Angular Acceleration

• Angular Acceleration Definition

  • Angular acceleration corresponds with linear acceleration.

  • Variable notation: Greek letter alpha (α).

  • Appearance: resembles a fish.

• Calculation of Angular Acceleration

  • Formula:
    $\alpha = \frac{\Delta \omega}{\Delta t}$ where:

  • $\Delta \omega$ = change in angular velocity

  • $\Delta t$ = change in time

  • Units:

  • Radians per second squared ($\text{rad/s}^2$)

  • Degrees per second squared ($\text{°/s}^2$)

  • Angular acceleration is a vector; direction is significant.

  • Preferred units: radians per second squared.

• Example Problem 1: Ice Skater

  • Scenario: Ice skater increases angular velocity from 450 degrees/s to 610 degrees/s over 2.3 seconds.

  • Known values:

  • Initial angular velocity ($\omega_i$) = 450 degrees/s

  • Final angular velocity ($\omega_f$) = 610 degrees/s

  • Time ($\Delta t$) = 2.3 seconds

  • Conversion of units:

  • Convert degrees to radians:

    • $\omega_i = \frac{450 \times \pi}{180} = 7.85 \text{ rad/s}$

    • $\omega_f = \frac{610 \times \pi}{180} = 10.65 \text{ rad/s}$

  • Change in angular velocity:

  • $\Delta \omega = \omega_f - \omega_i = 10.65 - 7.85 = 2.8 \text{ rad/s}$

  • Angular acceleration calculation:

  • $\alpha = \frac{\Delta \omega}{\Delta t} = \frac{2.8}{2.3} = 1.22 \text{ rad/s}^2$

Tangential Acceleration

• Tangential Acceleration Definition

  • Denoted as $a_t$, where $a$ is regular linear acceleration.

  • Tangential acceleration relates directly to angular acceleration and radius.

• Formula for Tangential Acceleration

  • $a_t = \alpha \times r$ where:

  • $\alpha$ = angular acceleration (in $\text{rad/s}^2$)

  • $r$ = radius (in meters)

  • Units: $\text{m/s}^2$

• Example Problem 2: Olympic Softball Pitcher

  • Scenario: Pitcher changes angular velocity from 1000 degrees/s to 2190 degrees/s over 0.1 seconds, with a radius of 0.7 meters.

  • Known values:

  • Initial angular velocity = 1000 degrees/s (convert to radians: $\omega_i = \frac{1000 \times \pi}{180} = 17.45 \text{ rad/s}$)

  • Final angular velocity = 2190 degrees/s (convert to radians: $\omega_f = \frac{2190 \times \pi}{180} = 38.25 \text{ rad/s}$)

  • Time = 0.1 seconds

  • Radius = 0.7 m

  • Angular acceleration:

  • $\Delta \omega = \omega_f - \omega_i = 38.25 - 17.45 = 20.8 \text{ rad/s}$

  • $\alpha = \frac{\Delta \omega}{\Delta t} = \frac{20.8}{0.1} = 208 \text{ rad/s}^2$

  • Calculation of Tangential Acceleration:

  • $a_t = \alpha \times r = 208 \times 0.7 = 145.6 \text{ m/s}^2$

Radial or Centripetal Acceleration

• Radial Acceleration Definition

  • Also known as centripetal acceleration; implies 'center-seeking'.

  • Occurs when an object moves in a circular path, requiring a radial force.

• Conceptual Understanding of Radial Acceleration

  • Relationship with Newton's Second Law: $F = ma$ leads to:

  • $F_r = m imes a_r$

  • Illustrative Example: Turntable experiment with jars and cork.

  • Round motion example involving resistance of inertia.

• Radial Acceleration Calculation

  • Formula options:

  • $a_r = \frac{v_t^2}{r}$ (using linear velocity)

  • $a_r = \omega^2 \times r$ (using angular velocity)

  • Units: $\text{m/s}^2$

• Example Problem 3: Hammer Throw

  • Radial acceleration when the tangential velocity of the hammer is 28 m/s and radius is 2 m:

  • $a_r = \frac{28^2}{2} = 392 \text{ m/s}^2$

Practical Applications and Implications

• Centripetal Acceleration in Sports.

  • Example of Sean White in snowboarding:

  • Acceleration relating to speed and forces on the body.

  • Discuss the engineering behind half pipe design affecting athlete performance.

• Observations on Safety in Sports Engineering

  • Height and radius of half pipe influence centripetal forces and athlete safety.

  • Higher walls, larger radius reduce centripetal force but also increase air time with associated risk.