume ($V$).
  • Bulk Equation: $P = -B\times\frac{\Delta V}{V}$, where $B$ is the Bulk Modulus.
  • Note on the Negative Sign: The negative sign is conventionally inserted because compression results in a decrease in volume (a negative $\Delta V$), which allows the bulk modulus and pressure to remain positive values.
  • Mnemonic: The speaker suggests thinking of a "bulky winter coat" zipping up and squeezing you from all sides to remember bulk compression.