(455) HL Rotational equations of accelerated motion [IB Physics HL]

Rotational Equations for Accelerated Motion

Key Variables

• Angular Displacement ((\Delta \theta)): measured in radians.

• Initial Angular Velocity ((\Omega_i)): measured in radians per second.

• Final Angular Velocity ((\Omega_f)): measured in radians per second.

• Angular Acceleration ((\alpha)): measured in radians per second squared.

• Time ((T)): measured in seconds.

Rotational Equations of Motion

• First Equation:[ \Delta \theta = \frac{(\Omega_i + \Omega_f)}{2} \times T ](Substituting (s) with (\Delta \theta))

• Second Equation:[ \Omega_f = \Omega_i + \alpha \times T ]

• Third Equation:[ \Delta \theta = \Omega_i \times T + \frac{1}{2} \alpha \times T^2 ]

• Fourth Equation:[ \Omega_f^2 = \Omega_i^2 + 2\alpha \Delta \theta ]

Example Problem

• Given Values:

  • (\Omega_i = 22) radians/second

  • (\Omega_f = 49) radians/second

  • (T = 7.2) seconds

• Task: Find (\alpha) (Angular Acceleration)

  • Use the second equation:[ \Omega_f = \Omega_i + \alpha \times T ]

  • Substitute known values:[ 49 = 22 + \alpha \times 7.2 ]

  • Isolate (\alpha):[ \alpha = \frac{49 - 22}{7.2} ]

  • Calculate:[ \alpha \approx 3.75 \text{ radians/second}^2 ]

  • Adjust for significant figures:[ \alpha \approx 3.8 \text{ radians/second}^2 ]

Conclusion

• Reviewed variables and equations related to rotational motion.

• Solved an example applying the new equations for motion in a rotational context.