Rotational Motion Overview

Overview

This lecture covers rotational motion, focusing on angular and tangential quantities, acceleration, centripetal forces, and related problem-solving using equations of motion.

Rotational System Setup

• Mass of object: 0.350 kg.
• Radius from axis: 0.75 m.
• Angular acceleration: 1.5 rad/s² (constant).
• Object starts from rest.

Tangential Speed Problem (Part A)

• Find time when tangential speed ( v = 12 ) m/s.
• ( v = r\omega ), so ( \omega = v/r = 16 ) rad/s.
• Using ( \omega = \omega_0 + \alpha t ), solve for ( t ): ( t = (\omega - \omega_0)/\alpha = 16/1.5 = 10.67 ) s.

Centripetal Acceleration after 10 Revolutions (Part B)

• 10 revolutions = ( 20\pi ) radians.
• Use ( \omega^2 = 2\alpha\theta ) (since ( \omega_0 = 0 )).
• Centripetal acceleration: ( a_c = r \omega^2 = r(2\alpha\theta) ).
• Calculate: ( a_c = 0.75 \times 2 \times 1.5 \times 20\pi = 141.37 ) m/s².

Total Acceleration at ( t = 13 ) seconds (Part C)

• ( a_{t} ) (tangential): ( r\alpha = 0.75 \times 1.5 = 1.125 ) m/s².
• ( \omega = 1.5 \times 13 = 19.5 ) rad/s at ( t = 13 ) s.
• ( a_c = r\omega^2 = 0.75 \times (19.5)^2 = 285.19 ) m/s².
• Total acceleration: ( a_{total} = \sqrt{a_t^2 + a_c^2} = 285.21 ) m/s².
• Direction: ( \theta = \arctan(a_t/a_c) \approx 0.23^\circ ) from radial.

Maximum Centripetal Force and Speed (Part D)

• Max centripetal force: 170 N.
• ( F_c = m v^2 / r \implies v = \sqrt{F_c r / m} ).
• ( v_{max} = \sqrt{170 \times 0.75 / 0.350} = 19.09 ) m/s.
• Time to reach ( v_{max} ): ( t = v_{max} / a_t = 19.09 / 1.125 = 16.97 ) s._

Time When Total Force Reaches 170 N

• Total force: ( F_{total} = m a_{total} ).
• ( a_{total} = F_{total} / m = 170 / 0.350 = 485.71 ) m/s².
• Set ( a_{total} = \sqrt{a_t^2 + (r\omega^2)^2} ), solve for ( \omega ), then for ( t ).
• Find ( \omega = 25.44 ) rad/s; ( t = \omega / \alpha = 25.44 / 1.5 = 16.96 ) s._

Key Terms & Definitions

• Angular acceleration ((\alpha)) — Change in angular velocity per unit time (rad/s²).
• Tangential speed (v) — Linear speed along the circular path (m/s).
• Centripetal acceleration ((a_c)) — Acceleration toward the center, ( a_c = r\omega^2 ) or ( v^2/r ).
• Tangential acceleration ((a_t)) — Linear acceleration along the tangent, ( a_t = r\alpha ).
• Total acceleration ((a_{total})) — Vector sum of ( a_c ) and ( a_t )._

Action Items / Next Steps

• Review equations of motion for constant angular acceleration.
• Practice problems involving tangential and centripetal acceleration.
• Read up on vector addition of acceleration components.